Three Bounded Solutions for a Degenerate Nonlinear Dirichlet Problem

Abstract

The existence of at least three bounded weak solutions is established for a nonlinear elliptic equation with unbounded coefficients. The approach is based on variational methods and critical point theorems.

1. Introduction

Let $\Omega \subset {\mathbb{R}}^N$ be a bounded domain with Lipschtz boundary $\partial \Omega$, $N\ge 2$, $1<q<p<\infty$ and $\lambda$ be a positive parameter, with which we want to study the following Dirichlet problem:

$$\left\{\begin{array}{cc}-\mathrm{div}\left(g\left(x\right)\right|\nabla u{|}^{p-2}\nabla u+h\left(x\right)|\nabla u{|}^{q-2}\nabla u)=\lambda f(x,u)\hfill & \mathrm{in}\Omega ,\hfill \\ \\ u=0\hfill & \mathrm{on}\partial \Omega ,\hfill \end{array}\right.$$

(1)

where $f:\Omega \times \mathbb{R}\to \mathbb{R}$ is a suitable Carathéodory function that satisfies subcritical growth condition and a certain behavior at $+\infty$ and at 0. This paper was inspired by a study by [1], who were the first to introduce the differential operator ${\Delta }_{pq}^{gh}\left(u\right)={\Delta }_{g,p}\left(u\right)+{\Delta }_{h,q}\left(u\right)={\mathrm{div}\left(g\left(x\right)\right|\nabla u|}^{p-2}\nabla u+{h\left(x\right)|\nabla u|}^{q-2}\nabla u)$, called the degenerate $(p,q)$-Laplacian with non-negative weight functions $g,h\in L^1(\Omega )$. The interest in Equation (1) started long ago when the question of stability in some non-smooth minimization problems was raised during the imaging of electrical conductivity (see, for example, refs. [2,3,4] and therein references). Here, we extend the results of [5] in which the authors studied Problem (1) with $h=0$, known as the weighted p-Laplacian case, and we obtain the existence and location of at least three weak solutions.

We refer the reader to [6,7,8] for an overview on nonlinear degenerate elliptic equations. I would like to mention related works dealing with existence results for degenerate p-Laplacian or $(p,q)$-Laplacian and convection terms via different methods, like the pseudomonotone operator theory or sub-super solution methods that are referred to in [9,10,11,12,13,14] or like the case of symmetric solutions among them in [15,16,17,18].

In this paper, we focus on the existence, boundedness and multiplicity of solutions for problems driven by a degenerate $(p,q)$-Laplacian operator with the weight functions g and h. We point out that, if $g=h=1$, we come back to a nonlinear elliptic problem driven by a classical $(p,q)$-Laplacian. On the other hand, if $g=1$, $h\in L^{\infty }(\Omega )$, we obtain the double phase operator.

For the weight function $g:\Omega \to [g_0,+\infty [$ with $g_0>0$, we assume that a further condition holds:

There exists a positive constant $s>max\left\{{\displaystyle \frac{N}{p}},{\displaystyle \frac{1}{p-1}}\right\}$ such that

$$g^{-s}\in L^1(\Omega ).$$

(2)

The main novelty of this work is the presence of an unbounded coefficient in the equation that is used to overcome this difficulty. The framework is adapted to an appropriate weighted Sobolev space, taking the constraints into account in an essential way.

This paper is organized as follows: In Section 2, we introduce weighted Sobolev spaces and recall some facts about them. In Section 3, we prove our main result (Theorem 2) and two special cases (Theorems 3 and 4).

2. Basic Results on Weighted Sobolev Spaces

We start by introducing the weighted Sobolev spaces that we use throughout this paper, and we present some of their properties. For a comprehensive treatment of weighted Sobolev spaces, we refer to [19,20].

The weighted Banach space with a weight function $g\in L_{loc}^1(\Omega )$ satisfying Condition (2) is defined by

$$L^p(g,\Omega )=\{u:\Omega \to \mathbb{R},\mathrm{measurable}:{\int }_{\Omega }{g\left(x\right)\left|u\right|}^{p}dx<\infty \},$$

endowed with the following norm:

$${\parallel u\parallel }_{L^p(g,\Omega )}={\left({\int }_{\Omega }g\left(x\right){\left|u\right|}^{p}dx\right)}^{{\displaystyle \frac{1}{p}}}.$$

The weighted Sobolev space with a weight function $g\in L_{loc}^1(\Omega )$ satisfying Condition (2) is defined by

$$W^{1,p}(g,\Omega )=\{u\in L^p(\Omega ):|\nabla u|\in L^p(g,\Omega )\},$$

endowed with the following norm:

$${\parallel u\parallel }_{W^{1,p}(g,\Omega )}={\left({\int }_{\Omega }{\left|u\right|}^p+g\left(x\right){|\nabla u|}^{p}dx\right)}^{{\displaystyle \frac{1}{p}}}.$$

$W^{1,p}(g,\Omega )$ is a uniformly convex (hence reflexive) Banach space and $C_0^{\infty }(\Omega )$ is an its subset (see [14]). $W_0^{1,p}(g,\Omega )$ is the closure of $C_0^{\infty }(\Omega )$ with respect to the norm ${\parallel ·\parallel }_{W^{1,p}(g,\Omega )}$.

• On $W_0^{1,p}(g,\Omega )$, we consider the equivalent norm

  $$\parallel u\parallel ={\left({\int }_{\Omega }g\left(x\right){|\nabla u|}^{p}dx\right)}^{{\displaystyle \frac{1}{p}}}.$$

If we consider s as given by (2) and the exponent $p_s={\displaystyle \frac{ps}{s+1}}$, we obtain

$${\parallel u\parallel }_{W_0^{1,p_s}(\Omega )}\le \parallel g^{-s}{\parallel }_{L^1(\Omega )}^{{\displaystyle \frac{1}{ps}}}\parallel u\parallel \mathrm{for}\mathrm{all}u\in W_0^{1,p}(g,\Omega ).$$

(3)

Then, the following embedding is continuous:

$$W_0^{1,p}(g,\Omega )\hookrightarrow W_0^{1,p_s}(\Omega ).$$

The critical exponent of $p_s$ is given by

$$p_s^{*}=\left\{\begin{array}{cc}{\displaystyle \frac{p_{s}N}{N-p_s}}\hfill & \mathrm{if}ps<N(s+1)\hfill \\ +\infty \hfill & \mathrm{if}ps\ge N(s+1)\hfill \end{array}\right..$$

Through this paper, we assume that $ps<N(s+1)$. Using the Sobolev embedding theorem and the Rellich–Kondrachov compact embedding theorem, we have that $W_0^{1,p}(g,\Omega )$ is a continuous embedding in $L^{\tau }(\Omega )$ if $1\le \tau \le p_s^{*}$, and the embedding is compact if $1\le \tau <p_s^{*}$. Taking into account (3),

$${\parallel u\parallel }_{L^{p_s^{*}}(\Omega )}\le c_{p_s^{*}}\parallel g^{-s}{\parallel }_{L^1(\Omega )}^{{\displaystyle \frac{1}{ps}}}\parallel u\parallel \mathrm{for}\mathrm{all}u\in W_0^{1,p}(g,\Omega ),$$

(4)

where $c_{p_s^{*}}$ is the best Sobolev constant for the embedding $W_0^{1,p_s}(\Omega )\to L^{p_s^{*}}(\Omega )$ (see [21,22]).

From Holder’s inequality and (4), we obtain

$${\parallel u\parallel }_{L^{\tau }(\Omega )}\le c_{p_s^{*}}{|\Omega |}^{{\displaystyle \frac{p_s^{*}-\tau }{p_s^{*}\tau }}}\parallel g^{-s}{\parallel }_{L^1(\Omega )}^{{\displaystyle \frac{1}{ps}}}\parallel u\parallel \mathrm{for}\mathrm{all}u\in W_0^{1,p}(g,\Omega ),$$

(5)

where $|\Omega |$ is the Lebesgue measure of $\Omega$ in ${\mathbb{R}}^N$. In particular, we observe that, since $s>{\displaystyle \frac{N}{p}}$, we have $p<p_s^{*}$. Then, Equation (5) also holds for $\tau =p$.

Using the other non-negative weight $h\in L^1(\Omega )$, we have the corresponding degenerate space $L^q(h,\Omega )$ and the semi-norm

$${\parallel v\parallel }_{L^q(h,\Omega )}={\left({\int }_{\Omega }h\left(x\right){\left|v\right|}^q\right)}^{{\displaystyle \frac{1}{q}}}\forall v\in L^q(h,\Omega ).$$

The following proposition shows the relation between the spaces (see proposition 1 of [1]).

Proposition 1.

Let $1<q<p<\infty$ and assume that

$$g^{-{\displaystyle \frac{q}{p-q}}}{h}^{{\displaystyle \frac{p}{p-q}}}\in L^1(\Omega ).$$

(6)

Then, there is a continuous embedding $W_0^{1,p}(g,\Omega )\to W_0^{1,q}(h,\Omega )$.

Denote the topological dual space of $W_0^{1,p}(g,\Omega )$ by ${\left(W_0^{1,p}(g,\Omega )\right)}^{*}$ and the duality bracket by $\langle ·,·\rangle$. Thanks to Proposition 1 and Condition (2), the (negative) degenerate $(p,q)$-Laplacian $-{\Delta }_{pq}^{gh}:W_0^{1,p}(g,\Omega )\to {\left(W_0^{1,p}(g,\Omega )\right)}^{*}$

$$\langle -{\Delta }_{pq}^{gh}\left(u\right),v\rangle ={\int }_{\Omega }{\left(g\left(x\right)\right|\nabla u|}^{p-2}\nabla u+{h\left(x\right)|\nabla u|}^{q-2}\nabla u)\nabla vdx$$

(7)

is well defined. It is a coercive operator, that is, ${lim}_{\parallel u\parallel \to +\infty }{\displaystyle \frac{\langle -{\Delta }_{pq}^{gh}\left(u\right),u\rangle }{\parallel u\parallel }}=+\infty$, and it has the $S_{+}$ property, i.e., any sequence $\left\{u_n\right\}\subset W_0^{1,p}(g,\Omega )$ that satisfies $u_n\rightharpoonup u$ in $W_0^{1,p}(g,\Omega )$ and

$$\underset{n\to +\infty }{lim\; sup}\langle -{\Delta }_{pq}^{gh}\left(u_n\right),u_n-u\rangle \le 0.$$

Then, $u_n\to u$ in $W_0^{1,p}(g,\Omega )$. For the proof of these properties, we refer to [1].

3. Bounded Weak Solutions

In this section, we prove the existence of at least three bounded weak solutions for every $\lambda$ in an appropriate interval. These solutions are obtained as critical points of an energy functional related to Problem (1) taking into account Proposition 1,

$$I\left(u\right)={\displaystyle \frac{1}{p}}{\parallel u\parallel }^p+{\displaystyle \frac{1}{q}}{\parallel |\nabla u|\parallel }_{L^q(h,\Omega )}^q-\lambda {\int }_{\Omega }F(x,u\left(x\right))dx,\forall u\in W_0^{1,p}(g,\Omega ),$$

(8)

where $F(x,t)={\displaystyle {\int }_0^{t}f(x,\xi )d\xi }$ for all $(x,t)\in \Omega \times \mathbb{R}$.

The (negative) degenerate p-Laplacian with the weight $g\in L^1(\Omega )$ satisfying Condition (2) is the operator $-{\Delta }_{g,p}:W_0^{1,p}(g,\Omega )\to {\left(W_0^{1,p}(g,\Omega )\right)}^{*}$ defined by

$$\langle -{\Delta }_{g,p}\left(u\right),v\rangle ={\int }_{\Omega }g\left(x\right){|\nabla u|}^{p-2}\nabla u·\nabla vdx,\forall u,v\in W_0^{1,p}(g,\Omega ).$$

(9)

We also use the first eigenvalue ${\lambda }_1$ of the operator $-{\Delta }_{g,p}:W_0^{1,p}(g,\Omega )\to {\left(W_0^{1,p}(g,\Omega )\right)}^{*}$

$${\lambda }_1=\underset{u\in W_0^{1,p}(g,\Omega ),u\ne 0}{inf}{\displaystyle \frac{{\int }_{\Omega }g\left(x\right){|\nabla u|}^{p}dx}{{\int }_{\Omega }{\left|u\right|}^{p}dx}}$$

(10)

(see Lemma 3.1 of [19]).

We start assuming that the nonlinearity f satisfies a subcritical growth condition

$\left(H\right)$

There exist two non-negative constants, $a_1$ and $a_2$, and a constant $\gamma \in [1,p]$ such that

$$\left|f(x,t)\right|\le a_1+a_2{\left|t\right|}^{\gamma -1}\mathrm{for}\mathrm{all}(x,t)\in \Omega \times \mathbb{R}.$$

For later use, we have the following condition:

$\left(H1\right)$

There exists one non-negative constant $b_1$ with $b_1{\lambda }_1^{-1}<1$, a function $\varrho \in L^1(\Omega )$ such that

$$f(x,t)t\le \varrho +b_1{\left|t\right|}^p\mathrm{for}\mathrm{all}(x,t)\in \Omega \times \mathbb{R}.$$

A weak solution of Problem (1) is any $u\in W_0^{1,p}(g,\Omega )$ such that

$${\int }_{\Omega }{g\left(x\right)|\nabla u\left(x\right)|}^{p-2}\nabla u\left(x\right)·\nabla v\left(x\right)dx+{\int }_{\Omega }h\left(x\right){|\nabla u\left(x\right)|}^{q-2}\nabla u\left(x\right)·\nabla v\left(x\right)dx=\lambda {\int }_{\Omega }f(x,u\left(x\right))v\left(x\right)dx,$$

(11)

for all $v\in W_0^{1,p}(g,\Omega )$.

Notice that, under Assumptions (2), (6) and $\left(H\right)$, the integrals exist by the definition of a weak solution.

We are now in a position to prove the following a priori estimate for the solutions of Problem (1):

Lemma 1.

Under Assumptions (H) and (H1), the set of solutions to Problem (1) is bounded in $W_0^{1,p}(g,\Omega )$.

Proof. 

Acting on (11) with the test function $v=u\in W_0^{1,p}(g,\Omega )$ results in

$${\parallel u\parallel }^p\le {\int }_{\Omega }f(x,u)udx.$$

Hypothesis (H1), in conjunction with (10), ensures that

$${\parallel u\parallel }^p\le d_1{\lambda }_1^{-1}{\parallel u\parallel }^p+{\parallel \varrho \parallel }_{L^1(\Omega )}.$$

Taking into account that $d_1{\lambda }_1^{-1}<1$, a conclusion can be drawn. □

In this paper, we want to prove the existence of at least three weak solutions. Our main tool that we use is the following variational theorem obtained in [23] (see also [24]).

Theorem 1.

Let X be a real Banach space and let Φ, $\Psi :X\to \mathbb{R}$ be two continuously Gateaux differentiable functionals with Φ bounded below. Assume that $\Phi \left(0\right)=\Psi \left(0\right)=0$. Assume that there are $r\in \mathbb{R}$ and $\tilde{u}\in X$, with $\Phi \left(\tilde{u}\right)>r$, such that

$${\displaystyle \frac{{\displaystyle \underset{u\in {\Phi }^{-1}\left(\left]-\infty ,r\right]\right)}{sup}\Psi \left(u\right)}}{r}}<{\displaystyle \frac{\Psi \left(\tilde{u}\right)}{\Phi \left(\tilde{u}\right)}},$$

(12)

and, for each

$$\lambda \in \Lambda =\left]{\displaystyle \frac{\Phi \left(\tilde{u}\right)}{\Psi \left(\tilde{u}\right)}},{\displaystyle \frac{r}{{\displaystyle \underset{u\in {\Phi }^{-1}\left(\left]-\infty ,r\right]\right)}{sup}\Psi \left(u\right)}}}\right[,$$

the functional $\Phi -\lambda \Psi$ is bounded from below and satisfies Condition $\left(PS\right)$.

Then, for each $\lambda \in \Lambda$, the functional $\Phi -\lambda \Psi$ admits at least three critical points.

In order to study Problem (1), we split the energy functional given by (8) into two functionals $\Phi ,\Psi :W_0^{1,p}(g,\Omega )\to \mathbb{R}$ defined by

$$\Phi \left(u\right)={\displaystyle \frac{1}{p}}{\parallel u\parallel }^p+{\displaystyle \frac{1}{q}}{\parallel |\nabla u|\parallel }_{L^q(h,\Omega )}^q\mathrm{and}\Psi \left(u\right)={\int }_{\Omega }F(x,u\left(x\right))dx,\forall u\in W_0^{1,p}(g,\Omega ).$$

(13)

$\Phi$ and $\Psi$ are continuously Gâteaux differentiable functionals, whose Gâteaux derivatives at point $u\in W_0^{1,p}(g,\Omega )$ are given by

$$\langle {\Phi }^{\prime }\left(u\right),v\rangle =\langle -{\Delta }_{pq}^{gh}\left(u\right),v\rangle ={\int }_{\Omega }{g\left(x\right)|\nabla u\left(x\right)|}^{p-2}\nabla u\left(x\right)·\nabla v\left(x\right)dx+{\int }_{\Omega }h\left(x\right){|\nabla u\left(x\right)|}^{q-2}\nabla u\left(x\right)·\nabla v\left(x\right)dx,$$

and

$$\langle {\Psi }^{\prime }\left(u\right),v\rangle ={\int }_{\Omega }f(x,u\left(x\right))v\left(x\right)dx,$$

for every $v\in W_0^{1,p}(g,\Omega )$.

Remark 1.

The functional Φ is sequentially weakly lower semicontinuous since Φ is continuous and convex on $W_0^{1,p}(g,\Omega )$ (see [21,25]).

Important properties on functionals $\Phi$ and $\Psi$ are listed in the following propositions:

Proposition 2.

Assume that Hypothesis (H) holds, $\left\{u_n\right\}\subset W_0^{1,p}(g,\Omega )$ and $u\in W_0^{1,p}(g,\Omega )$ with $u_n\rightharpoonup u$. Then,

$$\underset{n\to +\infty }{lim}\langle {\Psi }^{\prime }\left(u_n\right),u_n-u\rangle =0.$$

Proof. 

Hypothesis $\left(H\right)$ enables us to assert that $u_n\to u$ in $L^{\gamma }(\Omega )$ and in $L^1(\Omega )$. Therefore,

$$|\langle {\Psi }^{\prime }\left(u_n\right),u_n-u\rangle |=|{\int }_{\Omega }f(x,u_n)(u_n-u)dx|\le a_1\parallel u_n{-u\parallel }_{L^1(\Omega )}+a_2\parallel u_n{\parallel }_{L^{\gamma }(\Omega )}^{\gamma -1}{\parallel u_n-u\parallel }_{L^{\gamma }(\Omega )}\to 0,$$

for $n\to +\infty$. □

Proposition 3.

Assume that Hypothesis (H) holds. The functional $\Psi :W_0^{1,p}(g,\Omega )\to \mathbb{R}$ is sequentially weakly continuous in $W_0^{1,p}(g,\Omega )$.

Proof. 

The operator ${\Psi }^{\prime }$ is the composition of the continuous and bounded mapping $u\to f(x,u)$, which takes a value in $L^{p_s^{\prime }}(\Omega )$ and the linear embedding $L^{p_s^{\prime }}(\Omega )\to {\left(W_0^{1,p}(a,\Omega )\right)}^{*}$, which is compact, as it is the adjoint of the compact embedding $W_0^{1,p}(g,\Omega )\to L^{p_s}(\Omega )$. Therefore, ${\Psi }^{\prime }$ is a compact operator.

Taking into account Corollary 41.9 of [26], since $W_0^{1,p}(g,\Omega )$ is reflexive, $\Psi$ is sequentially weakly continuous. □

Since $\Omega$ is a bounded domain in ${\mathbb{R}}^N$, we can find $x_0\in \Omega$ such that $B(x_0,R)$, the ball of center $x_0$ and radius $R:={sup}_{x\in \Omega }\mathrm{dist}(x,\partial \Omega )$, which is contained in $\Omega$. In the following, we use the constants

$$\begin{array}{c}{M}_1={\displaystyle \frac{2^q{c}_{p_s^{*}}^p{\parallel g^{-s}\parallel }_{L^1(\Omega )}^{\frac{1}{s}}}{{|\Omega |}^{\frac{p}{p_s^{*}}}}}min\left\{{\displaystyle \frac{{\parallel g\parallel }_{L^1\left(\mathcal{B}\right)}}{R^p}},{\displaystyle \frac{{\parallel h\parallel }_{L^1\left(\mathcal{B}\right)}}{R^q}}\right\}\hfill \\ M_2={\displaystyle \frac{2^p{c}_{p_s^{*}}^p{\parallel g^{-s}\parallel }_{L^1(\Omega )}^{\frac{1}{s}}}{{|\Omega |}^{\frac{p}{p_s^{*}}}}}max\left\{{\displaystyle \frac{{\parallel g\parallel }_{L^1\left(\mathcal{B}\right)}}{R^p}},{\displaystyle \frac{{\parallel h\parallel }_{L^1\left(\mathcal{B}\right)}}{R^q}}\right\}.\hfill \end{array}$$

where $\mathcal{B}:=B\left(x_0,R\right)\setminus B\left(x_0,{\displaystyle \frac{R}{2}}\right).$

Now, we formulate and prove our main results.

Theorem 2.

Assume (2), (6), Hypotheses (H) and (H1), and that there exist two positive constants c and d, with $c<M_1^{{\displaystyle \frac{1}{p}}}d$, such that

(i) 

$F(x,t)\ge 0,$ for all a.e. $x\in B\left(x_0,R\right)$ and for every $t\in [0,d]$.

(ii) 

$$a_1{c}^{1-p}+{\displaystyle \frac{a_2}{\gamma }}{c}^{\gamma -p}<{\displaystyle \frac{q{\int }_{B\left(x_0,\frac{R}{2}\right)}F(x,d)dx}{p{M}_2|\Omega |(d^p+d^q)}},$$

where $a_1$, $a_2$, and γ are given by $\left(H\right)$.

Then, for each

$$\lambda \in \tilde{\Lambda }=\left]{\displaystyle \frac{{|\Omega |}^{\frac{p}{p_s^{*}}}{M}_2}{q{c}_{p_s^{*}}^p{\parallel a^{-s}\parallel }_{L^1(\Omega )}^{\frac{1}{s}}}}{\displaystyle \frac{d^p+d^q}{{\displaystyle {\int }_{B\left(x_0,\frac{R}{2}\right)}F\left(x,d\right)dx}}},{\displaystyle \frac{1}{p{c}_{p_s^{*}}^p\parallel a^{-s}{\parallel }_{L^1(\Omega )}^{\frac{1}{s}}{|\Omega |}^{\frac{p_s^{*}-p}{p_s^{*}}}\left(a_1{c}^{1-p}+{\displaystyle \frac{a_2}{\gamma }}{c}^{\gamma -p}\right)}}\right[,$$

Problem (1) admits at least three bounded weak solutions.

Proof. 

Consider the energy functional I given by (8) and $\Phi$, $\Psi$ as in (13). Taking into account Remark 1 and Proposition 3, the energy functional I is sequentially weakly lower semicontinuous.

Consider $\tilde{u}\in W_0^{1,p}(g,\Omega )$ defined by

$$\tilde{u}\left(x\right)=\left\{\begin{array}{cc}0\hfill & \mathrm{if}x\in \Omega \setminus B(x_0,R),\hfill \\ {\displaystyle \frac{2d}{R}}\left(R-|x-x_0|\right)\hfill & \mathrm{if}x\in \mathcal{B},\hfill \\ d\hfill & \mathrm{if}x\in B\left(x_0,{\displaystyle \frac{R}{2}}\right).\hfill \end{array}\right.$$

Next, we obtain

$$\Phi \left(\tilde{u}\right)={\displaystyle \frac{1}{p}}{\left({\displaystyle \frac{2d}{R}}\right)}^p{\parallel g\parallel }_{L^1\left(\mathcal{B}\right)}+{\displaystyle \frac{1}{q}}{\left({\displaystyle \frac{2d}{R}}\right)}^q{\parallel h\parallel }_{L^1\left(\mathcal{B}\right)},$$

(14)

and, taking into account (i),

$$\Psi \left(\tilde{u}\right)\ge {\int }_{B\left(x_0,{\displaystyle \frac{R}{2}}\right)}F\left(x,d\right)dx.$$

(15)

Then,

$$\Phi \left(\tilde{u}\right)\ge {\displaystyle \frac{2^q}{p}}min\left\{{\displaystyle \frac{{\parallel g\parallel }_{L^1\left(\mathcal{B}\right)}}{R^p}},{\displaystyle \frac{{\parallel h\parallel }_{L^1\left(\mathcal{B}\right)}}{R^q}}\right\}(d^p+d^q)={\displaystyle \frac{{|\Omega |}^{\frac{p}{p_s^{*}}}{M}_1}{p{c}_{p_s^{*}}^p{\parallel g^{-s}\parallel }_{L^1(\Omega )}^{\frac{1}{s}}}}(d^p+d^q);$$

$$\Phi \left(\tilde{u}\right)\le {\displaystyle \frac{2^p}{q}}max\left\{{\displaystyle \frac{{\parallel g\parallel }_{L^1\left(\mathcal{B}\right)}}{R^p}},{\displaystyle \frac{{\parallel h\parallel }_{L^1\left(\mathcal{B}\right)}}{R^q}}\right\}(d^p+d^q)={\displaystyle \frac{{|\Omega |}^{\frac{p}{p_s^{*}}}{M}_2}{q{c}_{p_s^{*}}^p{\parallel g^{-s}\parallel }_{L^1(\Omega )}^{\frac{1}{s}}}}(d^p+d^q);$$

$$\begin{array}{c}\hfill {\displaystyle \frac{\Psi \left(\tilde{u}\right)}{\Phi \left(\tilde{u}\right)}}\ge {\displaystyle \frac{q{c}_{p_s^{*}}^p{\parallel g^{-s}\parallel }_{L^1(\Omega )}^{\frac{1}{s}}}{{|\Omega |}^{\frac{p}{p_s^{*}}}{M}_2}}{\displaystyle \frac{{\displaystyle {\int }_{B\left(x_0,\frac{R}{2}\right)}F\left(x,d\right)dx}}{d^p+d^q}}.\end{array}$$

(16)

Set $r={\displaystyle \frac{1}{p}}{\displaystyle \frac{{|\Omega |}^{\frac{p}{p_s^{*}}}}{c_{p_s^{*}}^p{\parallel g^{-s}\parallel }_{L^1(\Omega )}^{\frac{1}{s}}}}{c}^p$. From $c<M_1^{{\displaystyle \frac{1}{p}}}d$, one has $\Phi \left(\tilde{u}\right)>r$.

For all $u\in {\Phi }^{-1}\left(\left]-\infty ,r\right]\right)$, we have $∥u∥\le {\left(pr\right)}^{{\displaystyle \frac{1}{p}}}$. Then,

$$\begin{array}{ccc}\hfill {\displaystyle \frac{{\displaystyle \underset{u\in {\Phi }^{-1}\left(\left]-\infty ,r\right]\right)}{sup}\Psi \left(u\right)}}{r}}& \le & {\displaystyle \frac{{\displaystyle \underset{∥u∥\le {\left(pr\right)}^{\frac{1}{p}}}{sup}\Psi \left(u\right)}}{r}}\le {\displaystyle \frac{{\displaystyle \underset{∥u∥\le {\left(pr\right)}^{\frac{1}{p}}}{sup}\left(a_1{c}_{p_s^{*}}{|\Omega |}^{\frac{p_s^{*}-1}{p_s^{*}}}\parallel g^{-s}{\parallel }_{L^1(\Omega )}^{\frac{1}{ps}}\parallel u\parallel +{\displaystyle \frac{a_2}{\gamma }}{c}_{p_s^{*}}^{\gamma }{|\Omega |}^{\frac{p_s^{*}-\gamma }{p_s^{*}}}\parallel g^{-s}{\parallel }_{L^1(\Omega )}^{\frac{\gamma }{ps}}{\parallel u\parallel }^{\gamma }\right)}}{r}}\hfill \\ & \le & a_1{c}_{p_s^{*}}{|\Omega |}^{{\displaystyle \frac{p_s^{*}-1}{p_s^{*}}}}\parallel g^{-s}{\parallel }_{L^1(\Omega )}^{{\displaystyle \frac{1}{ps}}}{p}^{{\displaystyle \frac{1}{p}}}{r}^{{\displaystyle \frac{1-p}{p}}}+{\displaystyle \frac{a_2}{\gamma }}{c}_{p_s^{*}}^{\gamma }\parallel g^{-s}{\parallel }_{L^1(\Omega )}^{{\displaystyle \frac{\gamma }{ps}}}{|\Omega |}^{{\displaystyle \frac{p_s^{*}-\gamma }{p_s^{*}}}}{p}^{{\displaystyle \frac{\gamma }{p}}}{r}^{{\displaystyle \frac{\gamma -p}{p}}}\hfill \\ & =& p{c}_{p_s^{*}}^p{|\Omega |}^{{\displaystyle \frac{p_s^{*}-p}{p_s^{*}}}}{\parallel g^{-s}\parallel }_{L^1(\Omega )}^{{\displaystyle \frac{1}{s}}}\left[a_1{\left({\displaystyle \frac{c_{p_s^{*}}^p{\parallel g^{-s}\parallel }_{L^1(\Omega )}^{\frac{1}{s}}}{{|\Omega |}^{\frac{p}{p_s^{*}}}}}pr\right)}^{{\displaystyle \frac{1-p}{p}}}+{\displaystyle \frac{a_2}{\gamma }}{\left({\displaystyle \frac{c_{p_s^{*}}^p{\parallel g^{-s}\parallel }_{L^1(\Omega )}^{\frac{1}{s}}}{{|\Omega |}^{\frac{p}{p_s^{*}}}}}pr\right)}^{{\displaystyle \frac{\gamma -p}{p}}}\right]\hfill \\ & =& p{c}_{p_s^{*}}^p\parallel g^{-s}{\parallel }_{L^1(\Omega )}^{{\displaystyle \frac{1}{s}}}{|\Omega |}^{{\displaystyle \frac{p_s^{*}-p}{p_s^{*}}}}\left[a_1{c}^{1-p}+{\displaystyle \frac{a_2}{\gamma }}{c}^{\gamma -p}\right].\hfill \end{array}$$

Our aim is to apply Theorem 1. The functionals $\Phi$ and $\Psi$ satisfy all regularity assumptions requested in Theorem 1.

On the other hand, $\Phi \left(u\right)=\Phi \left(0\right)=\Psi \left(0\right)=0$.

Fix $\lambda \in \tilde{\Lambda }$. From (ii), we observe that one has $\tilde{\Lambda }\ne \varnothing$, moreover for each $u\in W_0^{1,p}(g,\Omega )$, one has

$$I\left(u\right)=\Phi \left(u\right)-\lambda \Psi \left(u\right)\ge {\displaystyle \frac{1}{p}}{\parallel u\parallel }^p-\lambda {\displaystyle \frac{a_2}{\gamma }}{c}_{p_s^{*}}^{\gamma }{|\Omega |}^{{\displaystyle \frac{p_s^{*}-\gamma }{p_s^{*}}}}\parallel g^{-s}{\parallel }_{L^1(\Omega )}^{{\displaystyle \frac{\gamma }{ps}}}{\parallel u\parallel }^{\gamma }-\lambda a_1{c}_{p_s^{*}}{|\Omega |}^{{\displaystyle \frac{p_s^{*}-1}{p_s^{*}}}}{\parallel g^{-s}\parallel }_{L^1(\Omega )}^{{\displaystyle \frac{1}{ps}}}∥u∥.$$

(17)

We claim that the operator I is coercive. If $\gamma \in [1,p[$ from (17), we have ${lim}_{\parallel u\parallel \to +\infty }{\displaystyle \frac{I\left(u\right)}{\parallel u\parallel }}=+\infty$ for every $\lambda \ge 0$. If $\gamma =p$, Condition (17) becomes

$$I\left(u\right)=\Phi \left(u\right)-\lambda \Psi \left(u\right)\ge {\displaystyle \frac{1}{p}}\left[1-\lambda a_2{c}_{p_s^{*}}^p{|\Omega |}^{{\displaystyle \frac{p_s^{*}-p}{p_s^{*}}}}{\parallel g^{-s}\parallel }_{L^1(\Omega )}^{{\displaystyle \frac{1}{s}}}\right]{\parallel u\parallel }^p-\lambda a_1{c}_{p_s^{*}}{|\Omega |}^{{\displaystyle \frac{p_s^{*}-1}{p_s^{*}}}}{\parallel g^{-s}\parallel }_{L^1(\Omega )}^{{\displaystyle \frac{1}{ps}}}∥u∥,$$

which implies that ${lim}_{\parallel u\parallel \to +\infty }{\displaystyle \frac{I\left(u\right)}{\parallel u\parallel }}=+\infty$ being

$$\lambda <{\displaystyle \frac{1}{p{c}_{p_s^{*}}^p\parallel g^{-s}{\parallel }_{L^1(\Omega )}^{\frac{1}{s}}{|\Omega |}^{\frac{p_s^{*}-p}{p_s^{*}}}\left(a_1{c}^{1-p}+{\displaystyle \frac{a_2}{p}}\right)}}\le {\displaystyle \frac{1}{c_{p_s^{*}}^p\parallel g^{-s}{\parallel }_{L^1(\Omega )}^{\frac{1}{s}}{|\Omega |}^{\frac{p_s^{*}-p}{p_s^{*}}}{a}_2}}.$$

Now, we have that I is bounded from below because it is coercive and sequentially weakly lower semicontinuous. Now, we are going to prove that I satisfies Condition $\left(PS\right)$. Consider a sequence $\left\{u_n\right\}\subset W_0^{1,p}(g,\Omega )$ such that ${\left\{I\left(u_n\right)\right\}}_{n\in \mathbb{N}}$ is bounded and that ${\left\{I^{\prime }\left(u_n\right)\right\}}_{n\in \mathbb{N}}$ converges to 0 in ${\left(W_0^{1,p}(g,\Omega )\right)}^{*}$. Since I is coercive, the sequence $\left\{u_n\right\}$ is bounded and, up to a subsequence, $u_n\rightharpoonup u$ with $u\in W_0^{1,p}(g,\Omega )$. Hence,

$$\langle {\Phi }^{\prime }\left(u_n\right),u_n-u\rangle =\langle I^{\prime }\left(u_n\right),u_n-u\rangle +\lambda \langle {\Psi }^{\prime }\left(u_n\right),u_n-u\rangle ,$$

and, from Proposition 3, we have

$$\underset{n\to \infty }{lim\; sup}\langle {\Phi }^{\prime }\left(u_n\right),u_n-u\rangle \le 0.$$

By the $S_{+}-$ property of ${\Phi }^{\prime }$ in $W_0^{1,p}(g,\Omega )$ (we recall that ${\Phi }^{\prime }=-{\Delta }_{pq}^{gh}$), we have that $u_n\to u$ in $W_0^{1,p}(g,\Omega )$. Hence, I fulfills Condition $\left(PS\right)$.

All assumptions of Theorem 1 are verified. Thus, for each $\lambda \in \tilde{\Lambda }\subset \left]{\displaystyle \frac{\Phi \left(\tilde{u}\right)}{\Psi \left(\tilde{u}\right)}},{\displaystyle \frac{r}{{\displaystyle \underset{u\in {\Phi }^{-1}\left(\left]-\infty ,r\right]\right)}{sup}\Psi \left(u\right)}}}\right[,$ Problem (1) admits at least three weak solutions. Moreover, taking Lemma 1 into account, the solutions are bounded. □

Let us now consider the special case when f is a positive continuous function.

Theorem 3.

If Conditions (2) and (6) hold, let $f:\Omega \times \mathbb{R}\to \mathbb{R}$ be a positive continuous function, satisfying $\left(H\right)$ with $1\le \gamma <p$$\left(H1\right)$ . Moreover, assume that

$$\underset{t\to 0}{lim}{\displaystyle \frac{f(x,t)}{{\left|t\right|}^{p-1}}}=0\mathit{uniformly}a.e.x\mathit{in}\Omega .$$

(18)

Then, there exists $\overline{\lambda }>0$ such that for every $\lambda >\overline{\lambda }$, Problem (1) admits at least three non-negative bounded weak solutions.

Proof. 

Using a similar argument as in [27] (Lemma 3.3) and [28] (Theorem 3.3) from (18), we have

$$\underset{r\to 0^{+}}{lim}{\displaystyle \frac{{\displaystyle \underset{u\in {\Phi }^{-1}\left(\left]-\infty ,r\right]\right)}{sup}\Psi \left(u\right)}}{r}}=0.$$

(19)

Set $\overline{\lambda }={\displaystyle \frac{{|\Omega |}^{\frac{p}{p_s^{*}}}{M}_2}{q{c}_{p_s^{*}}^p{\parallel g^{-s}\parallel }_{L^1(\Omega )}^{\frac{1}{s}}}}{inf}_{d>0}\left\{{\displaystyle \frac{d^p+d^q}{{\displaystyle {\int }_{B\left(x_0,\frac{R}{2}\right)}F\left(x,d\right)dx}}}\right\}$, where $x_0$, R and $M_2$ are given as in Theorem 2. For each $\lambda >\overline{\lambda }$, there is $\delta >0$ such that $\lambda >{\displaystyle \frac{{|\Omega |}^{\frac{p}{p_s^{*}}}{M}_2}{q{c}_{p_s^{*}}^p{\parallel g^{-s}\parallel }_{L^1(\Omega )}^{\frac{1}{s}}}}{\displaystyle \frac{{\delta }^p+{\delta }^q}{{\displaystyle {\int }_{B\left(x_0,\frac{R}{2}\right)}F\left(x,\delta \right)dx}}}$. Using $\delta$, we define the function $\tilde{u}$ as in Theorem 2. From (19), there is $\overline{r}\in \left]0,\Phi \left(\tilde{u}\right)\right[$, such that

$${\displaystyle \frac{{\displaystyle \underset{u\in {\Phi }^{-1}\left(\left]-\infty ,\overline{r}\right]\right)}{sup}\Psi \left(u\right)}}{\overline{r}}}\le {\displaystyle \frac{1}{\lambda }}<{\displaystyle \frac{q{c}_{p_s^{*}}^p{\parallel a^{-s}\parallel }_{L^1(\Omega )}^{\frac{1}{s}}}{{|\Omega |}^{\frac{p}{p_s^{*}}}{M}_2}}{\displaystyle \frac{{\displaystyle {\int }_{B\left(x_0,\frac{R}{2}\right)}F\left(x,\delta \right)dx}}{{\delta }^p+{\delta }^q}}<{\displaystyle \frac{\Psi \left(\tilde{u}\right)}{\Phi \left(\tilde{u}\right)}}.$$

Then, Condition (12) of Theorem 1 is satisfied.

Moreover, in the proof of Theorem 2, we have already showed that, when $1\le \gamma <p$, the functional I is coercive, bounded from below and satisfies Condition $\left(PS\right)$ for each $\lambda \ge 0$.

Since of all assumptions of Theorem 1 are verified, for each $\lambda >\overline{\lambda }$, Problem (1) admits at least three weak solutions. To prove that the weak solution u is non-negative, we use $-u^{-}=-max\{-u,0\}\in W_0^{1,p}(g,\Omega )$ as a test function in (11). Moreover, taking Lemma 1 into account, the solutions are bounded. Thus, the proof is complete. □

We obtain another result for an autonomous case.

Theorem 4.

Let $f:\mathbb{R}\to \mathbb{R}$ be a continuous function with $f\not\equiv 0$ and set $F\left(t\right)={\int }_0^{t}f\left(s\right)ds$ with $F\ge 0$ a.e. in Ω. If

$$\underset{\left|t\right|\to +\infty }{lim}{\displaystyle \frac{F\left(t\right)}{{\left|t\right|}^p}}=0$$

(20)

and

$$\underset{t\to 0}{lim}{\displaystyle \frac{F\left(t\right)}{{\left|t\right|}^p}}=0,$$

(21)

then there exists $\underline{\lambda }$ such that for every $\lambda >\underline{\lambda }$, Problem (1) admits at least three weak solutions.

Proof. 

Set $\underline{\lambda }={\displaystyle \frac{2^N\Gamma (1+\frac{N}{2}){|\Omega |}^{\frac{p}{p_s^{*}}}{M}_2}{{\pi }^{\frac{N}{2}}{R}^N{c}_{p_s^{*}}^p{\parallel g^{-s}\parallel }_{L^1(\Omega )}^{\frac{1}{s}}}}{inf}_{d>0}\left\{{\displaystyle \frac{d^p+d^q}{F\left(d\right)}}\right\}$, where $x_0$, R and $M_1$ are given as in Theorem 2, and $\Gamma$ is the gamma function. Fix a positive number $\lambda >\underline{\lambda }$. Then, there is $\eta >0$ such that $\lambda >{\displaystyle \frac{2^N\Gamma (1+\frac{N}{2}){|\Omega |}^{\frac{p}{p_s^{*}}}{M}_2}{{\pi }^{\frac{N}{2}}{R}^N{c}_{p_s^{*}}^p{\parallel g^{-s}\parallel }_{L^1(\Omega )}^{\frac{1}{s}}}}{\displaystyle \frac{{\eta }^p+{\eta }^q}{F\left(\eta \right)}}$. We consider the operator I as in (8), and we claim that it is coercive. In fact, we observe that from (20), when we fix $0<ϵ<{\displaystyle \frac{1}{\lambda p{c}_{p_s^{*}}^p{|\Omega |}^{\frac{p_s^{*}-p}{p_s^{*}}}{\parallel g^{-s}\parallel }_{L^1(\Omega )}^{\frac{1}{s}}}}$, there exists $\nu >0$ such that

$$F\left(t\right)\le {ϵ\left|t\right|}^p+\nu ,$$

for all $t\in \mathbb{R}$. Then, using (5), one has

$$\Psi \left(u\right)={\int }_{\Omega }F\left(u\right)dx\le {\int }_{\Omega }\left({ϵ\left|u\right|}^p+\nu \right)dx=ϵ{∥u∥}_{L^p(\Omega )}^p+\nu |\Omega |\le ϵc_{p_s^{*}}^p{|\Omega |}^{{\displaystyle \frac{p_s^{*}-p}{p_s^{*}}}}\parallel g^{-s}{\parallel }_{L^1(\Omega )}^{{\displaystyle \frac{1}{s}}}{\parallel u\parallel }^p+\nu |\Omega |.$$

(22)

Then,

$$I\left(u\right)=\Phi \left(u\right)-\lambda \Psi \left(u\right)\ge {\displaystyle \frac{1}{p}}{\parallel u\parallel }^p-\lambda ϵc_{p_s^{*}}^p{|\Omega |}^{{\displaystyle \frac{p_s^{*}-p}{p_s^{*}}}}\parallel g^{-s}{\parallel }_{L^1(\Omega )}^{{\displaystyle \frac{1}{s}}}{\parallel u\parallel }^p-\lambda \nu |\Omega |,$$

this relation leads to the coercivity of I. Moreover, I is bounded from below, owing to the fact that it is coercive and sequentially weakly lower semicontinuous (see Remark 1 and Proposition 3). Finally, as we have just proved in the proof of Theorem 2, I satisfies Condition $\left(PS\right)$. Arguing as in the proof of Theorem 3, we prove that Condition (12) of Theorem 1 holds.

All assumptions of Theorem 1 are verified. Then, for each $\lambda >\underline{\lambda }$, Problem (1) admits at least three weak solutions. If, additionally, f is non-negative, then the weak solutions are non-negative. If Condition $\left(H1\right)$ holds, the solutions are bounded. □

Now, we provide a simple example for which Theorem 4 applies, as follows:

Example 1.

We consider the following problem

$$\left\{\begin{array}{cc}-{\mathit{div}\left(\right|x|}^{{\displaystyle \frac{1}{3}}}{|\nabla u|+(1-|x|}^2)\nabla u)=\lambda f\left(u\right)\hfill & \mathit{in}\Omega ,\hfill \\ \\ u=0\hfill & \mathit{on}\partial \Omega .\hfill \end{array}\right.$$

(23)

where $\Omega =\{x\in {\mathbb{R}}^3:|x|<1\}$ is the unit ball and f is the derivative of the function $F:\mathbb{R}\to [0,+\infty [$ defined by $F\left(t\right)=t^4{e}^{-\left|t\right|}$ . We study the problem (23) with $N=3$, $p=3$, $q=2$, $g\left(x\right)={\left|x\right|}^{{\displaystyle \frac{1}{3}}}$ and $h\left(x\right)=1-{\left|x\right|}^2$ . The assumption (6) is verified because

$$g^{-{\displaystyle \frac{q}{p-q}}}{b}^{{\displaystyle \frac{p}{p-q}}}={\left|x\right|}^{-{\displaystyle \frac{2}{3}}}{(1-|x|}^2{)}^3\in L^1(\Omega ).$$

We choose $s=2\in \left]{\displaystyle \frac{N}{p}},+\infty \right[\cap \left[{\displaystyle \frac{1}{p-1}},+\infty \right[$, and we have $p_s=2$ and $p_s^{*}=6$. Thus,

$${∥g^{-s}∥}_{L^1(\Omega )}={\int \int \int }_{\Omega }{\left|x\right|}^{-{\displaystyle \frac{2}{3}}}dx<\infty .$$

Then, (2) and Condition $ps<N(s+1)$ are satisfied. We observe that

$${\displaystyle \underset{t\to 0}{lim}{\displaystyle \frac{F\left(t\right)}{t^p}}=\underset{t\to +\infty }{lim}{\displaystyle \frac{F\left(t\right)}{t^p}}=0.}$$

Then, owing to Theorem 4, there exists $\underline{\lambda }={\displaystyle \frac{72(2+3\sqrt{2})e^{\sqrt{2}}}{7}}$ such that the problem (23) admits at least three non-negative weak solutions for each $\lambda >{\displaystyle \frac{72(2+3\sqrt{2})e^{\sqrt{2}}}{7}}$.