Anisotropic (p, q) Equation with Partially Concave Terms

Abstract

We consider a Dirichlet problem driven by the anisotropic $(p,q)$ Laplacian. In the reaction, we have a parametric partially concave term plus a “superlinear” perturbation (convex term) which need not satisfy the Ambrosetti–Rabinowitz condition. Using variational tools, we show that for all small values of the parameter $\lambda >0$, the problem has at least two nontrivial smooth solutions.

1. Introduction

Let $\mathsf{\Omega }\subseteq {\mathbb{R}}^N$ be a bounded domain with a $C^2$ boundary $\partial \mathsf{\Omega }$. In this paper, we study the following parametric anisotropic $(p.q)$ equation

$$(P_{\lambda })\hspace{1em}\hspace{1em}\left\{\begin{array}{c}-{\Delta }_{p(z)}u(z)-{\Delta }_{q(z)}u(z)=\lambda u{(z)}^{\tau (z)-2}u(z)+f(z,u(z))\mathrm{in}\mathsf{\Omega },\hfill \\ {u|}_{\partial \mathsf{\Omega }}=0,u>0,\lambda >0.\hfill \end{array}\right.$$

We define $E_1=\{r\in C(\overline{\mathsf{\Omega }}):1<\underset{\overline{\mathsf{\Omega }}}{min}r\}$, and for $r\in E_1$, we set

$$r_{-}=\underset{\overline{\mathsf{\Omega }}}{min}r\mathrm{and}{r}_{+}=\underset{\overline{\mathsf{\Omega }}}{max}r.$$

For $r\in E_1$, by ${\Delta }_{r(z)}$ we define the anisotropic r-Laplacian differential operator (also known as the $r(z)$ Laplacian), which is defined by

$${\Delta }_{r(z)}u={\mathrm{div}(|Du|}^{r(z)-2}Du)\forall u\in W_0^{1,r(z)}(\mathsf{\Omega }).$$

In contrast to the standard r Laplacian (that is, r is constant), the anisotropic operator is not homogeneous, and this is the source of difficulties in the analysis of anisotropic problems. Problem $(P_{\lambda })$ is driven by the sum of two such operators with distinct variable exponents ($(p,q)$ equation). In the reaction (right-hand side) of $(P_{\lambda })$, we have the combined effect of two terms with distinct behavior. One is the parametric power term $x⟼{\lambda \left|x\right|}^{\tau (z)-2}x$ with $1<{\tau }_{-}<q_{-}<{\tau }_{+}<p_{+}$. These conditions imply that this term is partially “concave”. The second term (the perturbation) $f(z,x)$ is a Carathéodory function (that is, for all $x\in \mathbb{R}$, $z⟼f(z,x)$ is measurable and for almost all $z\in \mathsf{\Omega }$, $x⟼f(z,x)$ is continuous), which exhibits $(p_{+}-1)$-superlinear growth as $x\to \pm \infty$ but without satisfying the common in such settings Ambrosetti–Rabinowitz condition (the AR condition for short). Therefore, problem $(P_{\lambda })$ is a variant of the classical “concave–convex” problem, in which we have only a partially concave contribution. Concave–convex problems were investigated starting with the work of Ambrosetti et al. [1] who considered equations driven by the Laplace differential operator. Their work was extended to equations driven by the p Laplacian by García et al. [2] and Guo and Zhang [3]. Recently, Papageorgiou et al. [4] extended the aforementioned works to anisotropic $(p,q)$ equations with an indefinite potential term. All these works prove existence and multiplicity results which are global in the parameter $\lambda >0$. There are no works on problems with partially concave terms. Here, we prove a multiplicity theorem producing two nontrivial smooth solutions when the parameter $\lambda >0$ is small. However, our result here is not global in $\lambda >0$. It is an interesting open problem if our multiplicity theorem can be improved to be global in $\lambda >0$ (a bifurcation-type theorem). Finally, we mention the recent work of Leonardi and Papageorgiou [5] on anisotropic double phase equations. Their multiplicity result too is not global in $\lambda >0$.

2. Materials and Methods

The analysis of $(P_{\lambda })$ is based on variable Lebesgue and Sobolev spaces. Details about these spaces can be found in the book of Diening et al. [6].

Let $L^0(\mathsf{\Omega })$ be the space of all measurable functions $u:\mathsf{\Omega }⟶\mathbb{R}$. As usual, we identify two such functions which differ only on a Lebesgue-null set. Let $r\in E_1$. The variable Lebesgue space $L^{r(z)}(\mathsf{\Omega })$ is defined by

$$L^{r(z)}(\mathsf{\Omega })=\left\{u\in L^0(\mathsf{\Omega }):{\varrho }_r(u)={\int }_{\mathsf{\Omega }}{\left|u\right|}^{r(z)}dz<+\infty \right\}.$$

We call ${\varrho }_r$ the “modular function” corresponding to the exponent $r\in E_1$. This space is endowed with the so-called “Luxemburg norm” ${\parallel ·\parallel }_{r(z)}$ defined by

$${\parallel u\parallel }_{r(z)}=inf\left\{\mu >0:{\int }_{\mathsf{\Omega }}{\left({\displaystyle \frac{\left|u\right|}{\mu }}\right)}^{r(z)}dz⩽1\right\}\forall u\in L^{r(z)}(\mathsf{\Omega }).$$

Furnished with this norm, $L^{r(z)}(\mathsf{\Omega })$ becomes a Banach space which is separable and reflexive. In fact, it is uniformly convex, since the density of $x⟼{\left|x\right|}^{r(z)}$ of the modular function is uniformly convex. We define $r^{\prime }(z)={\displaystyle \frac{r(z)}{r(z)-1}}$ for all $z\in \overline{\mathsf{\Omega }}$. Then, $r^{\prime }\in E_1$ and ${\displaystyle \frac{1}{r(z)}}+{\displaystyle \frac{1}{r^{\prime }(z)}}=1$ for all $z\in \overline{\mathsf{\Omega }}$. We have

$$L^{r^{\prime }(z)}(\mathsf{\Omega })=L^{r(z)}{(\mathsf{\Omega })}^{\ast }$$

and the following Hölder-type inequality holds

$${\int }_{\mathsf{\Omega }}\left|uv\right|dz⩽\left({\displaystyle \frac{1}{r_{-}}}+{\displaystyle \frac{1}{r_{-}^{\prime }}}\right){\parallel u\parallel }_{r(z)}{\parallel v\parallel }_{r^{\prime }(z)}\forall u\in L^{r(z)}(\mathsf{\Omega }),v\in L^{r^{\prime }(z)}(\mathsf{\Omega }).$$

Moreover, if $r,s\in E_1$ and $r(z)⩽s(z)$ for all $z\in \mathsf{\Omega }$, then the embedding $L^{s(z)}(\mathsf{\Omega })\subseteq L^{r(z)}(\mathsf{\Omega })$ is continuous.

Using the variable Lebesgue spaces, we can define the corresponding variable Sobolev spaces. So, given $r\in E_1$, the variable Sobolev space $W^{1,r(z)}(\mathsf{\Omega })$ is defined by

$$W^{1,r(z)}(\mathsf{\Omega })=\left\{u\in L^{r(z)}(\mathsf{\Omega }):\left|Du\right|\in L^{r(z)}(\mathsf{\Omega })\right\}$$

with $Du$ being the weak gradient of u. The norm ${\parallel ··\parallel }_{1,r(z)}$ of this space is defined by

$${\parallel u\parallel }_{1,r(z)}={\parallel u\parallel }_{r(z)}+{\parallel Du\parallel }_{r(z)}\forall u\in W^{1,r(z)}(\mathsf{\Omega }),$$

with ${\parallel Du\parallel }_{r(z)}={\parallel \left|Du\right|\parallel }_{r(z)}$.

In addition, we define the Dirichlet variable Sobolev space $W_0^{1,r(z)}(\mathsf{\Omega })$ by

$$W_0^{1,r(z)}(\mathsf{\Omega })={\overline{C_c^{\infty }(\overline{\mathsf{\Omega }})}}^{{\parallel ·\parallel }_{1,r(z)}}.$$

Then, for this space, the Poincaré inequality holds; namely, we can find $\widehat{c}>0$ such that

$${\parallel u\parallel }_{r(z)}⩽\widehat{c}{\parallel Du\parallel }_{r(z)}\forall u\in W_0^{1,r(z)}(\mathsf{\Omega }).$$

Therefore, on $W_0^{1,r(z)}(\mathsf{\Omega })$, we can consider the equivalent norm $\parallel ·\parallel$ defined by

$$\parallel u\parallel ={\parallel Du\parallel }_{r(z)}\forall u\in W_0^{1,r(z)}(\mathsf{\Omega }).$$

The variable Sobolev spaces $W^{1,r(z)}(\mathsf{\Omega })$ and $W_0^{1,r(z)}(\mathsf{\Omega })$ are both Banach spaces which are separable and reflexive (in fact uniformly convex).

The modular function ${\varrho }_r$ and the norm $\parallel ·\parallel$ are closely related.

Proposition 1.

Suppose that $r\in E_1$. Then, the following statements hold

(a) 

$\parallel u\parallel =\mu$⟺${\varrho }_r({\displaystyle \frac{Du}{\mu }})=1$.

(b) 

$\parallel u\parallel <1$ (resp. $=1$, $>1$) ⟺ ${\varrho }_r(Du)<1$ (resp. $=1$, $>1$).

(c) 

$\parallel u\parallel <1$⟹${\parallel u\parallel }^{r_{+}}⩽{\varrho }_r(Du)⩽{\parallel u\parallel }^{r_{-}}$.

(d) 

$\parallel u\parallel >1$⟹${\parallel u\parallel }^{r_{-}}⩽{\varrho }_r(Du)⩽{\parallel u\parallel }^{r_{+}}$.

(e) 

$\parallel u\parallel \to 0$ (resp. $\to +\infty$) ⟺ ${\varrho }_r(Du)\to 0$ (resp. $\to +\infty$).

Given $r\in E_1$, we define

$$r^{\ast }(z)=\left\{\begin{array}{cc}{\displaystyle \frac{Nr(z)}{N-r(z)}}\hfill & \mathrm{if}r(z)<N,\hfill \\ +\infty \hfill & \mathrm{if}N⩽r(z)\hfill \end{array}\right.\forall z\in \overline{\mathsf{\Omega }}$$

(the variable critical Sobolev exponent). The variable Lebesgue and Sobolev spaces satisfy certain useful embeddings.

Proposition 2.

Suppose that $r\in E_1$ and $s\in C(\overline{\mathsf{\Omega }})$ with $1⩽s_{-}$. Then,

(a) 

$s(z)⩽r^{\ast }(z)$ for all $z\in \overline{\mathsf{\Omega }}$⟹$W_0^{1,r(z)}(\mathsf{\Omega })\subseteq L^{s(z)}(\mathsf{\Omega })$ continuously.

(b) 

$s(z)<r^{\ast }(z)$ for all $z\in \overline{\mathsf{\Omega }}$⟹$W_0^{1,r(z)}(\mathsf{\Omega })\subseteq L^{s(z)}(\mathsf{\Omega })$ compactly.

Also, we have

$$W_0^{1,r(z)}{(\mathsf{\Omega })}^{\ast }=W^{-1,r^{\prime }(z)}(\mathsf{\Omega }).$$

For $r\in E_1$, consider the operator

$$A_r:W_0^{1,r(z)}(\mathsf{\Omega })⟶W^{-1,r^{\prime }(z)}(\mathsf{\Omega })=W_0^{1,r(z)}{(\mathsf{\Omega })}^{\ast }$$

defined by

$$⟨A_r(u),h⟩={\int }_{\mathsf{\Omega }}{\left|Du\right|}^{r(z)-2}(Du,Dh)\forall u,h\in W_0^{1,r(z)}(\mathsf{\Omega }).$$

This operator has the following properties (see Fan-Zhang [7]).

Proposition 3.

For $r\in E_1$, the operator $A_r:W_0^{1,r(z)}(\mathsf{\Omega })⟶W^{-1,r^{\prime }(z)}(\mathsf{\Omega })$ defined above has the following conditions:

• Bounded (that is, maps bounded sets to bounded sets);
• Continuous and strictly monotone (thus, it is maximal monotone, too);
• Of type ${(S)}_{+}$, that is, if $u_n\stackrel{w}{⟶}u$ in $W_0^{1,r(z)}(\mathsf{\Omega })$ and $\underset{n\to +\infty }{lim\; sup}⟨A_r(u_n),u_n-u⟩⩽0$, then $u_n⟶u$ in $W_0^{1,r(z)}(\mathsf{\Omega })$.

Let X be a Banach space and $\phi \in C^1(X)$. We define

$$K_{\phi }=\{u\in X:{\phi }^{\prime }(u)=0\}$$

(the critical set of $\phi$). We say that $\phi$ satisfies the C condition if it has the following property:

“Every sequence ${\{u_n\}}_{n\in \mathbb{N}}\subseteq X$ such that ${\{\phi (u_n)\}}_{n\in \mathbb{N}}\subseteq \mathbb{R}$ is bounded and $(1+\parallel u_n{\parallel }_X){\phi }^{\prime }(u_n)\to 0$ in $X^{\ast }$ as $n\to \infty$ has a strongly convergent subsequence.”

This is a compactness-type condition on $\phi$, which compensates for the fact that the ambient space is not locally compact (being in general infinite dimensional).

Let $C^{0,1}(\overline{\mathsf{\Omega }})$ denote the space of all Lipschitz continuous functions defined on $\overline{\mathsf{\Omega }}$. Our hypotheses on the item of problem $(P_{\lambda })$ are the following:

$H_0$: 

$p,q\in C^{0,1}(\overline{\mathsf{\Omega }})\cap E_1$, $\tau \in E_1$ and $1<{\tau }_{-}<q_{-}<{\tau }_{+}<p_{+}$, $p_{-}<N$.

$H_1$: 

$f:\mathsf{\Omega }\times \mathbb{R}⟶\mathbb{R}$ is a Carathéodory function such that

(i) 

$\left|f(z,x)\right|⩽a(z)(1+|x{|}^{r-1})$ for a.a. $z\in \overline{\mathsf{\Omega }}$, all $x\in \mathbb{R}$ with $a\in L^{\infty }(\mathsf{\Omega })$ and $p_{+}<r<p_{-}^{\ast }={\displaystyle \frac{N{p}_{-}}{N-p_{-}}}$;

(ii) 

If $F(z,x)={\int }_0^{x}f(z,s)sz$, then $\underset{x\to \pm \infty }{lim}{\displaystyle \frac{F(z,x)}{{\left|x\right|}^{p_{+}}}}=+\infty$ uniformly for a.a. $z\in \mathsf{\Omega }$;

(iii) 

There exists $\vartheta \in C(\overline{\mathsf{\Omega }})$ such that ${\tau }_{+}<{\vartheta }_{-}$ and $\vartheta (z)\in ((r-p_{-}){\displaystyle \frac{N}{p_{-}^{\ast }}},p^{\ast }(z))$ for all $z\in \overline{\mathsf{\Omega }}$ and

$$0<{\beta }_0⩽\underset{x\to \pm \infty }{lim\; inf}{\displaystyle \frac{f(z,x)x-p_{+}F(z,x)}{{\left|x\right|}^{\vartheta (z)}}}$$

uniformly for a.a. $z\in \mathsf{\Omega }$;

(iv) 

There exists ${\delta }_0>0$ such that $0⩽F(z,x)$ for a.a. $z\in \mathsf{\Omega }$, all $\left|x\right|⩽{\delta }_0$ and $\underset{x\to 0}{lim}{\displaystyle \frac{F(z,x)}{{\left|x\right|}^{q_{+}}}}=0$ uniformly for a.a. $z\in \mathsf{\Omega }$.

Remark 1.

These hypotheses imply that $f(z,0)=0$ for a.a. $z\in \mathsf{\Omega }$. Also, hypotheses $H_1(ii)$ and $(iii)$ imply that

$$\underset{x\to \pm \infty }{lim}{\displaystyle \frac{f(z,x)}{{\left|x\right|}^{p_{+}-2}x}}=+\infty$$

uniformly for a.a. $z\in \mathsf{\Omega }$. Therefore, the perturbation $f(z,·)$ is $(p_{+}-1)$-superlinear.

We introduce the energy functional ${\phi }_{\lambda }:W_0^{1,p(z)}(\mathsf{\Omega })⟶\mathbb{R}$ for problem $(P_{\lambda })$ defined by

$${\phi }_{\lambda }(u)={\int }_{\mathsf{\Omega }}{\displaystyle \frac{1}{p(z)}}{\left|Du\right|}^{p(z)}dz+{\int }_{\mathsf{\Omega }}{\displaystyle \frac{1}{q(z)}}{\left|Du\right|}^{q(z)}dz-{\int }_{\mathsf{\Omega }}{\displaystyle \frac{\lambda }{\tau (z)}}{\left|u\right|}^{\tau (z)}dz-{\int }_{\mathsf{\Omega }}F(z,u)dz$$

for all $u\in W^{1,p(z)}(\mathsf{\Omega })$.

We have that ${\phi }_{\lambda }\in C^1(W_0^{1,p(z)}(\mathsf{\Omega }))$ and

$$⟨{\phi }_{\lambda }^{\prime }(u),h⟩=⟨V(v),h⟩-{\int }_{\mathsf{\Omega }}\lambda {\left|u\right|}^{\tau (z)-2}uhdz-{\int }_{\mathsf{\Omega }}f(z,u)hdz$$

for all $u,h\in W_0^{1,p(z)}(\mathsf{\Omega })$, with $V=A_p+A_q$. On account of Proposition 3, the operator V is bounded, continuous, strictly monotone (thus maximal monotone too), and of type ${(S)}_{+}$.

3. Main Result and Discussion

In this section, we show that for all $\lambda >0$ small, problem $(P_{\lambda })$ has at least two nontrivial smooth solutions. Our approach uses variational tools from the critical point theory.

Proposition 4.

If hypotheses $H_0$ and $H_1$ hold and $\lambda >0$, then ${\phi }_{\lambda }$ satisfies the C condition.

Proof. 

We consider a sequence ${\{u_n\}}_{n\in \mathbb{N}}\subseteq W_0^{1,p(z)}(\mathsf{\Omega })$ such that

$$|{\phi }_{\lambda }(u_n)|⩽c_1\forall n\in \mathbb{N},$$

(1)

for some $c_1>0$ and

$$(1+\parallel u_n\parallel ){\phi }_{\lambda }^{\prime }(u_n)\to 0\mathrm{in}{W}^{-1,p^{\prime }(z)}(\mathsf{\Omega })\mathrm{as}n\to \infty .$$

(2)

From (2), we have

$$|⟨V(u_n),h⟩-{\int }_{\mathsf{\Omega }}\lambda |u_n{|}^{\tau (z)-2}{u}_{n}hdz-{\int }_{\mathsf{\Omega }}f(z,u_n)hdz|⩽{\displaystyle \frac{{\epsilon }_n\parallel h\parallel }{1+\parallel u_n\parallel }})$$

(3)

for all $h\in W_0^{1,p(z)}(\mathsf{\Omega })$, $n\in \mathbb{N}$, with ${\epsilon }_n\to 0^{+}$.

In (3), we choose $h=u_n\in W_0^{1,p(z)}(\mathsf{\Omega })$ and obtain

$$-{\varrho }_p(D{u}_n)-{\varrho }_q(D{u}_n)+\lambda {\varrho }_{\tau }(u_n)+{\int }_{\mathsf{\Omega }}f(z,u_n)u_{n}dz⩽{\epsilon }_n\forall n\in \mathbb{N}.$$

(4)

Also, from (1), we have

$${\displaystyle \frac{1}{p_{+}}}\left({\varrho }_p(D{u}_n)+{\varrho }_q(D{u}_n)\right)-{\displaystyle \frac{1}{{\tau }_{-}}}{\varrho }_{\tau }(D{u}_n)-{\int }_{\mathsf{\Omega }}F(z,u_n)dz⩽c_1,$$

so

$${\varrho }_p(D{u}_n)+{\varrho }_q(D{u}_n)-{\displaystyle \frac{p_{+}}{{\tau }_{-}}}{\varrho }_{\tau }(D{u}_n)-{\int }_{\mathsf{\Omega }}{p}_{+}F(z,u_n)dz⩽p_{+}{c}_1\forall n\in \mathbb{N}.$$

(5)

We add (4) and (5) and obtain

$${\int }_{\mathsf{\Omega }}\left(f(z,u_n)u_n-p_{+}F(z,u_n)\right)dz⩽\left({\displaystyle \frac{p_{+}}{{\tau }_{-}}}-1\right){\varrho }_{\tau }(u_n)+c_2\forall n\in \mathbb{N},$$

(6)

for some $c_2>0$.

Hypotheses $H_1(i)$ and $(iii)$ imply that we can find ${\beta }_1\in (0,{\beta }_0)$ and $c_3>0$ such that

$${\beta }_1{\left|x\right|}^{\vartheta (z)}-c_3⩽f(z,x)x-p_{+}F(z,x)\mathrm{for}\mathrm{a}.\mathrm{a}.z\in \mathsf{\Omega },\mathrm{all}x\in \mathbb{R}.$$

(7)

We use (7) in (6) and obtain

$${\beta }_1{\varrho }_{\vartheta }(u_n)⩽\left({\displaystyle \frac{p_{+}}{{\tau }_{-}}}-1\right){\varrho }_{\tau }(u_n)+c_4\forall n\in \mathbb{N},$$

(8)

for some $c_4>0$.

Recall that ${\tau }_{+}<{\vartheta }_{-}$ (see hypothesis $H_1(iii)$). We claim that the sequence ${\{u_n\}}_{n\in \mathbb{N}}\subseteq L^{\vartheta (z)}(\mathsf{\Omega })$ is bounded. Assume without any loss of generality that $\parallel u_n{\parallel }_{\vartheta (z)}⩾1$, $\parallel u_n{\parallel }_{\tau (z)}⩾1$ for all $n\in \mathbb{N}$. Then, from (8) and Proposition 1, we have

$${\beta }_1\parallel u_n{\parallel }_{\vartheta (z)}^{{\vartheta }_{-}}⩽\left({\displaystyle \frac{p_{+}}{{\tau }_{-}}}-1\right){\parallel u_n\parallel }_{\tau (z)}^{{\tau }_{+}}+c_4\forall n\in \mathbb{N},$$

so

$${\beta }_1\parallel u_n{\parallel }_{\vartheta (z)}^{{\vartheta }_{-}}⩽c_5(1+\parallel u_n{\parallel }_{\vartheta (z)}^{{\tau }_{+}})\forall n\in \mathbb{N},$$

for some $c_5>0$ (recall that the embedding $L^{\vartheta (z)}(\mathsf{\Omega })\subseteq L^{\tau (z)}(\mathsf{\Omega })$ is continuous), so

$${\{{\mathrm{u}}_{\mathrm{n}}\}}_{\mathrm{n}\in \mathbb{N}}\subseteq {\mathrm{L}}^{\vartheta }(\mathsf{\Omega })\mathrm{is}\mathrm{bounded}$$

(9)

(since ${\tau }_{+}<{\vartheta }_{-}$).

From hypothesis $H_1(iii)$, we see that we may assume that $\vartheta (z)<r<p_{-}^{+}$ for all $z\in \overline{\mathsf{\Omega }}$. Therefore, we can find $t\in (0,1)$ such that

$${\displaystyle \frac{1}{r}}={\displaystyle \frac{1-t}{{\vartheta }_{-}}}+{\displaystyle \frac{t}{p_{-}^{\ast }}}.$$

(10)

Using the interpolation inequality (see Papageorgiou-Winkert [8], p. 116), we have

$$\parallel u_n{\parallel }_r⩽\parallel u_n{\parallel }_{{\vartheta }_{-}}^{1-t}{\parallel u_n\parallel }_{p_{-}^{\ast }}^t,$$

so

$$\parallel u_n{\parallel }_r^r⩽c_6{\parallel u_n\parallel }_{p_{-}^{\ast }}^{tr}\forall n\in \mathbb{N},$$

(11)

for some $c_6>0$ (see (9)).

We know that $u_n\in W_0^{1,p(z)}(\mathsf{\Omega })\subseteq W_0^{1,p_{-}}(\mathsf{\Omega })$ for all $n\in \mathbb{N}$. Then, from (11) and the Sobolev embedding theorem, we have

$$\parallel u_n{\parallel }_r^r⩽c_7{\parallel u_n\parallel }^{tr}\forall n\in \mathbb{N},$$

(12)

for some $c_7>0$.

In (3), we choose $h=n_n\in W_0^{1,p(z)}(\mathsf{\Omega })$ and obtain

$$\begin{array}{ccc}\hfill {\varrho }_p(D{u}_n)+{\varrho }_q(D{u}_n)& ⩽& {\epsilon }_n+\lambda {\varrho }_{\tau }(u_n)+{\int }_{\mathsf{\Omega }}f(z,u_n)u_{n}dz\hfill \\ & ⩽& c_8\left(1+\lambda +\parallel u_n{\parallel }_r^r\right)\hfill \\ & ⩽& c_9\left(1+\lambda +\parallel u_n{\parallel }^{tr}\right)\forall n\in \mathbb{N},\hfill \end{array}$$

for some $c_8,c_9>0$ (see hypothesis $H_1(i)$; see (9), (12) and recall that ${\tau }_{+}<{\vartheta }_{-}$).

Assume that $\parallel u_n\parallel ⩾1$ for all $n\in \mathbb{N}$. Then, we have

$$\parallel u_n{\parallel }^{p_{-}}⩽c_9(1+\lambda +\parallel u_n{\parallel }^{tr}).$$

(13)

From (10), we have

$$tr={\displaystyle \frac{(r-{\vartheta }_{-})N{p}_{-}}{N{p}_{-}-N{\vartheta }_{-}+{\vartheta }_{-}{p}_{-}}}<p_{-}$$

(see hypothesis $H_1(iii)$). Therefore, from (13), it follows that the sequence ${\{u_n\}}_{n\in \mathbb{N}}\subseteq W_0^{1,p(z)}(\mathsf{\Omega })$ is bounded.

So, we may assume that

$$u_n\stackrel{w}{⟶}u\mathrm{in}{W}_0^{1,p(z)}(\mathsf{\Omega }),u_n⟶u\mathrm{in}{L}^r\mathsf{\Omega }$$

(14)

(see Proposition 2).

In (3), we choose the test function $h=u_n-u\in W_0^{1,p(z)}(\mathsf{\Omega })$, pass to the limit as $n\to +\infty$ and use (14). Then,

$$\underset{n\to +\infty }{lim}⟨V(u_n),u_n-u⟩=0,$$

so

$$u_n⟶u\mathrm{in}{W}_0^{1,p(z)}(\mathsf{\Omega })$$

(by the ${(S)}_{+}$-property of V).

This proves that ${\phi }_{\lambda }$ satisfies the C condition. □

Proposition 5.

If hypotheses $H_0$ and $H_1$ hold, then there exists ${\lambda }^{\ast }>0$ such that for all $\lambda \in (0,{\lambda }^{\ast })$, we can find ${\varrho }_{\lambda }>0$ such that

$${\phi }_{\lambda }(u)⩾m_{\lambda }>0\forall u\in W_0^{1,p(z)}(\mathsf{\Omega }),\parallel u\parallel ={\varrho }_{\lambda }.$$

Proof. 

On account of hypotheses $H_1(i)$ and $(iv)$, given $\epsilon >0$, we can find $c_{10}=c_{10}(\epsilon )>0$ such that

$$F(z,x)⩽{\displaystyle \frac{\epsilon }{q_{+}}}{\left|x\right|}^{q_{+}}+c_{10}{\left|x\right|}^r\mathrm{for}\mathrm{a}.\mathrm{a}.z\in \mathsf{\Omega },\mathrm{all}x\in \mathbb{R}.$$

(15)

Then, for $u\in W_0^{1,p(z)}(\mathsf{\Omega })$, we have

$${\phi }_{\lambda }|(u)⩾{\displaystyle \frac{1}{p_{+}}}{\varrho }_p(Du)+{\displaystyle \frac{1}{q_{+}}}\left({\varrho }_q(Du)-\epsilon {\parallel u\parallel }_{q_{+}}^{q_{+}}\right)-{\displaystyle \frac{\lambda }{{\tau }_{-}}}{\varrho }_{\tau }(u)-c_{10}{\parallel u\parallel }_r^r$$

(see (15)).

Suppose that ${\parallel u\parallel ,\parallel u\parallel }_{1,q(z)},{\parallel u\parallel }_{\tau (z)}<1$. Then,

$${\phi }_{\lambda }(u)⩾{\displaystyle \frac{1}{u_{+}}}{\parallel u\parallel }^{p_{+}}+{\displaystyle \frac{1}{q_{+}}}\left({\parallel Du\parallel }_{q(z)}^{q_{+}}-\epsilon {\parallel u\parallel }_{q_{+}}^{q_{+}}\right)-c_{11}\left({\lambda \parallel u\parallel }^{{\tau }_{-}}+{\parallel u\parallel }^r\right),$$

for some $c_{11}>0$ (see Propositions 1 and 2).

From Proposition 2, we know that the embedding $W_0^{1,q(z)}(\mathsf{\Omega })\subseteq L^{q_{+}}(\mathsf{\Omega })$ is compact. Hence,

$${\parallel Du\parallel }_{q(z)}^{q_{+}}-{\epsilon \parallel u\parallel }_{q_{+}}^{q_{+}}⩾(c_{12}-\epsilon ){\parallel u\parallel }_{q_{+}}^{q_{+}},$$

for some $c_{12}>0$.

Then, choosing $\epsilon \in (0,c_{12})$, we obtain

$$\begin{array}{ccc}\hfill {\phi }_{\lambda }(u)& ⩾& {\displaystyle \frac{1}{p_{+}}}{\parallel u\parallel }^{p_{+}}-c_{11}\left({\lambda \parallel u\parallel }^{{\tau }_{-}}+{\parallel u\parallel }^r\right)\hfill \\ & =& \left({\displaystyle \frac{1}{p_{+}}}-c_{11}\left({\lambda \parallel u\parallel }^{{\tau }_{-}-p_{+}}+{\parallel u\parallel }^{r-p_{+}}\right)\right){\parallel u\parallel }^{p_{+}}.\hfill \end{array}$$

(16)

Consider the function

$${\xi }_{\lambda }(t)=\lambda t^{{\tau }_{-}-p_{+}}+t^{r-p_{+}}\forall t>0.$$

Evidently, ${\xi }_{\lambda }\in C^1(0,\infty )$, ${\xi }_{\lambda }⩾0$. Since ${\tau }_{-}<p_{+}<r$, we see that

$${\xi }_{\lambda }(t)⟶+\infty \mathrm{as}t\to 0^{+}\mathrm{and}\mathrm{as}t\to +\infty .$$

So, we can find $t_0>0$ such that

$${\xi }_{\lambda }(t_0)=\underset{t>0}{min}{\xi }_{\lambda }(t),$$

so

$${\xi }_{\lambda }^{\prime }(t_0)=0,$$

thus

$$\lambda (p_{+}-{\tau }_{-})t_0^{{\tau }_{-}-p_{+}-1}=(r-p_{+})t_0^{r-p_{+}-1}$$

and hence

$$t_0={\left({\displaystyle \frac{\lambda (p_{+}-{\tau }_{-})}{r-p_{+}}}\right)}^{{\displaystyle \frac{1}{r-{\tau }_{-}}}}.$$

Then, we have

$${\xi }_{\lambda }(t_0)=\lambda {\left({\displaystyle \frac{r-p_{+}}{\lambda (p_{+}-{\tau }_{-})}}\right)}^{{\displaystyle \frac{p_{+}-{\tau }_{-}}{r-{\tau }_{-}}}}+{\left({\displaystyle \frac{\lambda (p_{+}-{\tau }_{-})}{r-p_{+}}}\right)}^{{\displaystyle \frac{r-p_{+}}{r-{\tau }_{-}}}}.$$

Note that ${\displaystyle \frac{p_{+}-{\tau }_{-}}{r-{\tau }_{-}}}<1$. Therefore,

$${\xi }_{\lambda }(t_0)\to 0\mathrm{as}\lambda \to 0^{+}.$$

It follows that we can find ${\lambda }^{\ast }>0$ such that

$${\xi }_{\lambda }(t_0)<{\displaystyle \frac{1}{c_{11}{p}_{+}}}$$

(see (16)) and

$$t_0=t_0(\lambda )<1\forall \lambda \in (0,{\lambda }^{\ast })$$

(recall $\parallel u\parallel <1$). From (16), we infer that

$${\phi }_{\lambda }(u)⩾m_{\lambda }>0\forall u\in W_0^{1,p(z)}(\mathsf{\Omega }),\parallel u\parallel ={\varrho }_{\lambda }=t_0(\lambda )\forall \lambda \in (0,{\lambda }^{\ast }).$$

□

Proposition 6.

If hypotheses $H_0$ and $H_1$ hold and $\lambda >0$, then we can find $\widehat{u}\in C_0^1(\overline{\mathsf{\Omega }})\setminus \{0\}$, $\widehat{u}⩾0$ and $t\in (0,1)$ is small such that ${\phi }_{\lambda }(t\widehat{u})<0$.

Proof. 

By hypotheses $H_0$, we have $\epsilon =q_{-}-{\tau }_{-}>0$ and $\tau \in C(\overline{\mathsf{\Omega }})$. So, we can find $U\subseteq \mathsf{\Omega }$ open such that ${\left|U\right|}_N⩽1$ (${|·|}_N$ is the Lebesgue measure on ${\mathbb{R}}^N$) and $|\tau (z)-{\tau }^{\ast }| <\epsilon$ for all $z\in U$. Hence,

$$\tau (z)<\epsilon +{\tau }_{-}<q_{-}\forall z\in U.$$

(17)

We consider a bump function $\widehat{u}\in C_0^1(\overline{\mathsf{\Omega }})$ such that

$$\widehat{u}{|}_U=1\mathrm{and}0⩽\widehat{u}(z)⩽1\forall z\in \overline{\mathsf{\Omega }}.$$

(18)

Choose $t\in (0,1)$ to be small such that

$$0⩽t\widehat{u}(z)⩽{\delta }_0\forall z\in \overline{\mathsf{\Omega }},$$

(19)

with ${\delta }_0>0$ as postulated by hypothesis $H_1(iv)$. Then, we have

$${\phi }_{\lambda }(t\widehat{u})⩽{\displaystyle \frac{t^{q_{-}}}{q_{-}}}\left({\varrho }_p(D\widehat{u})+\varrho (D\widehat{u})\right)-{\displaystyle \frac{\lambda }{{\tau }_{+}}}{\int }_{\mathsf{\Omega }}{\left|t\widehat{u}\right|}^{\tau (z)}dz⩽c_{13}{t}^{q_{-}}-\lambda c_{14}{t}^{{\tau }_{-}+\epsilon }$$

for some $c_{13},c_{14}>0$ (see (17)–(19) and hypothesis $H_1(iv)$).

But ${\tau }_{-}+\epsilon <q_{-}$ (see (17)). So, choosing $t\in (0,1)$ even smaller, we have

$${\phi }_{\lambda }(t\widehat{u})<0.$$

□

Theorem 1.

If hypotheses $H_0$ and $H_1$ hold, then for all $\lambda >0$, the small problem $(P_{\lambda })$ has at least two nontrivial smooth solutions $u_0,u^{\ast }\in C_0^1(\overline{\mathsf{\Omega }})$.

Proof. 

Let $\lambda \in (0,{\lambda }^{\ast })$ and let ${\varrho }_{\lambda }$ be from Proposition 5. We consider the following minimization problem

$$inf\left\{{\phi }_{\lambda }(u):u\in {\overline{B}}_{{\varrho }_{\lambda }}\right\}={\widehat{m}}_{\lambda },$$

(20)

with ${\overline{B}}_{{\varrho }_{\lambda }}=\{u\in W_0^{1,p(z)}(\mathsf{\Omega }):\parallel u\parallel ⩽{\varrho }_{\lambda }\}$. The reflexivity of $W_0^{1,p(z)}(\mathsf{\Omega })$ and the Eberlein–Smulian theorem imply that ${\overline{B}}_{{\varrho }_{\lambda }}$ is sequentially weakly compact. Also, using Proposition 2, we see that ${\phi }_{\lambda }$ is sequentially weakly lower semicontinuous. So, by the Weierstrass–Tonelli theorem, we can find $u_0\in W_0^{1,p(z)}(\mathsf{\Omega })$ such that

$${\phi }_{\lambda }(u_0)={\widehat{m}}_{\lambda }$$

(21)

(see (20)). From the proof of Proposition 6, it is clear that we can have

$$\parallel t\widehat{u}\parallel <{\varrho }_{\lambda },$$

so

$${\phi }_{\lambda }(u_0)<0={\phi }_{\lambda }(0)$$

(see Proposition 6 and (21)). On the other hand, from Proposition 5, we know that

$${\phi }_{\lambda }{|}_{\partial B_{{\varrho }_{\lambda }}}>0.$$

Therefore, we infer that

$$0<\parallel u_0\parallel <{\varrho }_{\lambda },$$

so

$$⟨{\phi }_{\lambda }^{\prime }(u_0),h⟩=0\forall h\in W_0^{1,p(z)}(\mathsf{\Omega })$$

(see (21)).

Since $u_0\in K_{{\phi }_{\lambda }}$, we have that $u_0$ is a nontrivial solution of $(P_{\lambda })$ and the anisotropic regularity theory of Fan [9] implies $u_0\in C_0^1(\overline{\mathsf{\Omega }})$.

If $y\in C_0^1(\overline{\mathsf{\Omega }})$ with $y(z)>0$ for all $z\in \mathsf{\Omega }$, then on account of hypothesis $H_1(iii)$, we have

$${\phi }_{\lambda }(ty)⟶-\infty \mathrm{as}t\to +\infty .$$

(22)

Then, (22) together with Propositions 4 and 5 permit the use of the mountain pass theorem. So, we can find $u^{\ast }\in W_0^{1,p(z)}(\mathsf{\Omega })$ such that

$$u^{\ast }\in K_{{\phi }_{\lambda }}\mathrm{and}{\phi }_{\lambda }(u_0)<0<\widehat{m}⩽{\phi }_{\lambda }(u^{\ast }).$$

So, $u^{\ast }$ is a second nontrivial solution of $(P_{\lambda })$ ($0<\lambda <{\lambda }^{\ast }$) distinct from $u_0$. As before, the anisotropic regularity theory of Fan [9] implies that $u^{\ast }\in C_0^1(\overline{\mathsf{\Omega }})$. □

4. Conclusions

We have considered an anisotropic extension of the classical “concave–convex problem” and assumed only partial concavity of the parametric term. We proved that for all small values of the parameter, the problem has at least two nontrivial smooth solutions. It is an interesting open question whether we can have an existence and multiplicity result which will be global in the parameter $\lambda >0$ (see for example Papageorgiou et al. [4]).