A Multiplicity Theorem for Superlinear Double Phase Problems

Abstract

We consider a nonlinear Dirichlet problem driven by the double phase differential operator and with a superlinear reaction which need not satisfy the Ambrosetti–Rabinowitz condition. Using the Nehari manifold, we show that the problem has at least three nontrivial bounded solutions: nodal, positive and by the symmetry of the behaviour at $+\infty$ and $-\infty$ also negative.

1. Introduction

Let $\mathrm{\Omega }\subseteq {\mathbb{R}}^N$ be a bounded domain (that is, a bounded connected set in the N-dimensional Euclidean space) with a smooth boundary $\partial \mathrm{\Omega }$. In this paper, we study the following double phase problem

$$\left(P\right)\left\{\begin{array}{c}-{\mathrm{\Delta }}_p^{a}u\left(z\right)-{\mathrm{\Delta }}_{q}u\left(z\right)=f(z,u\left(z\right))\mathrm{in}\mathrm{\Omega },\hfill \\ {u|}_{\partial \mathrm{\Omega }}=0,1<q<p<+\infty \hfill \end{array}\right.$$

(see [1]), with the unknown $u:\mathrm{\Omega }⟶\mathbb{R}$. If $a\in L^{\infty }\left(\mathrm{\Omega }\right)\setminus \left\{0\right\}$, $a\left(z\right)⩾0$ for almost all $z\in \mathrm{\Omega }$ and $1<r<+\infty$, then by ${\mathrm{\Delta }}_r^a$ we denote the weighted r-Laplacian differential operator defined by

$${\mathrm{\Delta }}_r^{a}u={\mathrm{div}\left(a\left(z\right)\right|Du|}^{r-2}Du).$$

If $a\equiv 1$, then we have the usual r-Laplacian. In problem $\left(P\right)$ we have the sum of two such operators. So, the differential operator (left hand side) in $\left(P\right)$ is not homogeneous. The differential operator of $\left(P\right)$ is related to the so called “double phase” integral functional defined by

$$J\left(u\right)={\int }_{\mathrm{\Omega }}\left({a\left(z\right)\left|Du\right|}^p+{\left|Du\right|}^q\right)dz.$$

The integrand of this functional is the function

$$\vartheta (z,y)={a\left(z\right)\left|y\right|}^p+{\left|y\right|}^q\forall (z,y)\in \mathrm{\Omega }\times {\mathbb{R}}^N.$$

(1)

We do not assume that the weight a is bounded away from zero (that is, we do not assume that $\underset{\mathrm{\Omega }}{\mathrm{ess}\inf }a>0$). So, the function $\vartheta (z,·)$ exhibits unbalanced growth, namely we have

$${\left|y\right|}^q⩽\vartheta (z,x)⩽c_0\left({\left|y\right|}^p+{\left|y\right|}^q\right)\mathrm{for}\mathrm{a}.\mathrm{a}.z\in \mathrm{\Omega },\mathrm{all}y\in {\mathbb{R}}^N,$$

for some $c_0>0$. Such functionals were first investigated by Marcellini [2,3] and Zhikov [4,5] in the context of problems of the calculus of variations and of nonlinear elasticity theory. The unbalanced growth of $\vartheta (z,·)$ requires the use of Musielak–Orlicz spaces for the treatment of problem $\left(P\right)$. The interest for this kind of problems, was revived recently with the work of Mingione and coworkers who produced interesting interior regularity results for local minimizers of such functionals. We refer to the paper of Baroni–Colombo–Mingione [6] and the references therein. We also mention the very recent work of Ragusa–Tachikawa [7] who extended the interior regularity results to anisotropic double phase functionals. However, a global (that is, up to the boundary) regularity theory for these problems, is not yet available and this makes their study difficult.

The reaction $f(z,x)$ in problem $\left(P\right)$ is a Carathéodory function, which is $(p-1)$-superlinear at $+\infty$ and symmetrically at $-\infty$. This problem was examined recently by Gasiński–Papageorgiou [8] and Liu-Dai [9], but under stronger conditions on $f(z,·)$. Here using the Nehari method, we show that problem $\left(P\right)$ has at least three nontrivial solutions all with sign information (nodal (that is sign-changing) solution, positive solution and by the symmetry of the behaviour of the data also negative solution) and minimum energy (ground state solutions). The use of the Nehari manifold helps us overcome the difficulties that originate from the fact that for double phase problems we have no global regularity theory and so many of the tools and techniques of “balanced” problems cannot be used. Our multiplicity result improves considerably Theorem 4.9 of Gasiński–Papageorgiou [8] and Theorem 1.4 of Liu–Dai [9]. Normally multiplicity results for superlinear problems are obtained by a combination of critical point theory and Morse theory, see the works of Wang [10] (semilinear equations) and Papageorgiou–Rǎdulescu [11] (nonlinear, nonhomogeneous equations). The lack of global regularity theory for double phase problems, leads to a different approach based on the Nehari manifold, as this was developed by Brown–Wu [12], Szulkin–Weth [13] and Willem [14]. Other existence and multiplicity results for double phase problems can be found in the works of Gasiński–Winkert [15] (coercive equations), Colasuonno–Squassina [16], Ge–Wang–Lu [17] (eigenvalue problems), Gasiński–Winkert [18], Papageorgiou–Vetro–Vetro [19] (Robin problems). For the nonlinear problems related to the nonlinear frequency shift phenomena we refer to Kalyabin et al. [20] and Sadovnikov et al. [21]. Finally we mention the two recent informative survey articles by Mingione–Rǎdulescu [1] and Rǎdulescu [22].

2. Mathematical Background—The Nehari Manifold

As we already mentioned in the Introduction, the appropriate functional framework for the analysis of double phase problems, is provided by the so called Musielak–Orlicz spaces. For a comprehensive presentation of the theory of these spaces, we refer to the book of Harjulehto–Hästö [23].

We introduce the following hypotheses (denoted by H1) on the weight a and the exponents $q,p$, which will be used in the sequel.

Hypothesis 1.

$a\in C^{0,1}\left(\overline{\mathrm{\Omega }}\right)\setminus \left\{0\right\}$ (that is, $a:\overline{\mathrm{\Omega }}⟶\mathbb{R}$ is nonzero and Lipschitz continuous), $a\left(z\right)⩾0$ for all $z\in \mathrm{\Omega }$, $1<q<p<N$ and $\frac{p}{q}<1+\frac{1}{N}$.

Remark 1.

The last condition on the exponents $q,p$ is common in Dirichlet double phase problems and guarantees that the Poincaré inequality is valid for the corresponding Musielak–Orlicz–Sobolev space (see Harjulehto–Hästö ([23], p. 138)). Note that the inequality $\frac{p}{q}<1+\frac{1}{N}$ implies that $p<q^{*}$ and as we will see later in this section, this guarantees useful compact embeddings among the relevant spaces.

We consider the Carathéodory integrand

$$\vartheta (z,x)={a\left(z\right)\left|x\right|}^p+{\left|x\right|}^q$$

and let

$$M\left(\mathrm{\Omega }\right)=\{u:\mathrm{\Omega }⟶\mathbb{R}\mathrm{measurable}\}.$$

As usual we identify two such functions which differ only on a Lebesgue-null subset of $\mathrm{\Omega }$. Then the Musielak–Orlicz space $L^{\vartheta }\left(\mathrm{\Omega }\right)$ is defined by

$$L^{\vartheta }\left(\mathrm{\Omega }\right)=\{u\in M\left(\mathrm{\Omega }\right):{\varrho }_{\vartheta }\left(u\right)<\infty \},$$

with ${\varrho }_{\vartheta }$ being the modular function defined by

$${\varrho }_{\vartheta }\left(u\right)={\int }_{\mathrm{\Omega }}\vartheta (z,u\left(z\right))dz,$$

where $\vartheta :\mathrm{\Omega }\times \mathbb{R}⟶\mathbb{R}$ is defined by (2). We equip $L^{\vartheta }\left(\mathrm{\Omega }\right)$ with the so called “Luxembourg norm”

$${\parallel u\parallel }_{\vartheta }=inf\left\{\mu >0:{\varrho }_{\vartheta }\left(\frac{u}{\mu }\right)⩽1\right\}$$

(see Adams [24]). Then $L^{\vartheta }\left(\mathrm{\Omega }\right)$ becomes a Banach space which is separable and reflexive (in fact uniformly convex). Using the space $L^{\vartheta }\left(\mathrm{\Omega }\right)$, we can define the corresponding Musielak–Orlicz–Sobolev space $W^{1,\vartheta }\left(\mathrm{\Omega }\right)$ by

$$W^{1,\vartheta }\left(\mathrm{\Omega }\right)=\{u\in L^{\vartheta }:\left|Du\right|\in L^{\vartheta }\left(\mathrm{\Omega }\right)\}.$$

Here and in the sequel $Du$ denotes the weak gradient of $u\in L^{\vartheta }\left(\mathrm{\Omega }\right)$. We equip $W^{1,\vartheta }\left(\mathrm{\Omega }\right)$ with the norm

$${\parallel u\parallel }_{1,\vartheta }={\parallel u\parallel }_{\vartheta }+{\parallel Du\parallel }_{\vartheta }\forall u\in W^{1,\vartheta }\left(\mathrm{\Omega }\right),$$

with ${\parallel Du\parallel }_{\vartheta }={\parallel \left|Du\right|\parallel }_{\vartheta }$. We set

$$W_0^{1,\vartheta }\left(\mathrm{\Omega }\right)={\overline{C_c^{\infty }\left(\mathrm{\Omega }\right)}}^{{\parallel ·\parallel }_{1,\vartheta }}.$$

For this space the Poincaré inequality holds (see Harjulehto–Hästö ([23], p. 138)), that is, there exists $\widehat{c}>0$ such that

$${\parallel u\parallel }_{\vartheta }⩽\widehat{c}{\parallel Du\parallel }_{\vartheta }\forall u\in W_0^{1,\vartheta }\left(\mathrm{\Omega }\right).$$

Therefore on $W_0^{1,\vartheta }\left(\mathrm{\Omega }\right)$ we can consider the equivalent norm

$$\parallel u\parallel ={\parallel Du\parallel }_{\vartheta }\forall u\in W_0^{1,\vartheta }\left(\mathrm{\Omega }\right).$$

The spaces $W^{1,\vartheta }\left(\mathrm{\Omega }\right)$ and $W_0^{1,\vartheta }\left(\mathrm{\Omega }\right)$ are Banach spaces which are separable and reflexive (in fact uniformly convex).

There is a close relation between the norm $\parallel ·\parallel$ and the modular function ${\varrho }_{\vartheta }$.

Proposition 1.

(a) If $u\ne 0$, then $\parallel u\parallel =\mu$⟺${\varrho }_{\vartheta }\left(\frac{u}{\mu }\right)=1$.

(b) $\parallel u\parallel <1$ (resp. $=1$, $>1$) ⟺${\varrho }_{\vartheta }\left(u\right)<1$ (resp. $=1$, $>1$).

(c) $\parallel u\parallel <1$⟹${\parallel u\parallel }^p⩽{\varrho }_{\vartheta }\left(u\right)⩽{\parallel u\parallel }^q$.

(d) $\parallel u\parallel >1$⟹${\parallel u\parallel }^q⩽{\varrho }_{\vartheta }\left(u\right)⩽{\parallel u\parallel }^p$.

(e) $\parallel u\parallel \to 0$⟺${\varrho }_{\vartheta }\left(u\right)\to 0$ and $\parallel u\parallel \to +\infty$⟺${\varrho }_{\vartheta }\left(u\right)\to +\infty$.

Proposition 2.

(a) $L^{\vartheta }\left(\mathrm{\Omega }\right)\subseteq L^r\left(\mathrm{\Omega }\right)$ and $W_0^{1,\vartheta }\left(\mathrm{\Omega }\right)\subseteq W_0^{1,r}\left(\mathrm{\Omega }\right)$ continuously for every $1⩽r⩽q$.

(b) $W_0^{1,\vartheta }\left(\mathrm{\Omega }\right)\subseteq L^r\left(\mathrm{\Omega }\right)$ continuously (resp. compactly), if $1⩽r⩽q^{*}$ (resp. $1⩽r<q^{*}$), where $q^{*}=\frac{Nq}{N-q}$.

(c) $L^p\left(\mathrm{\Omega }\right)\subseteq L^{\vartheta }\left(\mathrm{\Omega }\right)$ continuously.

Let $V:W_0^{1,\vartheta }\left(\mathrm{\Omega }\right)\to W_0^{1,\vartheta }{\left(\mathrm{\Omega }\right)}^{*}$ be the nonlinear map defined by

$$⟨V\left(u\right),h⟩={\int }_{\mathrm{\Omega }}\left({a\left(z\right)\left|Du\right|}^{p-2}{(Du,Dh)}_{{\mathbb{R}}^N}+{\left|Du\right|}^{q-2}{(Du,Dh)}_{{\mathbb{R}}^N}\right)dz$$

for all $u,h\in W_0^{1,\vartheta }\left(\mathrm{\Omega }\right)$ (by $⟨·,·⟩$ we denote the duality brackets for the pair $(W_0^{1,\vartheta }\left(\mathrm{\Omega }\right),W_0^{1,\vartheta }{\left(\mathrm{\Omega }\right)}^{*})$). This operator has the following properties (see Liu-Dai ([9], Proposition 3.1)).

Proposition 3.

The operator $V:W_0^{1,\vartheta }\left(\mathrm{\Omega }\right)⟶W_0^{1,\vartheta }{\left(\mathrm{\Omega }\right)}^{*}$ is bounded (that is, maps bounded sets into bounded sets), continuous, strictly monotone (thus maximal monotone too) and of type ${\left(S\right)}_{+}$, that is, V has the following property: “if $u_n\stackrel{w}{⟶}u$ in $W_0^{1,\vartheta }\left(\mathrm{\Omega }\right)$ and

$$\underset{n\to +\infty }{lim\; sup}⟨V\left(u_n\right),u_n-u⟩⩽0,$$

then $u_n\to u$ in $W_0^{1,\vartheta }\left(\mathrm{\Omega }\right)$.”

In the sequel by ${\varrho }_a$ we denote the modular function defined by

$${\varrho }_a\left(Du\right)={\int }_{\mathrm{\Omega }}a\left(z\right){\left|Du\right|}^{p}dz\forall u\in W_0^{1,\vartheta }\left(\mathrm{\Omega }\right).$$

For every $x\in \mathbb{R}$, let

$$x^{+}=max\{x,0\},x^{-}=max\{-x,0\}.$$

Then for every $u\in M\left(\mathrm{\Omega }\right)$, we set

$$u^{\pm }\left(z\right)=u{\left(z\right)}^{\pm }\forall z\in \mathrm{\Omega }.$$

If $u\in W_0^{1,\vartheta }\left(\mathrm{\Omega }\right)$, we know that

$$u^{\pm }\in W_0^{1,\vartheta }\left(\mathrm{\Omega }\right),u=u^{+}-u^{-},\left|u\right|=u^{+}+u^{-}.$$

By ${\widehat{\lambda }}_1\left(q\right)$ we denote the principal eigenvalue of $(-{\mathrm{\Delta }}_q,W_0^{1,q}\left(\mathrm{\Omega }\right))$, that is, we consider the nonlinear eigenvalue problem

$$\left\{\begin{array}{c}-{\mathrm{\Delta }}_{q}u\left(z\right)=\widehat{\lambda }{\left|u\left(z\right)\right|}^{q-2}u\left(z\right)\mathrm{in}\mathrm{\Omega },\hfill \\ {u|}_{\partial \mathrm{\Omega }}=0.\hfill \end{array}\right.$$

This problem has a smallest eigenvalue ${\widehat{\lambda }}_1\left(q\right)>0$, which has the following variational characterization

$${\widehat{\lambda }}_1\left(q\right)=inf\left\{\frac{{\parallel Du\parallel }_q^q}{{\parallel u\parallel }_q^q}:u\in W_0^{1,q}\left(\mathrm{\Omega }\right),u\ne 0\right\}.$$

(2)

This eigenvalue is simple and isolated and the corresponding eigenfunctions have fixed sign and belong in $C^1\left(\overline{\mathrm{\Omega }}\right)$. In fact this is the only eigenvalue with eigenfunctions of fixed sign. All the other eigenvalues have eigenfunctions which are nodal. Moreover, if $C_{+}$ is the positive cone of

$$C_0^1\left(\overline{\mathrm{\Omega }}\right)=\{u\in C^1\left(\overline{\mathrm{\Omega }}\right):u{|}_{\partial \mathrm{\Omega }}=0\}$$

that is

$$C_{+}=\{u\in C_0^1\left(\overline{\mathrm{\Omega }}\right):u\left(z\right)⩾0\mathrm{for}\mathrm{all}z\in \overline{\mathrm{\Omega }}\}$$

and $\widehat{u}\in C_0^1\left(\overline{\mathrm{\Omega }}\right)$ is a positive eigenfunction corresponding to ${\widehat{\lambda }}_1\left(q\right)>0$, then

$$\widehat{u}\in \mathrm{int}{C}_{+}=\left\{u\in C_{+}:u\left(z\right)>0\mathrm{for}\mathrm{all}z\in \mathrm{\Omega },\frac{\partial u}{\partial n}{|}_{\partial \mathrm{\Omega }}<\right\},$$

with n being the outward unit normal on $\partial \mathrm{\Omega }$ and $\frac{\partial u}{\partial n}={(Du,n)}_{{\mathbb{R}}^N}$.

The next lemma is an easy consequence of the above properties (see for example Mugnai–Papageorgiou ([25], Lemma 4.11)).

Lemma 1.

If $\beta \in L^{\infty }\left(\mathrm{\Omega }\right)$, $\beta \left(z\right)⩽{\widehat{\lambda }}_1\left(q\right)$ for almost all $z\in \mathrm{\Omega }$ and the inequality is strict on a set of positive Lebesgue measure, then there exists $\tilde{c}>0$ such that

$$\tilde{c}{\parallel Du\parallel }_q^q⩽{\parallel Du\parallel }_q^q-{\int }_{\mathrm{\Omega }}\beta \left(z\right){\left|u\right|}^{q}dz\forall u\in W_0^{1,q}\left(\mathrm{\Omega }\right).$$

Next we introduce our hypotheses on the reaction $f(z,x)$.

Hypothesis 2.

$f:\mathrm{\Omega }\times \mathbb{R}⟶\mathbb{R}$ is a Carathéodory function such that $f(z,0)=0$ for almost all $z\in \mathrm{\Omega }$ and

(i)

$\left|f(z,x)\right|⩽\widehat{a}{\left(z\right)(1+|x|}^{r-1})$ for almost all $z\in \mathrm{\Omega }$, all $x\in \mathbb{R}$ with $\widehat{a}\in L^{\infty }\left(\mathrm{\Omega }\right)$, $p<r<q^{*}$;

(ii)

${\displaystyle \underset{x\to \pm \infty }{lim}\frac{f(z,x)}{{\left|x\right|}^{p-2}x}=+\infty }$ uniformly for almost all $z\in \mathrm{\Omega }$;

(iii)

for almost all $z\in \mathrm{\Omega }$, the quotient function $x⟼\frac{f(z,x)}{{\left|x\right|}^{p-1}}$ is increasing on $\mathbb{R}\setminus \left\{0\right\}$;

(iv)

there exists $\widehat{\beta }\in L^{\infty }\left(\mathrm{\Omega }\right)$ and ${\widehat{c}}_0>0$ such that $\widehat{\beta }\not\equiv {\widehat{\lambda }}_1\left(q\right)$,

$$\widehat{\beta }\left(z\right)⩽{\widehat{\lambda }}_1\left(q\right)\mathrm{for}\mathrm{a}.\mathrm{a}.z\in \mathrm{\Omega }$$

and

$$-{\widehat{c}}_0\le \underset{x\to 0}{lim\; inf}\frac{f(z,x)}{{\left|x\right|}^{q-2}x}\le \underset{x\to 0}{lim\; sup}\frac{f(z,x)}{{\left|x\right|}^{q-2}x}\le \beta \left(z\right)\mathrm{uniformly}\mathrm{for}\mathrm{a}.\mathrm{a}.z\in \mathrm{\Omega }.$$

Remark 2.

Hypothesis H2 (ii) implies that for almost all $z\in \mathrm{\Omega }$, $f(z,·)$ is $(p-1)$-superlinear. However, we do not assume the usual in superlinear problems Ambrosetti–Rabinowitz condition (see, for example, Willem ([14], p. 46)). This condition is used by Liu–Dai [9] (see hypothesis $\left(f_4\right)$ in [9]). The Ambrosetti–Rabinowitz condition is a convenient but restrictive condition which permits the verification of the Palais–Smale condition (the compactness type condition on the energy functional), which is necessary in order to apply the minimax theorem of the critical point theory. Furthermore, Hypothesis H2 (iii) is a relaxed version of Nehari-type monotonicity condition which requires that the quotient function $x⟼\frac{f(z,x)}{{\left|x\right|}^{p-1}}$ is strictly increasing on $\mathbb{R}\setminus \left\{0\right\}$. This stronger version is used by Liu–Dai [9] (see hypothesis $\left(f_5\right)$ in [9]). Finally Hypothesis H2 (iv) describes the behaviour of $f(z,·)$ near zero and is less restrictive than the corresponding condition in Liu–Dai [9] (see hypothesis $\left(f_3\right)$ in [9]), who requires that $f(z,·)$ is strictly $(q-1)$-sublinear near zero. Hypotheses H2 are also less restrictive than those of Gasiński–Papageorgiou [8] (see hypotheses $H\left(f\right)$ in [8]).

Example 1.

The following functions satisfy Hypotheses H2, but fail to satisfy the hypotheses of Gasiński–Papageorgiou [8] and Liu-Dai [9]:

$$\begin{array}{ccc}\hfill f_1\left(x\right)& =& \left\{\begin{array}{ccc}{\left|x\right|}^{p-2}x\hfill & \mathrm{if}\hfill & \left|x\right|⩽1,\hfill \\ {\left|x\right|}^{p-2}x(ln|x|+1)\hfill & \mathrm{if}\hfill & 1<\left|x\right|,\hfill \end{array}\right.\hfill \\ \hfill f_2\left(x\right)& =& \left\{\begin{array}{ccc}{\beta \left|x\right|}^{q-2}x+{\eta \left|x\right|}^{p-2}xln(1+|x\left|\right)\hfill & \mathrm{if}\hfill & \left|x\right|⩽1,\hfill \\ {\left|x\right|}^{p-2}xln\left|x\right|+\widehat{\eta }{\left|x\right|}^{p-2}x\hfill & \mathrm{if}\hfill & 1<\left|x\right|,\hfill \end{array}\right.\hfill \end{array}$$

with $\beta <{\widehat{\lambda }}_1\left(q\right)$, $\eta ⩾2\beta (p-q)$ and $\widehat{\eta }=\beta +\eta ln2$. In these examples for the sake of simplicity we have dropped the z-dependence.

For $\eta >0$, we consider the following perturbation of the reaction

$$f_{\eta }(z,x)=f(z,x)+\eta r{\left|x\right|}^{r-2}x.$$

Note that $f_{\eta }$ satisfies the same hypotheses as f and in fact now we have that for almost all $z\in \mathrm{\Omega }$, the quotient function $x⟼\frac{f_{\eta }(z,x)}{{\left|x\right|}^{p-1}}$ is strictly increasing on $\mathbb{R}\setminus \left\{0\right\}$.

We consider the Dirichlet double phase problem with $f_{\eta }(z,x)$ as reaction. So, we consider the following problem

$$\left(P_{\eta }^{ }\right)\left\{\begin{array}{c}-{\mathrm{\Delta }}_p^{a}u\left(z\right)-{\mathrm{\Delta }}_{q}u\left(z\right)=f_{\eta }(z,u\left(z\right))\mathrm{in}\mathrm{\Omega },\hfill \\ {u|}_{\partial \mathrm{\Omega }}=0,1<q<p.\hfill \end{array}\right.$$

First we will prove a multiplicity result for problem $\left(P_{\eta }\right)$ and then let $\eta \to 0^{+}$ to have the multiplicity theorem for the original problem $\left(P\right)$.

We set

$$F_{\eta }(z,x)={\int }_0^x{f}_{\eta }(z,s)ds$$

and consider the energy (Euler) functional ${\phi }_{\eta }:W_0^{1,\vartheta }\left(\mathrm{\Omega }\right)⟶\mathbb{R}$ for problem $\left(P_{\eta }\right)$ defined by

$${\phi }_{\eta }\left(u\right)=\frac{1}{p}{\varrho }_a\left(Du\right)+\frac{1}{q}{\parallel Du\parallel }_q^q-{\int }_{\mathrm{\Omega }}{F}_{\eta }(z,u)dz\forall u\in W_0^{1,\vartheta }\left(\mathrm{\Omega }\right).$$

Evidently ${\phi }_{\eta }\in C^1\left(W_0^{1,\vartheta }\left(\mathrm{\Omega }\right)\right)$ and we have

$$⟨{\phi }_{\eta }^{\prime }\left(u\right),h⟩=⟨V\left(u\right),h⟩-{\int }_{\mathrm{\Omega }}{f}_{\eta }(z,u)hdz\forall h,u\in W_0^{1,\vartheta }\left(\mathrm{\Omega }\right).$$

Since we want to provide sign information for all the solutions produced, we introduce also the positive and negative truncations ${\phi }_{\eta }^{\pm }:W_0^{1,\vartheta }\left(\mathrm{\Omega }\right)⟶\mathbb{R}$ defined by

$${\phi }_{\eta }^{\pm }\left(u\right)=\frac{1}{p}{\varrho }_a\left(Du\right)+\frac{1}{q}{\parallel Du\parallel }_q^q-{\int }_{\mathrm{\Omega }}{F}_{\eta }(z,\pm u^{\pm })dz\forall u\in W_0^{1,\vartheta }\left(\mathrm{\Omega }\right).$$

Again we have ${\phi }_{\eta }^{\pm }\in C^1\left(W_0^{1,\vartheta }\left(\mathrm{\Omega }\right)\right)$.

We introduce the Nehari manifold $N^{\eta }$ for the functional ${\phi }_{\eta }$ defined by

$$N^{\eta }=\{u\in W_0^{1,\vartheta }\left(\mathrm{\Omega }\right):⟨{\phi }_{\eta }^{\prime }\left(u\right),u⟩=0,u\ne 0\}.$$

Evidently the Nehari manifold $N^{\eta }$ contains the nontrivial weak solutions of problem $\left(P_{\eta }\right)$. In order to produce constant sign solutions, we introduce the corresponding sets for the functionals ${\phi }_{\eta }^{\pm }$, namely the sets

$$\begin{array}{ccc}\hfill N_{+}^{\eta }& =& \{u\in W_0^{1,\vartheta }\left(\mathrm{\Omega }\right):⟨{\left({\phi }_{\eta }^{+}\right)}^{\prime }\left(u\right),u⟩=0,u⩾0,u\ne 0\},\hfill \\ \hfill N_{-}^{\eta }& =& \{u\in W_0^{1,\vartheta }\left(\mathrm{\Omega }\right):⟨{\left({\phi }_{\eta }^{-}\right)}^{\prime }\left(u\right),u⟩=0,u⩽0,u\ne 0\}.\hfill \end{array}$$

Evidently $N_{+}^{\eta },N_{-}^{\eta }\subseteq N^{\eta }$.

Finally for the purpose of producing nodal solutions, we introduce also the set

$$N_0^{\eta }=\{u\in W_0^{1,\vartheta }\left(\mathrm{\Omega }\right):u^{+}\in N^{\eta },-u^{-}\in N^{\eta }\}.$$

Given $u\in W_0^{1,\vartheta }\left(\mathrm{\Omega }\right)\setminus \left\{0\right\}$, by $k_u^{\eta }$ we denote the corresponding fibering function defined by

$$k_u^{\eta }\left(t\right)={\phi }_{\eta }\left(tu\right)\forall t⩾0.$$

Note that $tu\in N^{\eta }$ if and only if ${\left(k_u^{\eta }\right)}^{\prime }\left(t\right)=0$.

Proposition 4.

If Hypotheses H1, H2 hold and $u\in W_0^{1,\vartheta }\left(\mathrm{\Omega }\right)\setminus \left\{0\right\}$, then there exists unique $t_u>0$ such that $t_{u}u\in N^{\eta }$.

Proof. 

Consider the function

$${\mu }_u^{\eta }\left(t\right)=t^p{\varrho }_a\left(Du\right)+t^q{\parallel Du\parallel }_q^q-{\int }_{\mathrm{\Omega }}{f}_{\eta }(z,tu)\left(tu\right)dz.$$

(3)

Evidently ${\mu }_u^{\eta }\left(t\right)=t{\left(k_u^{\eta }\right)}^{\prime }\left(t\right)$ for all $t⩾0$.

Note that Hypothesis H2 (iii) is equivalent to saying that for almost all $z\in \mathrm{\Omega }$ the quotient map $x⟼\frac{f(z,x)x}{{\left|x\right|}^p}$ is increasing in $\left|x\right|\ne 0$. Therefore, for $x\ne 0$ and $t\in (0,1)$ we have

$$\frac{f_{\eta }(z,tx)\left(tx\right)}{t^p{\left|x\right|}^p}<\frac{f_{\eta }(z,x)x}{{\left|x\right|}^p}\mathrm{for}\mathrm{a}.\mathrm{a}.z\in \mathrm{\Omega },$$

so

$$f_{\eta }(z,tx)\left(tx\right)<t^p{f}_{\eta }(z,x)\mathrm{for}\mathrm{a}.\mathrm{a}.z\in \mathrm{\Omega }.$$

(4)

Using (4) in (3), we have

$${\mu }_u^{\eta }\left(t\right)⩾t^p{\varrho }_a\left(Du\right)+t^q{\parallel Du\parallel }_q^q-t^p{\int }_{\mathrm{\Omega }}{f}_{\eta }(z,u)udz.$$

Since $q<p$, we see that there exists $t_0\in (0,1)$ such that

$${\mu }_u^{\eta }\left(t\right)>0\mathrm{for}\mathrm{all}t\in (0,t_0).$$

(5)

On the other hand, we have

$$\frac{{\mu }_u^{\eta }\left(t\right)}{t^p}={\varrho }_a\left(Du\right)+\frac{1}{t^{p-q}}{\parallel Du\parallel }_q^q-{\int }_{\mathrm{\Omega }}\frac{f_{\eta }(z,tu)\left(tu\right)}{t^p}dz.$$

Using Hypothesis H2 (ii) we see that

$$\underset{t\to +\infty }{lim}\frac{{\mu }_u^{\eta }\left(t\right)}{t^p}=-\infty ,$$

so

$${\mu }_u^{\eta }\left(t\right)<0\mathrm{for}\mathrm{all}t>0\mathrm{large}.$$

(6)

Then (5), (6) and Bolzano’s theorem, imply that there exists $t_u>0$ such that

$${\mu }_u^{\eta }\left(t_u\right)=0,$$

so

$$t_u{\left(k_u^{\eta }\right)}^{\prime }\left(t_u\right)=0,$$

thus

$${\left(k_u^{\eta }\right)}^{\prime }\left(t_u\right)=0$$

and hence $t_{u}u\in N^{\eta }$. Note that

$${\mu }_u^{\eta }\left(t\right)=0⟺{\varrho }_a\left(Du\right)={\int }_{\mathrm{\Omega }}\frac{f_{\eta }(z,tu)\left(tu\right)}{t^p}dz-\frac{1}{t^{p-q}}{\parallel Du\parallel }_q^q$$

and observe that in the last equation the right hand side is strictly increasing in $t>0$. Therefore, $t_u>0$ is unique. □

The Nehari manifold is much smaller than the ambient space $W_0^{1,\vartheta }\left(\mathrm{\Omega }\right)$. So, ${\phi }_{\eta }{|}_{N^{\eta }}$ exhibits properties which fail to be true globally. In fact on account of Hypothesis H2 (ii), ${\phi }_{\eta }$ is unbounded below. However as we will show in the next proposition ${\phi }_{\eta }{|}_{N^{\eta }}$ is coercive, hence bounded below. This illustrates the usefulness of the Nehari manifold.

Proposition 5.

If Hypotheses H1, H2 hold, then ${\phi }_{\eta }{|}_{N^{\eta }}$ is coercive.

Proof. 

We argue by contradiction. So, suppose that we can find a sequence ${\left\{u_n\right\}}_{n\in \mathbb{N}}\subseteq N^{\eta }$ such that

$${\phi }_{\eta }\left(u_n\right)⩽c_1\mathrm{for}\mathrm{all}n\in \mathbb{N}\mathrm{and}\parallel u_n\parallel \to +\infty ,$$

(7)

for some $c_1>0$. Let $y_n=\frac{u_n}{\parallel u_n\parallel }$ for $n\in \mathbb{N}$. Then $\parallel y_n\parallel =1$ for all $n\in \mathbb{N}$ and so we may assume that

$$y_n\stackrel{w}{⟶}y\mathrm{in}{W}_0^{1,\vartheta }\left(\mathrm{\Omega }\right),y_n⟶y\mathrm{in}{L}^r\left(\mathrm{\Omega }\right)$$

(8)

(see Proposition 2). First we assume that $y\ne 0$. Let

$$\widehat{\mathrm{\Omega }}=\{z\in \mathrm{\Omega }:|y\left(z\right)|>0\}.$$

Then $|\widehat{\mathrm{\Omega }}{|}_N>0$, where by ${|·|}_N$ we denote the Lebesgue measure on ${\mathbb{R}}^N$. Hence we have

$$|u_n\left(z\right)|⟶+\infty \mathrm{for}\mathrm{a}.\mathrm{a}.z\in \widehat{\mathrm{\Omega }},$$

so

$$\frac{F_{\eta }(z,u_n\left(z\right))}{|u_n{\left(z\right)|}^p}⟶+\infty \mathrm{for}\mathrm{a}.\mathrm{a}.z\in \widehat{\mathrm{\Omega }}$$

(see Hypothesis H2 (ii)). By Fatou’s lemma, we have

$${\int }_{\widehat{\mathrm{\Omega }}}\frac{F_{\eta }(z,u_n)}{\parallel u_n{\parallel }^p}dz⟶+\infty$$

(recall that $|\widehat{\mathrm{\Omega }}{|}_N>0$). On account of Hypotheses H2 (i),(ii), there exists $M>0$ such that

$$F_{\eta }(z,x)⩾-M\mathrm{for}\mathrm{a}.\mathrm{a}.z\in \mathrm{\Omega },\mathrm{all}x\in \mathbb{R}.$$

So, it follows that

$${\int }_{\mathrm{\Omega }}\frac{F_{\eta }(z,u_n)}{\parallel u_n{\parallel }^p}dz⟶+\infty .$$

(9)

For $x\ne 0$ and $\tau ⩾0$, using the chain rule and Hypothesis H2 (iii), we have

$$\begin{array}{ccc}& & \frac{1-{\tau }^p}{p}{f}_{\eta }(z,x)x+F_{\eta }(z,\tau x)-F_{\eta }(z,x)\hfill \\ & =& {\int }_{\tau }^1{f}_{\eta }(z,x)x{s}^{p-1}ds-{\int }_{\tau }^1\frac{d}{ds}{F}_{\eta }(z,sx)ds\hfill \\ & =& {\int }_{\tau }^1{f}_{\eta }(z,x)x{s}^{p-1}ds-{\int }_{\tau }^1{f}_{\eta }(z,sx)xds\hfill \\ & =& {\int }_{\tau }^1\left(\frac{f_{\eta }(z,x)}{{\left|x\right|}^{p-1}}-\frac{f_{\eta }(z,sx)}{{\left(s\right|x\left|\right)}^{p-1}}\right)s^{p-1}x{\left|x\right|}^{p-1}ds\hfill \\ & >& 0,\hfill \end{array}$$

so

$$F_{\eta }(z,x)-F_{\eta }(z,\tau x)⩽\frac{1-{\tau }^p}{p}{f}_{\eta }(z,x)x\mathrm{for}\mathrm{a}.\mathrm{a}.z\in \mathrm{\Omega },\mathrm{all}x\in \mathbb{R},\tau ⩾0.$$

(10)

From (10) with $\tau =0$, we have

$$p{F}_{\eta }(z,x)⩽f_{\eta }(z,x)x\mathrm{for}\mathrm{a}.\mathrm{a}.z\in \mathrm{\Omega },\mathrm{all}x\in \mathbb{R}.$$

(11)

From (11) and (9) it follows that

$${\int }_{\mathrm{\Omega }}\frac{f_{\eta }(z,u_n)u_n}{\parallel u_n{\parallel }^p}dz⟶+\infty .$$

(12)

Since $u_n\in N^{\eta }$ for all $n\in \mathbb{N}$, we have

$${\varrho }_a\left(D{u}_n\right)+{\parallel D{u}_n\parallel }_q^q={\int }_{\mathrm{\Omega }}{f}_{\eta }(z,u_n)u_{n}dz\forall n\in \mathbb{N},$$

so

$${\int }_{\mathrm{\Omega }}\frac{f_{\eta }(z,u_n)u_n}{\parallel u_n{\parallel }^p}dz={\varrho }_a\left(D{y}_n\right)+\frac{1}{\parallel u_n{\parallel }^{p-q}}{\parallel D{y}_n\parallel }_q^q⩽c_2\forall n\in \mathbb{N},$$

(13)

for some $c_2>0$. Comparing (12) and (13) we have a contradiction.

Next we assume that $y=0$. Let $\xi ⩾1$ and introduce

$$v_n={\left(p\xi \right)}^{\frac{1}{p}}{y}_n\forall n\in \mathbb{N}.$$

Evidently we have

$$v_n\stackrel{w}{⟶}0\mathrm{in}{W}_0^{1,\vartheta }\left(\mathrm{\Omega }\right)\mathrm{and}{v}_n⟶0\mathrm{in}{L}^r\left(\mathrm{\Omega }\right)$$

(see (8) and recall that $y=0$). We obtain

$${\int }_{\mathrm{\Omega }}{F}_{\eta }(z,v_n)dz⟶0.$$

(14)

Let $t_n\in [0,1]$ be such that

$${\phi }_{\eta }\left(t_n{u}_n\right)=\underset{0⩽t⩽1}{max}{\phi }_{\eta }\left(t{u}_n\right).$$

(15)

On account of (7), we can find $n_0\in \mathbb{N}$ such that

$$0<\frac{{\left(p\xi \right)}^{\frac{1}{p}}}{\parallel u_n\parallel }⩽1\forall n⩾n_0.$$

(16)

From (15) and (16), we see that for $n⩾n_0$, we have

$$\begin{array}{ccc}\hfill {\phi }_{\eta }\left(t_n{u}_n\right)& ⩾& {\phi }_{\eta }\left(v_n\right)\hfill \\ & ⩾& \xi {\varrho }_a\left(D{y}_n\right)+\frac{{\xi }^{\frac{q}{p}}}{q}{\parallel D{y}_n\parallel }_q^q-{\int }_{\mathrm{\Omega }}{F}_{\eta }(z,v_n)dz\hfill \\ & ⩾& \frac{{\xi }^{\frac{q}{p}}}{q}{\varrho }_{\vartheta }\left(y_n\right)-{\int }_{\mathrm{\Omega }}{F}_{\eta }(z,v_n)dz\hfill \end{array}$$

(since $p>1$ and $\xi ⩾1$), so

$${\phi }_{\eta }\left(t_n{u}_n\right)⩾\frac{{\xi }^{\frac{q}{p}}}{2q}\forall n⩾n_1⩾n_0$$

(see (14) and since $\parallel y_n\parallel =1$; see Proposition 1).

However, $\xi ⩾1$ is arbitrary. So, we infer that

$${\phi }_{\eta }\left(t_n{u}_n\right)⟶+\infty \mathrm{as}n\to +\infty .$$

(17)

We know that

$${\phi }_{\eta }\left(0\right)=0\mathrm{and}{\phi }_{\eta }\left(u_n\right)⩽c_1\forall n\in \mathbb{N}$$

(see (7)). Then from (17) it follows that we can find $n_2\in \mathbb{N}$ such that

$$t_n\in (0,1)\forall n⩾n_2.$$

(18)

From (15) and (18) we have

$$\frac{d}{dt}{\phi }_{\eta }\left(t{u}_n\right){|}_{t=t_n}=0,$$

so

$$⟨{\phi }_{\eta }^{\prime }\left(t_n{u}_n\right),t_n{u}_n⟩=0)$$

(by the chain rule and (18)), so

$$t_n{u}_n\in N^{\eta }\forall n⩾n_2,$$

(19)

with $0<t_n<1$. However, we also have $u_n\in N^{\eta }$. Then on account of (19) and Proposition 4, we have a contradiction. □

On account of Hypotheses H2 (i),(ii), we can find $c_3>0$ such that

$${\phi }_{\eta }\left(u\right)⩽{\phi }_{\eta }^{\pm }\left(u\right)+c_3\forall u\in W^{1,\vartheta }\left(\mathrm{\Omega }\right).$$

Then from Proposition 5, we infer the following corollary.

Corollary 1.

If Hypotheses H1, H2 hold, then ${\phi }_{\eta }^{\pm }{|}_{N^{\eta }}$ are coercive.

Next we show that the elements of the Nehari manifold maximize the fibering function.

Proposition 6.

If Hypotheses H1, H2 hold and $u\in N^{\eta }$, then ${\phi }_{\eta }\left(tu\right)⩽{\phi }_{\eta }\left(u\right)$ for all $t⩾0$.

Proof. 

We consider the corresponding fibering function

$$k_u^{\eta }\left(t\right)={\phi }_{\eta }\left(tu\right)\forall t⩾0.$$

Recall that

$${\left(k_u^{\eta }\right)}^{\prime }\left(1\right)={\mu }_u^{\eta }\left(1\right)$$

(see (3)). Hypotheses H2 (i),(ii),(iv) imply that given $\beta >0$, we can find $c_4=c_4\left(\beta \right)>0$ such that

$$F_{\eta }(z,x)⩾\frac{\beta }{p}{\left|x\right|}^p-c_4{\left|x\right|}^q\mathrm{for}\mathrm{a}.\mathrm{a}.z\in \mathrm{\Omega },\mathrm{all}x\in \mathbb{R}.$$

(20)

Therefore we have

$$\begin{array}{ccc}\hfill k_u^{\eta }\left(t\right)& =& {\phi }_{\eta }\left(tu\right)=\frac{t^p}{p}{\varrho }_a\left(Du\right)+\frac{t^q}{q}{\parallel Du\parallel }_q^q-{\int }_{\mathrm{\Omega }}{F}_{\eta }(z,tu)dz\hfill \\ & ⩽& \frac{t^p}{p}\left({\varrho }_a\left(Du\right)-\beta {\parallel u\parallel }_p^p\right)+c_5{t}^q,\hfill \end{array}$$

for some $c_5>0$ (see (20)). Choosing $\beta >\frac{{\varrho }_a\left(Du\right)}{{\parallel u\parallel }_p^p}$, we see that

$$k_u^{\eta }\left(t\right)⩽c_5{t}^q-c_6{t}^p,$$

for some $c_6>0$, so

$$k_u^{\eta }\left(t\right)<0\mathrm{for}\mathrm{all}\mathrm{t}>0\mathrm{large}$$

(21)

(since $q<p$). On the other hand from Hypotheses H2 (i),(iv), we see that given $\epsilon >0$ we can find $c_7=c_7\left(\epsilon \right)>0$ such that

$$F_{\eta }(z,x)⩽\frac{1}{q}(\widehat{\beta }\left(z\right)+\epsilon ){\left|x\right|}^q+c_7{\left|x\right|}^r\mathrm{for}\mathrm{a}.\mathrm{a}.z\in \mathrm{\Omega },\mathrm{all}x\in \mathbb{R}.$$

(22)

Hence we have

$$\begin{array}{ccc}\hfill k_u^{\eta }\left(t\right)& =& {\phi }_{\eta }\left(tu\right)\hfill \\ & ⩾& \frac{t^p}{p}{\varrho }_a\left(Du\right)+\frac{t^q}{q}\left({\parallel Du\parallel }_q^q-{\int }_{\mathrm{\Omega }}\widehat{\beta }{\left(z\right)\left|u\right|}^{q}dz-\frac{\epsilon }{{\widehat{\lambda }}_1\left(q\right)}{\parallel Du\parallel }_q^q\right)-c_8{t}^r\hfill \\ & ⩾& \frac{t^p}{p}{\varrho }_a\left(Du\right)+\frac{t^q}{q}\left(c_9-\frac{\epsilon }{{\widehat{\lambda }}_1\left(q\right)}\right){\parallel Du\parallel }_q^q-c_8{t}^r\hfill \end{array}$$

for some $c_8,c_9>0$ (see (22), (21) and Lemma 1).

Choosing $\epsilon \in (0,{\widehat{\lambda }}_1\left(q\right)c_9)$, we have

$$k_u^{\eta }\left(t\right)⩾c_{10}{t}^p-c_8{t}^r,$$

for some $c_{10}>0$. Since $p<r$, it follows that

$$k_u^{\eta }>0\mathrm{for}\mathrm{t}\in (0,1)\mathrm{small}.$$

(23)

Then from (21) and (23), it follows that $k_u^{\eta }$ has a local maximizer ${\widehat{t}}_u>0$ and so we have

$${\left(k_u^{\eta }\right)}^{\prime }\left({\widehat{t}}_u\right)={\mu }_u^{\eta }\left({\widehat{t}}_u\right)=0.$$

Since $u\in N^{\eta }$, from Proposition 4 we infer that

$${\widehat{\mathrm{t}}}_{\mathrm{u}}=1\mathrm{and}\mathrm{this}\mathrm{is}\mathrm{a}\mathrm{global}\mathrm{maximizer}\mathrm{of}{\mathrm{k}}_{\mathrm{u}}^{\eta },$$

so

$$k_u^{\eta }\left(t\right)⩽k_u^{\eta }\left(1\right)\forall t⩾0$$

and thus

$${\phi }_{\eta }\left(tu\right)⩽{\phi }_{\eta }\left(u\right)\forall t⩾0.$$

□

We show that the elements of the Nehari manifold not only are nontrivial but in fact are bounded away from zero in norm.

Proposition 7.

If Hypotheses H1, H2 hold, then there exists $d_0>0$ such that $d_0⩽{\parallel u\parallel }_r,\parallel u\parallel$ for all $u\in N^{\eta }$.

Proof. 

Hypotheses H2 (i),(iv) imply that given $\epsilon >0$, we can find $c_{11}=c_{11}\left(\epsilon \right)>0$ such that

$$f(z,x)x⩽(\widehat{\beta }\left(z\right)+\epsilon ){\left|x\right|}^q+c_{11}{\left|x\right|}^r\mathrm{for}\mathrm{a}.\mathrm{a}.z\in \mathrm{\Omega },\mathrm{all}x\in \mathbb{R}.$$

(24)

Given $u\in N^{\eta }$, we have

$$\begin{array}{ccc}\hfill {\varrho }_a\left(Du\right)+{\parallel Du\parallel }_q^q& =& {\int }_{\mathrm{\Omega }}f(z,u)udz\hfill \\ & ⩽& {\int }_{\mathrm{\Omega }}\widehat{\beta }{\left(z\right)\left|u\right|}^{q}dz+\frac{\epsilon }{{\widehat{\lambda }}_1\left(q\right)}{\parallel Du\parallel }_q^q+c_{11}{\parallel u\parallel }_r^r\hfill \end{array}$$

(see (24) and (2)), so

$${\varrho }_a\left(Du\right)+\left({\parallel Du\parallel }_q^q-{\int }_{\mathrm{\Omega }}\widehat{\beta }{\left(z\right)\left|u\right|}^q-\frac{\epsilon }{{\widehat{\lambda }}_1\left(q\right)}{\parallel Du\parallel }_q^q\right)⩽c_{11}{\parallel u\parallel }_r^r,$$

thus

$${\varrho }_a\left(Du\right)+\left(c_{12}-\frac{\epsilon }{{\widehat{\lambda }}_1\left(q\right)}\right){\parallel Du\parallel }_q^q⩽c_{11}{\parallel u\parallel }_r^r,$$

for some $c_{12}>0$ (see Lemma 1).

Choosing $\epsilon \in (0,{\widehat{\lambda }}_1\left(q\right)c_{12})$, we obtain

$${\varrho }_{\vartheta }\left(u\right)⩽c_{13}{\parallel u\parallel }_r^r,$$

for some $c_{13}>0$, thus

$${min\{\parallel u\parallel }^q{,\parallel u\parallel }^p\}⩽c_{13}{\parallel u\parallel }_r^r$$

(see Proposition 1).

Since $q<p<r$ and $W_0^{1,\vartheta }\left(\mathrm{\Omega }\right)\subseteq L^r\left(\mathrm{\Omega }\right)$ (see Proposition 2), we conclude that there exists $d_0>0$ such that

$$d_0⩽{\parallel u\parallel }_r,\parallel u\parallel \forall u\in N^{\eta }.$$

□

Similar results can be proved for the functionals ${\phi }_{\eta }^{\pm }$ and the corresponding sets $N_{\pm }^{\eta }$ (recall that $N_{\pm }^{\eta }\subseteq N^{\eta }$).

Proposition 8.

If Hypotheses H1, H2 hold, then

(a)   for every $u\in W_0^{1,\vartheta }\left(\mathrm{\Omega }\right)\setminus \left\{0\right\}$, $u⩾0$ (resp. $u⩽0$) there exists unique $t_u^{+}>0$ (resp. $t_u^{-}<0$) such that

$$t_u^{+}u\in N_{+}^{\eta },t_u^{-}u\in N_{-}^{\eta };$$

(b)   we have

$$\begin{array}{ccc}\hfill {\phi }_{\eta }^{+}\left(tu\right)& ⩽& {\phi }_{\eta }^{+}\left(u\right)\forall u\in N_{+}^{\eta },t⩾0,\hfill \\ \hfill {\phi }_{\eta }^{-}\left(tu\right)& ⩽& {\phi }_{\eta }^{-}\left(u\right)\forall u\in N_{-}^{\eta },t⩾0;\hfill \end{array}$$

(c)   there exist $d_0^{\pm }>0$ such that

$$\begin{array}{ccc}& & d_0^{+}⩽{\parallel u\parallel }_r,\parallel u\parallel \forall u\in N_{+}^{\eta },\hfill \\ & & d_0^{-}⩽{\parallel u\parallel }_r,\parallel u\parallel \forall u\in N_{-}^{\eta }.\hfill \end{array}$$

3. Multiple Solutions for Problem $\left(P_{\eta }^{ }\right)$

Let

$$m_{\eta }^{+}=\underset{N_{+}^{\eta }}{inf}{\phi }_{\eta }^{+}\mathrm{and}{m}_{\eta }^{-}=\underset{N_{-}^{\eta }}{inf}{\phi }_{\eta }^{-}.$$

We show that both infima are realized.

Proposition 9.

If Hypotheses H1, H2 hold, then there exist $u_{\eta }^{*}\in N_{+}^{\eta }$ and ${\widehat{u}}_{\eta }\in N_{-}^{\eta }$ such that

$${\phi }_{\eta }^{+}\left(u_{\eta }^{*}\right)=m_{\eta }^{+}\mathrm{and}{\phi }_{\eta }^{-}\left({\widehat{u}}_{\eta }\right)=m_{\eta }^{-}.$$

Proof. 

We consider a minimizing sequence ${\left\{u_n\right\}}_{n\in \mathbb{N}}\subseteq N_{+}^{\eta }$ for ${\phi }_{\eta }^{+}{|}_{N_{+}^{\eta }}$. Therefore, we have

$${\phi }_{\eta }^{+}\left(u_n\right)\searrow m_{\eta }^{+}.$$

Corollary 1 implies that the sequence ${\left\{u_n\right\}}_{n\in \mathbb{N}}\subseteq W_0^{1,\vartheta }\left(\mathrm{\Omega }\right)$ is bounded. So, by passing to a subsequence if necessary, we may assume that

$$u_n\stackrel{w}{⟶}{u}_{\eta }^{*}\mathrm{in}{W}_0^{1,\vartheta }\left(\mathrm{\Omega }\right)\mathrm{and}{u}_n⟶u_{\eta }^{*}\mathrm{in}{L}^r\left(\mathrm{\Omega }\right).$$

(25)

Since $u_n\in N_{+}^{\eta }\subseteq N^{\eta }$, we have

$${\varrho }_a\left(D{u}_n\right)+{\parallel D{u}_n\parallel }_q^q={\int }_{\mathrm{\Omega }}{f}_{\eta }(z,u_n)u_{n}dz,u_n⩾0\forall n\in \mathbb{N}.$$

(26)

From (25) and (26), we have $u_{\eta }^{*}⩾0$ and we can say that $u_{\eta }^{*}\ne 0$. Indeed, if $u_{\eta }^{*}=0$, then from (26), we see that

$${\varrho }_{\vartheta }\left(u_n\right)⟶0,$$

so

$$u_n⟶0\mathrm{in}{W}_0^{1,\vartheta }\left(\mathrm{\Omega }\right)$$

(27)

(see Proposition 1). However, from Proposition 8, we know that

$$0<d_0^{+}⩽\parallel u_n\parallel \forall n\in \mathbb{N}.$$

(28)

Comparing (27) and (28) we have a contradiction. Therefore $u_{\eta }^{*}\ne 0$, $u_{\eta }^{*}⩾0$.

The modular function ${\varrho }_a$ is continuous, convex, hence it is sequentially weakly lower semicontinuous and so from (25) we have

$${\varrho }_a\left(D{u}_{\eta }^{*}\right)+{\parallel D{u}_{\eta }^{*}\parallel }_q^q⩽{\int }_{\mathrm{\Omega }}{f}_{\eta }(z,u_{\eta }^{*})u_{\eta }^{*}dz.$$

(29)

Suppose that the inequality (29) is strict, that is, we have

$${\varrho }_a\left(D{u}_{\eta }^{*}\right)+{\parallel D{u}_{\eta }^{*}\parallel }_q^q<{\int }_{\mathrm{\Omega }}{f}_{\eta }(z,u_{\eta }^{*})u_{\eta }^{*}dz.$$

(30)

Consider the function

$${\left({\mu }_{u_{\mu }^{*}}^{\eta }\right)}^{+}\left(t\right)=t^p{\varrho }_a\left(D{u}_{\eta }^{*}\right)+t^q{\parallel D{u}_{\eta }^{*}\parallel }_q^q-{\int }_{\mathrm{\Omega }}{f}_{\eta }(z,t{u}_{\eta }^{*})\left(t{u}_{\eta }^{*}\right)dz\forall t⩾0.$$

(31)

Furthermore, let

$$k_{u_{\eta }^{*}}^{\eta ,+}\left(t\right)={\phi }_{\eta }^{+}\left(t{u}_{\eta }^{*}\right)\forall t⩾0$$

be the fibering function for $u_{\eta }^{*}$. Then

$${\left({\mu }_{u_{\eta }^{*}}^{\eta }\right)}^{+}\left(t\right)=t{\left(k_{u_{\eta }^{*}}^{\eta ,+}\right)}^{\prime }\left(t\right).$$

As in the proof of Proposition 4, exploiting Hypothesis H2 (iv) and recalling that $q<p<r$, we see that

$${\left({\mu }_{u_{\eta }^{*}}^{\eta }\right)}^{+}\left(t\right)>0\mathrm{for}\mathrm{t}>0\mathrm{small},{\left({\mu }_{u_{\eta }^{*}}^{\eta }\right)}^{+}\left(1\right)<0$$

(see (30) and (31)). So, by Bolzano’s theorem, we can find ${\widehat{t}}_{u_{\eta }^{*}}\in (0,1)$ such that

$${\left({\mu }_{u_{\eta }^{*}}^{\eta }\right)}^{+}\left({\widehat{t}}_{u_{\eta }^{*}}\right)=0,$$

so

$${\widehat{t}}_{u_{\eta }^{*}}{\left(k_{u_{\eta }^{*}}^{\eta ,+}\right)}^{\prime }\left({\widehat{t}}_{u_{\eta }^{*}}\right)=0,$$

thus

$${\left(k_{u_{\eta }^{*}}^{\eta ,+}\right)}^{\prime }\left({\widehat{t}}_{u_{\eta }^{*}}\right)=0$$

and hence

$${\widehat{t}}_{u_{\eta }^{*}}{u}_{\eta }^{*}\in N_{+}^{\eta }.$$

Therefore, we have

$$\begin{array}{ccc}\hfill m_{\eta }^{*}& ⩽& {\phi }_{\eta }^{+}\left({\widehat{t}}_{u_{\eta }^{*}}{u}_{\eta }^{*}\right)\hfill \end{array}$$

(32)

$$\begin{array}{ccc}& =& \frac{1}{p}{\varrho }_a\left(D\left({\widehat{t}}_{u_{\eta }^{*}}{u}_{\eta }^{*}\right)\right)+\frac{1}{q}{\parallel D\left({\widehat{t}}_{u_{\eta }^{*}}{u}_{\eta }^{*}\right)\parallel }_q^q-{\int }_{\mathrm{\Omega }}{F}_{\eta }(z,{\widehat{t}}_{u_{\eta }^{*}}{u}_{\eta }^{*})dz\hfill \end{array}$$

$$\begin{array}{ccc}& =& \frac{1}{p}\left({\int }_{\mathrm{\Omega }}f(z,{\widehat{t}}_{u_{\eta }^{*}}{u}_{\eta }^{*})\left({\widehat{t}}_{u_{\eta }^{*}}{u}_{\eta }^{*}\right)dz-{\parallel D\left({\widehat{t}}_{u_{\eta }^{*}}{u}_{\eta }^{*}\right)\parallel }_q^q\right)\hfill \end{array}$$

$$\begin{array}{ccc}& & +\frac{1}{q}{\parallel D\left({\widehat{t}}_{u_{\eta }^{*}}{u}_{\eta }^{*}\right)\parallel }_q^q-{\int }_{\mathrm{\Omega }}{F}_{\eta }(z,{\widehat{t}}_{u_{\eta }^{*}}{u}_{\eta }^{*})dz\hfill \end{array}$$

$$\begin{array}{ccc}& =& \frac{1}{p}{\int }_{\mathrm{\Omega }}\left(f_{\eta }(z,{\widehat{t}}_{u_{\eta }^{*}}{u}_{\eta }^{*})\left({\widehat{t}}_{u_{\eta }^{*}}{u}_{\eta }^{*}\right)-p{F}_{\eta }(z,{\widehat{t}}_{u_{}^{*}}{u}_{\eta }^{*})\right)dz\hfill \end{array}$$

$$\begin{array}{ccc}& & +\left(\frac{1}{q}-\frac{1}{p}\right){\parallel D\left({\widehat{t}}_{u_{\eta }^{*}}{u}_{\eta }^{*}\right)\parallel }_q^q\hfill \end{array}$$

(since ${\widehat{t}}_{u_{\eta }^{*}}{u}_{\eta }^{*}\in N_{+}^{\eta }\subseteq N^{\eta }$). From (10) we have

$$\frac{s^p}{p}{f}_{\eta }(z,x)x-F_{\eta }(z,sx)⩽\frac{1}{p}{f}_{\eta }(z,x)x-F_{\eta }(z,x)$$

for almost all $z\in \mathrm{\Omega }$, all $x\in \mathbb{R}$, all $s⩾0$. If $s\in (0,1)$, from (4), we have

$$f_{\eta }(z,sx)\left(sx\right)<s^p{f}_{\eta }(z,x)x\mathrm{for}\mathrm{a}.\mathrm{a}.z\in \mathrm{\Omega },\mathrm{all}x\in \mathbb{R}.$$

Therefore,

$$\frac{1}{p}{f}_{\eta }(z,sx)\left(sx\right)-F_{\eta }(z,sx)<\frac{1}{p}{f}_{\eta }(z,x)x-F_{\eta }(z,x)$$

(33)

for almost all $z\in \mathrm{\Omega }$, all $x\in \mathbb{R}$, all $s\in (0,1)$. Using (33), (32) and (25), we obtain

$$\begin{array}{ccc}\hfill m_{\eta }^{+}& <& \frac{1}{p}{\int }_{\mathrm{\Omega }}\left(f_{\eta }(z,u_{\eta }^{*})u_{\eta }^{*}-p{F}_{\eta }(z,u_{\eta }^{*})\right)dz+\left(\frac{1}{q}-\frac{1}{p}\right){\parallel D{u}_{\eta }^{*}\parallel }_q^q\hfill \\ & ⩽& \underset{n\to +\infty }{lim\; inf}\left(\frac{1}{p}{\int }_{\mathrm{\Omega }}\left(f_{\eta }(z,u_n)u_n-p{F}_{\eta }(z,u_n)\right)dz+\left(\frac{1}{q}-\frac{1}{p}\right){\parallel D{u}_n\parallel }_q^q\right)\hfill \\ & =& \underset{n\to +\infty }{lim\; inf}\left(\frac{1}{p}{\varrho }_a\left(D{u}_n\right)+\frac{1}{q}{\parallel D{u}_n\parallel }_q^q-{\int }_{\mathrm{\Omega }}{F}_{\eta }(z,u_n)dz\right)\hfill \\ & =& \underset{n\to +\infty }{lim\; inf}{\phi }_{\eta }^{+}\left(u_n\right)=m_{\eta }^{+}\hfill \end{array}$$

(since $u_n\in N_{+}^{\eta }\subseteq N^{\eta }$), a contradiction.

This proves that the strict inequality (30) cannot happen and so from (29) we have

$${\varrho }_a\left(D{u}_{\eta }^{*}\right)+{\parallel D{u}_{\eta }^{*}\parallel }_q^q={\int }_{\mathrm{\Omega }}f(z,u_{\eta }^{*})u_{\eta }^{*}dz.$$

Since

$${\int }_{\mathrm{\Omega }}f(z,u_n)u_{n}dz⟶{\int }_{\mathrm{\Omega }}f(z,u_{\eta }^{*})u_{\eta }^{*}dz$$

(see (25)), we infer that

$${\varrho }_{\vartheta }\left(u_n\right)⟶{\varrho }_{\vartheta }\left(u_{\eta }^{*}\right).$$

From this and the uniform convexity of the double phase integrand $\vartheta (z,·)$, we obtain

$${\varrho }_{\vartheta }(u_n-u_{\eta }^{*})⟶0$$

(see Harjulehto–Hästö ([23], p. 65)), so

$$u_n⟶u_{\eta }^{*}\mathrm{in}{W}_0^{1,\vartheta }\left(\mathrm{\Omega }\right)$$

(see Proposition 1). It follows that

$$u_{\eta }^{*}\in N_{+}^{\eta }\mathrm{and}{m}_{\eta }^{+}={\phi }_{\eta }^{+}\left(u_{\eta }^{*}\right).$$

Similarly, working with the set $N_{-}^{\eta }$ and the functional ${\phi }_{\eta }^{-}$, we produce ${\widehat{u}}_{\eta }\in W_0^{1,\vartheta }\left(\mathrm{\Omega }\right)$ such that

$${\widehat{u}}_{\eta }\in N_{-}^{\eta }\mathrm{and}{m}_{\eta }^{-}={\phi }_{\eta }^{-}\left({\widehat{u}}_{\eta }\right).$$

□

Reasoning as in Willem ([14], p. 74) and in Szulkin–Weth ([13], p. 611), via the quantitative deformation lemma of Willem [14], we can show that

$$u_{\eta }^{*}\in K_{{\phi }_{\eta }^{+}}=\{u\in W_0^{1,\vartheta }\left(\mathrm{\Omega }\right):{\left({\phi }_{\eta }^{+}\right)}^{\prime }\left(u\right)=0\}$$

(the critical set of ${\phi }_{\eta }^{+}$) and

$${\widehat{u}}_{\eta }\in K_{{\phi }_{\eta }^{-}}=\{u\in W_0^{1,\vartheta }\left(\mathrm{\Omega }\right):{\left({\phi }_{\eta }^{-}\right)}^{\prime }\left(u\right)=0\}$$

(the critical set of ${\phi }_{\eta }^{-}$). So, $N_{+}^{\eta }$ is a natural constraint for ${\phi }_{\eta }^{+}$, while $N_{-}^{\eta }$ is a natural constraint for ${\phi }_{\eta }^{-}$ (see Papageorgiou–Rǎdulescu–Repovš ([26], p. 425)).

Proposition 10.

If Hypotheses H1 and H2 hold and $u_{\eta }^{*}\in N_{+}^{\eta }$, ${\widehat{u}}_{\eta }\in N_{-}^{\eta }$ are as in Proposition 9, then $u_{\eta }^{*}\in K_{{\phi }_{\eta }^{+}}$, ${\widehat{u}}_{\eta }\in K_{{\phi }_{\eta }^{-}}$.

Proof. 

Let $\tau \in (0,+\infty )\setminus \left\{1\right\}$. We have

$${\phi }_{\eta }^{+}\left(\tau u_{\eta }^{*}\right)<{\phi }_{\eta }^{+}\left(u_{\eta }^{*}\right)=m_{\eta }^{+}$$

(34)

(see Proposition 8 and recall that $u_{\eta }^{*}\in N_{+}^{\eta }$). Arguing by contradiction, suppose that

$${\left({\phi }_{\eta }^{+}\right)}^{\prime }\left(u_{\eta }^{*}\right)\ne 0.$$

Then we can find $\delta >0$ and $l>0$ such that

$$\mathrm{if}\parallel \mathrm{u}-{\mathrm{u}}_{\eta }^{*}\parallel ⩽3\delta ,\mathrm{then}\parallel {\left({\phi }_{\eta }^{+}\right)}^{\prime }\left(\mathrm{u}\right){\parallel }_{*}⩾\mathrm{l}.$$

Consider the interval $D_{+}=(\frac{1}{2},\frac{2}{3})$ and note that

$$m_{+}^{*}=\underset{\tau \in \partial D_{+}}{max}{\phi }_{\eta }^{+}\left(\tau u_{\eta }^{*}\right)<m_{\eta }^{+}$$

(35)

(see (34)). We can always choose $\delta >0$ small so that

$$\frac{l\delta }{8}⩽\frac{m_{\eta }^{+}-m_{+}^{*}}{4}.$$

Then we apply the quantitative deformation lemma of Willem ([14], p. 38), with

$$\epsilon =\frac{l\delta }{8}\mathrm{and}S={\overline{B}}_{\delta }\left(u_{\eta }^{*}\right)$$

(here ${\overline{B}}_{\delta }\left(u_{\eta }^{*}\right)=\{u\in W_0^{1,\vartheta }\left(\mathrm{\Omega }\right):\parallel u-u_{\eta }^{*}\parallel ⩽\delta \}$). In what follows for $c\in \mathbb{R}$,

$${\left({\phi }_{\eta }^{+}\right)}^c=\{u\in W_0^{1,\vartheta }\left(\mathrm{\Omega }\right):{\phi }_{\eta }^{+}\left(u\right)⩽c\}.$$

So, there is a deformation $h:[0,1]\times W_0^{1,\vartheta }\left(\mathrm{\Omega }\right)⟶W_0^{1,\vartheta }\left(\mathrm{\Omega }\right)$ such that

•$h(1,u)=u$ if ${\phi }_{\eta }^{+}\notin [m_{\eta }^{+}-2\epsilon ,m_{\eta }^{+}+2\epsilon ]\cap S_{3\epsilon }$;

•$h(1,{\left({\phi }_{\eta }^{+}\right)}^{m_{\eta }^{*}+\epsilon }\cap {\overline{B}}_{\delta }\left(u_{\eta }^{*}\right))\subseteq {\left({\phi }_{\eta }^{+}\right)}^{m_{\eta }^{+}-\epsilon }$;

•${\phi }_{\eta }^{+}\left(h(1,u)\right)⩽{\phi }_{\eta }^{+}\left(u\right)$ for all $u\in W_0^{1,\vartheta }\left(\mathrm{\Omega }\right)$.

Then we have

$$\underset{\tau \in {\overline{D}}_{+}}{max}{\phi }_{\eta }^{+}\left(h(1,\tau u_{\eta }^{*})\right)<m_{\eta }^{+}.$$

(36)

We show that

$$h(1,D_{+}{u}_{\eta }^{*})\cap N_{+}^{\eta }\ne \varnothing .$$

Then this nonempty intersection and (36) lead to a contradiction. To show that this intersection is nonempty we argue as follows. Let $\zeta \left(\tau \right)=h(1,\tau u_{\eta }^{*})$ and consider the functions

$$\begin{array}{ccc}\hfill {\lambda }_0\left(\tau \right)& =& ⟨{\left({\phi }_{\eta }^{+}\right)}^{\prime }\left(\tau u_{\eta }^{*}\right),u_{\eta }^{*}⟩,\hfill \\ \hfill {\lambda }_1\left(\tau \right)& =& \frac{1}{\tau }⟨{\left({\phi }_{\eta }^{+}\right)}^{\prime }\left(\zeta \left(\tau \right)\right),\zeta \left(\tau \right)⟩.\hfill \end{array}$$

Since $u_{\eta }^{*}\in N_{+}^{\eta }$, we have

$$\begin{array}{ccc}\hfill ⟨{\left({\phi }_{\eta }^{+}\right)}^{\prime }\left(\tau u_{\eta }^{*}\right),u_{\eta }^{*}⟩& >& 0\mathrm{for}0<\tau <1,\hfill \\ \hfill ⟨{\left({\phi }_{\eta }^{+}\right)}^{\prime }\left(\tau u_{\eta }^{*}\right),u_{\eta }^{*}⟩& <& 0\mathrm{for}1<\tau \hfill \end{array}$$

(see Proposition 8).

Let $d_B(·,·,·)$ denote the Brouwer degree. From Lloyd ([27], p. 20), we have

$$d_B({\lambda }_0,D_{+},0)=-1.$$

(37)

Note that from the choice of $\epsilon >0$ and the properties of the deformation we have

$${\zeta |}_{\partial D_{+}}=\mathrm{id}.$$

Therefore,

$${\lambda }_0{|}_{\partial D_{+}}={\lambda }_1{|}_{\partial D_{+}}$$

(see (35) and the properties of the deformation h). So, by the boundary value dependence property of the degree (see Lloyd ([27], p. 25)), we have

$$d_B({\lambda }_1,D_{+},0)=d_B({\lambda }_0,D_{+},0)=-1$$

(see (37)). Then the solution property of the degree (see Lloyd ([27], p. 23)) implies that we can find $\tau \in D_{+}$ such that

$${\lambda }_1\left(\tau \right)=0,$$

so

$$\zeta \left(\tau \right)=h(1,\tau u_{\tau }^{*})\in N_{+}^{\eta },$$

thus

$$h(1,D_{+}{u}_{\eta }^{*})\cap N_{+}^{\eta }\ne Ø.$$

However, as we already stated earlier, this contradicts (36). Therefore, we must have ${\left({\phi }_{\eta }^{+}\right)}^{\prime }\left(u_{\eta }^{*}\right)=0$ and so $u_{\eta }^{*}\in K_{{\phi }_{\eta }^{+}}$. Similarly we show that ${\widehat{u}}_{\eta }\in K_{{\phi }_{\eta }^{-}}$. □

So, for the approximate problem $\left(P_{\eta }\right)$ (with $\eta >0$), we can have two constant sign solutions.

Proposition 11.

If Hypotheses H1 and H2 hold and $\eta >0$, then problem $\left(P_{\eta }\right)$ has at least two constant sign solutions

$$\begin{array}{ccc}& & u_{\eta }^{*}\in W_0^{1,\vartheta }\left(\mathrm{\Omega }\right)\cap L^{\infty }\left(\mathrm{\Omega }\right),u_{\eta }^{*}\left(z\right)>0\mathrm{for}\mathrm{a}.\mathrm{a}.z\in \mathrm{\Omega },\hfill \\ & & {\widehat{u}}_{\eta }\in W_0^{1,\vartheta }\left(\mathrm{\Omega }\right)\cap L^{\infty }\left(\mathrm{\Omega }\right),{\widehat{u}}_{\eta }\left(z\right)<0\mathrm{for}\mathrm{a}.\mathrm{a}.z\in \mathrm{\Omega }.\hfill \end{array}$$

Proof. 

From Proposition 10, we already have two constant sign solutions

$$u_{\eta }^{*}\in N_{+}^{\eta }\mathrm{and}{\widehat{u}}_{\eta }\in N_{-}^{\eta }.$$

From Theorem 3.1 of Gasiński–Winkert [15], we have

$$u_{\eta }^{*},{\widehat{u}}_{\eta }\in W_0^{1,\vartheta }\left(\mathrm{\Omega }\right)\cap L^{\infty }\left(\mathrm{\Omega }\right).$$

Finally Proposition 2.4 of Papageorgiou–Vetro–Vetro [19] implies that

$${\widehat{u}}_{\eta }\left(z\right)<0<u_{\eta }^{*}\left(z\right)\mathrm{for}\mathrm{a}.\mathrm{a}.z\in \mathrm{\Omega }.$$

□

Using $N_0^{\eta }$, we can generate a nodal (sign changing) solution of problem $\left(P_{\eta }\right)$. We set

$$m_{\eta }^0=\underset{N_0^{\eta }}{inf}{\phi }_{\eta }.$$

Proposition 12.

If Hypotheses H1 and H2 hold and $\eta >0$, then we can find $y_{\eta }^{*}\in N_0^{\eta }$ such that

$${\phi }_{\eta }\left(y_{\eta }^{*}\right)=m_{\eta }^0.$$

Proof. 

We consider a minimizing sequence ${\left\{y_n\right\}}_{n\in \mathbb{N}}\subseteq N_0^{\eta }$ for ${\phi }_{\eta }{|}_{N_0^{\eta }}$, that is

$$y_n\in N_0^{\eta }\forall n\in \mathbb{N}\mathrm{and}{\phi }_{\eta }\left(y_n\right)\searrow m_{\eta }^0.$$

(38)

We have

$$\left\{\begin{array}{c}{\phi }_{\eta }\left(y_n\right)={\phi }_{\eta }(y_n^{+}-y_n^{-})=\phi \left(\eta \right)\left(y_n^{+}\right)+{\phi }_{\eta }(-y_n^{-}),\hfill \\ y_n^{+}\in N^{\eta },-y_n^{-}\in N^{\eta }\forall n\in \mathbb{N}.\hfill \end{array}\right.$$

(39)

From Proposition 5, we have that the sequences ${\left\{y_n^{+}\right\}}_{n\in \mathbb{N}}\subseteq W_0^{1,\vartheta }\left(\mathrm{\Omega }\right)$ and ${\left\{y_n^{-}\right\}}_{n\in \mathbb{N}}\subseteq W_0^{1,\vartheta }\left(\mathrm{\Omega }\right)$ are bounded. So, we may assume that

$$y_n^{+}\stackrel{w}{⟶}{y}_1^{*}\mathrm{and}{y}_n^{-}\stackrel{w}{⟶}{y}_2^{*}\mathrm{in}{W}_0^{1,\varrho }\left(\mathrm{\Omega }\right)$$

(40)

Suppose that $y_1^{*}=0$. Since $y_n^{+}\in N^{\eta }$, for $n\in \mathbb{N}$ (see (39)), we have

$${\varrho }_a\left(D{y}_n^{+}\right)+{\parallel D{y}_n^{+}\parallel }_q^q={\int }_{\mathrm{\Omega }}{f}_{\eta }(z,y_n^{+})y_n^{+}dz\forall n\in \mathbb{N},$$

so

$${\varrho }_{\theta }\left(y_n^{+}\right)⟶0$$

(see (40) and recall that $y_1^{*}=0$), thus

$$y_n^{+}⟶0\mathrm{in}{W}_0^{1,\vartheta }\left(\mathrm{\Omega }\right)$$

(see Proposition 1).

This then contradicts Proposition 7. Therefore $y_1^{*}\ne 0$.

Similarly we show that $y_2^{*}\ne 0$.

From Proposition 4, we know that there exist unique $t_{+},t_{-}>0$ such that

$$t_{+}{y}_1^{*}\in N^{\eta },t_{-}{y}_2^{*}\in N^{\eta }.$$

We set

$$y_{\eta }^{*}=t_{+}{y}_1^{*}-t_{-}{y}_2^{*}\in W_0^{1,\vartheta }\left(\mathrm{\Omega }\right),$$

so

$${\left(y_{\eta }^{*}\right)}^{+}=t_{+}{y}_1^{*}\mathrm{and}{\left(y_{\eta }^{*}\right)}^{-}=t_{-}{y}_2^{*}.$$

Then, using (38), (39) and Proposition 6, we have

$$\begin{array}{ccc}\hfill m_{\eta }^0& =& \underset{n\to +\infty }{lim}{\phi }_{\eta }\left(y_n\right)\hfill \\ & =& \underset{n\to +\infty }{lim}\left({\phi }_{\eta }\left(y_n^{+}\right)+{\phi }_{\eta }(-y_n^{-})\right)\hfill \\ & ⩾& \underset{n\to +\infty }{lim\; inf}\left({\phi }_{\eta }\left(t_n{y}_n^{+}\right)+{\phi }_{\eta }(-t_{-}{y}_n^{-})\right)\hfill \\ & ⩾& {\phi }_{\eta }\left(t_n{y}_1^{*}\right)+{\phi }_{\eta }(-t_{-}{y}_2^{*})\hfill \\ & =& {\phi }_{\eta }\left(y_{\eta }^{*}\right)\hfill \\ & ⩾& m_{\eta }^0\hfill \end{array}$$

(since $y_{\eta }^{*}\in N_0^{\eta }$), so

$$m_{\eta }^0={\phi }_{\eta }\left(y_{\eta }^{*}\right),$$

with $y_{\eta }^{*}\in N_0^{\eta }$. □

Arguing as in the proof of Proposition 10, we show that $N_0^{\eta }$ is a natural constraint for the functional ${\phi }_{\eta }$, that is, $y_{\eta }^{*}\in K_{{\phi }_{\eta }}$. In this case we replace the interval $D_{+}$ by the square

$$\widehat{D}=\left(\frac{1}{2},\frac{3}{2}\right)\times \left(\frac{1}{2},\frac{3}{2}\right)$$

and the functionals $\zeta ,{\lambda }_0,{\lambda }_1$ by

$$\begin{array}{ccc}\hfill \widehat{\zeta }(\tau ,t)& =& h(1,\tau y_1^{*}-t{y}_2^{*})\forall \tau ,t⩾0,\hfill \\ \hfill {\widehat{\lambda }}_0(\tau ,t)& =& \left(⟨{\phi }_{\eta }^{\prime }\left(\tau y_1^{*}\right),y_1^{*}⟩,⟨{\phi }_{\eta }^{\prime }(-t{y}_2^{*}),-y_2^{*}⟩\right),\hfill \\ \hfill {\widehat{\lambda }}_1(\tau ,t)& =& \left(\frac{1}{\tau }⟨{\phi }_{\eta }^{\prime }\left(\zeta {(\tau ,t)}^{+}\right),\zeta {(\tau ,t)}^{+}⟩,\frac{1}{\tau }⟨{\phi }_{\eta }^{\prime }(-\zeta {(\tau ,t)}^{-}),-\zeta {(\tau ,t)}^{-}⟩\right).\hfill \end{array}$$

Reasoning as in the proof of Proposition 10, via the quantitative deformation lemma of Willem ([14], p. 38), we obtain the following result showing that $N_0^{\eta }$ is a natural constraint for the functional ${\phi }_{\eta }$.

Proposition 13.

If Hypotheses H1 and H2 hold and $\eta >0$, then

$$y_{\eta }^{*}\in K_{{\phi }_{\eta }}=\{y\in W_0^{1,\vartheta }\left(\mathrm{\Omega }\right):{\phi }_{\eta }^{\prime }\left(y\right)=0\}$$

and so $y_{\eta }^{*}\in W_0^{1,\vartheta }\left(\mathrm{\Omega }\right)\cap L^{\infty }\left(\mathrm{\Omega }\right)$ is a nodal solution of $\left(P_{\eta }\right)$.

Summarizing the situation for problem $\left(P_{\eta }\right)$, we can state the following multiplicity result.

Proposition 14.

If Hypotheses H1 and H2 hold and $\eta >0$, then problem $\left(P_{\eta }\right)$ has at least three nontrivial solutions

$$\begin{array}{ccc}& & u_{\eta }^{*}\in W_0^{1,\vartheta }\left(\mathrm{\Omega }\right)\cap L^{\infty }\left(\mathrm{\Omega }\right),u_{\eta }^{*}\left(z\right)>0\mathrm{for}\mathrm{a}.\mathrm{a}.z\in \mathrm{\Omega },\hfill \\ & & {\widehat{u}}_{\eta }\in W_0^{1,\vartheta }\left(\mathrm{\Omega }\right)\cap L^{\infty }\left(\mathrm{\Omega }\right),{\widehat{u}}_{\eta }\left(z\right)<0\mathrm{for}\mathrm{a}.\mathrm{a}.z\in \mathrm{\Omega },\hfill \\ & & y_{\eta }^{*}\in W_0^{1,\vartheta }\left(\mathrm{\Omega }\right)\cap L^{\infty }\left(\mathrm{\Omega }\right)\mathrm{nodal}\hfill \end{array}$$

and

$${\phi }_{\eta }^{+}\left(u_{\eta }^{*}\right)=m_{+}^{\eta }<0<m_{-}^{\eta }={\phi }_{\eta }^{-}\left({\widehat{u}}_{\eta }\right),{\phi }_{\eta }\left(y_{\eta }^{*}\right)=m_0^{\eta }.$$

4. Multiple Solutions for Problem $\left(P\right)$

In this section, we use the solutions from Proposition 14 and let $\eta \to 0^{+}$ to produce multiple solutions with sign information for problem $\left(P\right)$.

Proposition 15.

If Hypotheses H1 and H2 hold, then there exists $d^{*}>0$ such that

$$0<d^{*}⩽{\phi }_{\eta }\left(u\right)\forall u\in N^{\eta },\eta \in (0,1).$$

Proof. 

On account of Hypotheses H2 (i),(iv), given $\epsilon >0$, we can find $c_{14}=c_{14}\left(\epsilon \right)>0$ such that

$$F(z,x)⩽\frac{1}{q}(\widehat{\beta }\left(z\right)+\epsilon ){\left|x\right|}^q+\frac{c_{14}}{r}{\left|x\right|}^r\mathrm{for}\mathrm{a}.\mathrm{a}.z\in \mathrm{\Omega },\mathrm{all}x\in \mathbb{R}.$$

(41)

If $u\in N^{\eta }$ ($\eta \in (0,1)$), then from Proposition 6, we have

$$\begin{array}{ccc}\hfill {\phi }_{\eta }\left(u\right)& =& \underset{t⩾0}{max}{\phi }_{\eta }\left(tu\right)\hfill \\ & ⩾& \underset{t⩾0}{max}\left(\frac{t^p}{p}{\varrho }_a\left(Du\right)+\frac{t^q}{q}{\parallel Du\parallel }_q^q-{\int }_{\mathrm{\Omega }}F(z,tu)dz-\frac{t^r}{r}{\parallel u\parallel }_r^r\right)\hfill \\ & ⩾& \underset{t⩾0}{max}(c_{15}{t}^p-c_{16}{t}^r),\hfill \end{array}$$

for some $c_{15},c_{16}>0$ (recall that $\eta \in (0,1)$, use (41), Lemma 1 and choose $\epsilon >0$ small), so

$${\phi }_{\eta }\left(u\right)⩾d^{*}>0\forall \eta \in (0,1)$$

(since $p<r$). □

In a similar fashion, using Proposition 8, we obtain the following result.

Proposition 16.

If Hypotheses H1 and H2 hold, then there exists $d_{\pm }^{*}>0$ such that

$$\begin{array}{ccc}\hfill 0<d_{+}^{*}⩽{\phi }_{\eta }^{+}\left(u\right)& & \forall u\in N_{+}^{\eta },\eta \in (0,1),\hfill \\ \hfill 0<d_{-}^{*}⩽{\phi }_{\eta }^{-}\left(u\right)& & \forall u\in N_{-}^{\eta },\eta \in (0,1).\hfill \end{array}$$

Now we are ready to pass to the limit as $\eta \to 0^{+}$ and have the multiplicity result for problem $\left(P\right)$.

In the sequel $\phi :W_0^{1,\vartheta }\left(\mathrm{\Omega }\right)⟶\mathbb{R}$ is the energy functional for problem $\left(P\right)$ defined by

$$\phi \left(u\right)=\frac{1}{p}{\varrho }_a\left(Du\right)+\frac{1}{q}{\parallel Du\parallel }_q^q-{\int }_{\mathrm{\Omega }}F(z,u)dz\forall u\in W_0^{1,\vartheta }\left(\mathrm{\Omega }\right).$$

We have that $\phi \in C^1\left(W_0^{1,\vartheta }\left(\mathrm{\Omega }\right)\right)$.

Furthermore, we consider the positive and negative truncations of $\phi$, namely the $C^1$-functionals ${\phi }_{\phi }:W_0^{1,\vartheta }\left(\mathrm{\Omega }\right)⟶\mathbb{R}$ defined by

$${\phi }_{\pm }\left(u\right)=\frac{1}{p}{\varrho }_q\left(Du\right)+\frac{1}{q}{\parallel Du\parallel }_q^q-{\int }_{\mathrm{\Omega }}F(z,\pm u^{\pm })dz\forall u\in W_0^{1,\vartheta }\left(\mathrm{\Omega }\right).$$

Again we have ${\phi }_{\pm }\in C^1\left(W_0^{1,\vartheta }\left(\mathrm{\Omega }\right)\right)$.

We introduce the corresponding Nehari manifolds

$$\begin{array}{ccc}\hfill N& =& \{u\in W_0^{1,\vartheta }\left(\mathrm{\Omega }\right):⟨{\phi }^{\prime }\left(u\right),u⟩=0,u\ne 0\},\hfill \\ \hfill N_{+}& =& \{u\in W_0^{1,\vartheta }\left(\mathrm{\Omega }\right):⟨{\phi }_{+}^{\prime }\left(u\right),u⟩=0,u⩾0,u\ne 0\},\hfill \\ \hfill N_{-}& =& \{u\in W_0^{1,\vartheta }\left(\mathrm{\Omega }\right):⟨{\phi }_{-}^{\prime }\left(u\right),u⟩=0,u⩽0,u\ne 0\},\hfill \\ \hfill N_0& =& \{u\in W_0^{1,\vartheta }\left(\mathrm{\Omega }\right):u^{+}\in N,-u^{-}\in N\}.\hfill \end{array}$$

Theorem 1.

If Hypotheses H1 and H2 hold, then problem $\left(P\right)$ has at least three nontrivial solutions

$$\begin{array}{ccc}& & u^{*}\in W_0^{1,\vartheta }\left(\mathrm{\Omega }\right)\cap L^{\infty }\left(\mathrm{\Omega }\right),u^{*}\left(z\right)>0\mathrm{for}\mathrm{a}.\mathrm{a}.z\in \mathrm{\Omega },\hfill \\ & & \widehat{u}\in W_0^{1,\vartheta }\left(\mathrm{\Omega }\right)\cap L^{\infty }\left(\mathrm{\Omega }\right),\widehat{u}\left(z\right)<0\mathrm{for}\mathrm{a}.\mathrm{a}.z\in \mathrm{\Omega },\hfill \\ & & y^{*}\in W_0^{1,\vartheta }\left(\mathrm{\Omega }\right)\cap L^{\infty }\left(\mathrm{\Omega }\right)\mathrm{nodal}.\hfill \end{array}$$

Proof. 

Let ${\eta }_n\to 0^{+}$. According to Proposition 14, we can find ${\overline{u}}_n={\overline{u}}_{{\eta }_n}\in N_0^{{\eta }_n}$ such that

$${\phi }_{{\eta }_n}\left({\overline{u}}_n\right)=m_0^{{\eta }_n}⟶m_{*}>0\mathrm{and}{\phi }_{{\eta }_n}^{\prime }\left({\overline{u}}_n\right)=0\forall n\in \mathbb{N}$$

(42)

(see Proposition 15).

Claim. The sequence ${\left\{{\overline{u}}_n\right\}}_{n\in \mathbb{N}}\subseteq W_0^{1,\vartheta }\left(\mathrm{\Omega }\right)$ is bounded.

We argue by contradiction. So, suppose that at least for a subsequence, we have

$$\parallel {\overline{u}}_n\parallel ⟶+\infty .$$

(43)

Let ${\overline{v}}_n=\frac{{\overline{u}}_n}{\parallel {\overline{u}}_n\parallel }$ for $n\in \mathbb{N}$. Then $\parallel {\overline{v}}_n\parallel =1$ for all $n\in \mathbb{N}$ and so we may assume that

$${\overline{v}}_n\stackrel{w}{⟶}\overline{v}\mathrm{in}{W}_0^{1,\vartheta }\left(\mathrm{\Omega }\right)\mathrm{and}{\overline{v}}_n⟶\overline{v}\mathrm{in}{L}^r\left(\mathrm{\Omega }\right).$$

(44)

First suppose that $\overline{v}=0$. Using (42) and Proposition 6, for $k>1$, we have

$$\begin{array}{ccc}\hfill m_0^{{\eta }_n}& =& {\phi }_{{\eta }_n}\left({\overline{u}}_n\right)\hfill \\ & ⩾& {\phi }_{{\eta }_n}\left(\frac{k}{\parallel {\overline{u}}_n\parallel }{\overline{u}}_n\right)\hfill \\ & =& {\phi }_{{\eta }_n}\left(k{\overline{v}}_n\right)\hfill \\ & =& \frac{k^p}{p}{\varrho }_a\left(D{\overline{v}}_n\right)+\frac{k^q}{q}\parallel D{\overline{v}}_n{\parallel }_q^q-{\int }_{\mathrm{\Omega }}F(z,k{\overline{v}}_n)dz-\eta k^r{\parallel {\overline{v}}_n\parallel }_r^r\hfill \\ & ⩾& \frac{k^q}{p}-{\int }_{\mathrm{\Omega }}F(z,k{\overline{v}}_n)dz-\eta k^r{\parallel {\overline{v}}_n\parallel }_r^r\hfill \end{array}$$

(since $k>0$, $q<p$, $\parallel {\overline{v}}_n\parallel =1$; see Proposition 1).

We pass to the limit as $n\to +\infty$ and use (44) (recalling that $\overline{v}=0$). Then

$$m_{*}⩾\frac{k^q}{p}>0$$

(see (42)).

Since $k>1$ is arbitrary, we let $k\to +\infty$ and have a contradiction.

Next suppose that $\overline{v}\ne 0$ and set

$$\widehat{\mathrm{\Omega }}=\{z\in \mathrm{\Omega }:v\left(z\right)\ne 0\}.$$

We have $|\widehat{\mathrm{\Omega }}{|}_N>0$ (here ${|·|}_N$ denotes the Lebesgue measure on ${\mathbb{R}}^N$) and $|u_n\left(z\right)|\to +\infty$ for a.a. $z\in \widehat{\mathrm{\Omega }}$. From (42) we have

$$\begin{array}{ccc}\hfill 0& <& \frac{m_0^{{\eta }_n}}{\parallel {\overline{u}}_n{\parallel }^p}=\frac{{\phi }_{{\eta }_n}\left({\overline{u}}_n\right)}{\parallel {\overline{u}}_n{\parallel }^p}\hfill \\ & ⩽& \frac{1}{p}{\varrho }_a\left(D{\overline{v}}_n\right)+\frac{1}{q\parallel {\overline{u}}_n{\parallel }^{p-q}}{\parallel D{\overline{v}}_n\parallel }_q^q-{\int }_{\mathrm{\Omega }}\frac{F(z,{\overline{u}}_n)}{\parallel {\overline{u}}_n{\parallel }^p}dz\hfill \\ & ⩽& \frac{1}{q}\left({\varrho }_a\left(D{\overline{v}}_n\right)+{\parallel D{\overline{v}}_n\parallel }_q^q\right)-{\int }_{\mathrm{\Omega }}\frac{F(z,{\overline{u}}_n)}{\parallel {\overline{u}}_n{\parallel }^p}dz\hfill \\ & =& \frac{1}{q}{\varrho }_{\vartheta }\left({\overline{v}}_n\right)-{\int }_{\mathrm{\Omega }}\frac{F(z,{\overline{u}}_n)}{\parallel {\overline{u}}_n{\parallel }^p}dz\hfill \\ & =& \frac{1}{q}-{\int }_{\mathrm{\Omega }}\frac{F(z,{\overline{u}}_n)}{\parallel {\overline{u}}_n{\parallel }^p}dz\hfill \end{array}$$

(45)

for all $n⩾n_1$ (see (43) and recall that $q<p$).

Hypothesis H2 (ii) implies that

$${\int }_{\widehat{\mathrm{\Omega }}}\frac{F(z,{\overline{u}}_n)}{\parallel {\overline{u}}_n{\parallel }^p}dz⟶+\infty$$

and so

$${\int }_{\mathrm{\Omega }}\frac{F(z,{\overline{u}}_n)}{\parallel {\overline{u}}_n{\parallel }^p}dz⟶+\infty .$$

Passing to the limit as $n\to +\infty$ in (45), we have a contradiction. This proves the Claim.

On account of the Claim, we may assume that

$${\overline{u}}_n\stackrel{w}{⟶}{y}^{*}\mathrm{in}{W}_0^{1,\vartheta }\left(\mathrm{\Omega }\right)\mathrm{and}{\overline{u}}_n⟶y^{*}\mathrm{in}{L}^r\left(\mathrm{\Omega }\right).$$

(46)

From (42), we have

$$⟨V\left({\overline{u}}_n\right),h⟩={\int }_{\mathrm{\Omega }}{f}_{\eta }(z,{\overline{u}}_n)hdz\forall h\in W_0^{1,\vartheta }\left(\mathrm{\Omega }\right).$$

We choose $h={\overline{u}}_n-y^{*}\in W_0^{1,\vartheta }\left(\mathrm{\Omega }\right)$, pass to the limit as $n\to +\infty$ and use (46). We obtain

$$\underset{n\to +\infty }{lim}⟨V\left({\overline{u}}_n\right),{\overline{u}}_n-y^{*}⟩=0,$$

so

$${\overline{u}}_n⟶y^{*}\mathrm{in}{W}_0^{1,\vartheta }\left(\mathrm{\Omega }\right)$$

(see Proposition 3), so

$${\overline{u}}_n^{+}⟶{\left(y^{*}\right)}^{+}\mathrm{and}{\overline{u}}_n^{-}⟶{\left(y^{*}\right)}^{-}\mathrm{in}{W}_0^{1,\vartheta }\left(\mathrm{\Omega }\right).$$

(47)

From Proposition 15, we have

$$\phi \left(0\right)=0<d^{*}⩽\phi \left({\left(y^{*}\right)}^{\pm }\right)$$

(see (47) and recall that ${\eta }_n\to 0^{+}$), so ${\left(y^{*}\right)}^{\pm }\ne 0$ and thus $y^{*}\in N_0$ (see (47)).

Moreover, we have

$$m_{*}=\phi \left(y_{*}\right)⩾m_0=\underset{N_0}{inf}\phi .$$

(48)

We will show that in fact equality holds in (48). To this end, given $\epsilon >0$ we choose $v_{\epsilon }\in N_0$ such that

$$\phi \left(v_{\epsilon }\right)⩽m_0+\epsilon .$$

(49)

According to Proposition 8, we can find ${\tau }_n,t_n>0$ such that

$${\tau }_n{v}_{\epsilon }^{+}\in N_{+}^{{\eta }_n},-t_n{v}_{\epsilon }^{-}\in N_{-}^{{\eta }_n},\mathrm{hence}{\tau }_n{v}_{\epsilon }^{+}-t_n{v}_{\epsilon }^{-}\in N_{-}^{{\eta }_n}\forall n\in \mathbb{N}.$$

(50)

We know that

$$\phi \left(u\right)={\phi }_{\eta }\left(u\right)+\eta {\parallel u\parallel }_r^r\forall u\in W_0^{1,\vartheta }\left(\mathrm{\Omega }\right),\eta >0.$$

(51)

Then,

$$\begin{array}{ccc}\hfill m_0+\epsilon & ⩾& \phi \left(v_{\epsilon }\right)=\phi \left(v_{\epsilon }^{+}\right)+\phi (-v_{\epsilon }^{-})\hfill \\ & ⩾& \phi \left({\tau }_n{v}_{\epsilon }^{+}\right)+\phi (-t_n{v}_{\epsilon }^{-})\hfill \\ & =& \phi ({\tau }_n{v}_{\epsilon }^{+}-t_n{v}_{\epsilon }^{-})\hfill \\ & ⩾& {\phi }_{{\eta }_n}({\tau }_n{v}_{\epsilon }^{+}-t_n{v}_{\epsilon }^{-})\hfill \\ & ⩾& m_0^{{\eta }_n}\forall n\in \mathbb{N}\hfill \end{array}$$

(see Proposition 6, (51), (50)), so

$$m_0+\epsilon ⩾m_{*}$$

(see (42)). Since $\epsilon >0$ is arbitrary, we let $\epsilon \to 0^{+}$ and obtain

$$m_{*}⩽m_0,$$

so

$$m_{*}=m_0$$

(see (48)). From (48), we have

$$\phi \left(y^{*}\right)=m_0,y^{*}\in N_0.$$

Since ${\phi }_{{\eta }_n}^{\prime }\left({\overline{u}}_n\right)=0$ for all $n\in \mathbb{N}$, we have ${\phi }^{\prime }\left(y^{*}\right)=0$ and so we conclude that $y^{*}$ is nodal solution of $\left(P\right)$ and $y^{*}\in W_0^{1,\vartheta }\left(\mathrm{\Omega }\right)\cap L^{\infty }\left(\mathrm{\Omega }\right)$.

Similarly, using Propositions 14 and 16, we produce $u^{*}\in N_{+}$ and $\widehat{u}\in N_{-}$ with

$$\begin{array}{ccc}& & {\phi }_{+}\left(u^{*}\right)=m_{+}=\underset{N_{+}}{inf}\phi ,u^{*}\in W_0^{1,\vartheta }\left(\mathrm{\Omega }\right)\cap L^{\infty }\left(\mathrm{\Omega }\right),u^{*}\left(z\right)>0\mathrm{a}.\mathrm{e}.\mathrm{in}\mathrm{\Omega },\hfill \\ & & {\phi }_{-}\left(\widehat{u}\right)=m_{-}=\underset{N_{-}}{inf}\phi ,\widehat{u}\in W_0^{1,\vartheta }\left(\mathrm{\Omega }\right)\cap L^{\infty }\left(\mathrm{\Omega }\right),\widehat{u}\left(z\right)<0\mathrm{a}.\mathrm{e}.\mathrm{in}\mathrm{\Omega }.\hfill \end{array}$$

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5. Conclusions

We have extended the classical multiplicity result to superlinear problems (see [10,25], three solutions theorem) to double-phase problems, under more general conditions on the reaction and we provided sign information for all the solutions produced.