Quasilinear Dirichlet Problems with Degenerated p-Laplacian and Convection Term

Abstract

The paper develops a sub-supersolution approach for quasilinear elliptic equations driven by degenerated p-Laplacian and containing a convection term. The presence of the degenerated operator forces a substantial change to the functional setting of previous works. The existence and location of solutions through a sub-supersolution is established. The abstract result is applied to find nontrivial, nonnegative and bounded solutions.

1. Introduction

In this paper, we study the following quasilinear elliptic problem

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

(P)

on a bounded domain $\mathsf{\Omega }\subset {\mathbb{R}}^N$ with $N\ge 2$ and $p\in (1,N)$. We assume that the boundary $\partial \mathsf{\Omega }$ of $\mathsf{\Omega }$ is locally Lipschitzian, i.e., each point of $\partial \mathsf{\Omega }$ has a neighborhood whose intersection with $\partial \mathsf{\Omega }$ is the graph of a Lipschitz continuous function. Throughout the text we denote by $|·|$ and · the standard Euclidean norm and scalar product on ${\mathbb{R}}^N$, respectively. A main feature of the present work is that the leading part of the equation in (P) is the differential operator in divergence form $\mathrm{div}\left(a\left(x\right)\right|\nabla u{|}^{p-2}\nabla u)$ known as the degenerated p-Laplacian with the weight $a\in L_{\mathrm{loc}}^1\left(\mathsf{\Omega }\right)$. It is supposed that the function a be positive almost everywhere in $\mathsf{\Omega }$ and that the following condition holds

$$a^{-s}\in L^1\left(\mathsf{\Omega }\right)\mathrm{for}\mathrm{some}s\in \left(\frac{N}{p},+\infty \right)\cap \left[\frac{1}{p-1},+\infty \right).$$

(1)

In the case where $a\left(x\right)\equiv 1$ we recover the ordinary p-Laplacian. Various examples of useful weights meeting the requirement (1) are given in [1]. For instance, it is obvious that defining $a\left(x\right)=\mathrm{dist}(x,S)$ for $x\in \mathsf{\Omega }$, with a nonempty closed subset S of $\partial \mathsf{\Omega }$, one obtains a function a on $\mathsf{\Omega }$ for which (1) holds true with any listed s.

The natural space associated with problem (P) is $W_0^{1,p}(a,\mathsf{\Omega })$ that is the closure of $C_0^{\infty }\left(\mathsf{\Omega }\right)$ in the weighted Sobolev space $W^{1,p}(a,\mathsf{\Omega })$. In Section 2 we briefly survey the spaces $W^{1,p}(a,\mathsf{\Omega })$ and $W_0^{1,p}(a,\mathsf{\Omega })$. The (negative) degenerated p-Laplacian with the weight $a\in L_{\mathrm{loc}}^1\left(\mathsf{\Omega }\right)$ under condition (1) is defined on $W_0^{1,p}(a,\mathsf{\Omega })$ and takes values in the dual space ${\left(W_0^{1,p}(a,\mathsf{\Omega })\right)}^{*}$.

Corresponding to the constant s in (1) we set

$$p_s={\displaystyle \frac{ps}{s+1}}$$

and the Sobolev critical exponent $p_s^{*}=\frac{N{p}_s}{N-p_s}$ (we note that $1\le p_s<N$). There is a continuous embedding $W^{1,p}(a,\mathsf{\Omega })\hookrightarrow L^{p_s^{*}}\left(\mathsf{\Omega }\right)$, so a continuous embedding $L^{{\left(p_s^{*}\right)}^{\prime }}\left(\mathsf{\Omega }\right)\hookrightarrow {\left(W_0^{1,p}(a,\mathsf{\Omega })\right)}^{*}$, where ${\left(p_s^{*}\right)}^{\prime }$ stands for the Hölder conjugate of $p_s^{*}$, i.e., ${\left(p_s^{*}\right)}^{\prime }=\frac{p_s^{*}}{p_s^{*}-1}$. In order to handle problem (P) the idea is to arrange that the right-hand side $f(x,u,\nabla u)$ become an element of $L^{{\left(p_s^{*}\right)}^{\prime }}\left(\mathsf{\Omega }\right)$, which basically will be achieved through an adequate growth condition (see Hypothesis 1). We emphasize that the nonlinearity $f(x,u,\nabla u)$ depends on the solution u and on its gradient $\nabla u$, which generally makes the variational methods be ineffective. Such a term $f(x,u,\nabla u)$ is often called convection. It is expressed by means of a function $f:\mathsf{\Omega }\times \mathbb{R}\times {\mathbb{R}}^N\to \mathbb{R}$ that is Carathéodory, i.e., $f(·,t,\xi )$ is measurable for every $(t,\xi )\in \mathbb{R}\times {\mathbb{R}}^N$ and $f(x,·,·)$ is continuous for a.e. $x\in \mathsf{\Omega }$.

The goal of our work is to build a systematical approach to problem (P) via the method of sub-supersolution. It is for the first time when the method of sub-supersolution is implemented for problem (P) involving the degenerated p-Laplacian and related convection. In this respect, the functional setting is adapted to the novel situation of degenerated operators relying in an essential way on the associated exponent $p_s$. For results on the method of sub-supersolution applied to problems exhibiting convection terms but not driven by degenerated differential operators we refer to [2,3,4,5,6].

By a (weak) solution to problem (P) we mean a function $u\in W_0^{1,p}(a,\mathsf{\Omega })$ such that $f(x,u,\nabla u)\in L^{{\left(p_s^{*}\right)}^{\prime }}\left(\mathsf{\Omega }\right)$ and

$${\int }_{\mathsf{\Omega }}a\left(x\right){|\nabla u\left(x\right)|}^{p-2}\nabla u\left(x\right)·\nabla v\left(x\right)dx={\int }_{\mathsf{\Omega }}f(x,u\left(x\right),\nabla u\left(x\right))v\left(x\right)dx,\forall v\in W_0^{1,p}(a,\mathsf{\Omega }).$$

(2)

A function $\underline{u}\in W^{1,p}(a,\mathsf{\Omega })$ is called a subsolution for problem (P) if $\underline{u}\le 0$ on $\partial \mathsf{\Omega }$ (in the sense of traces), $f(·,\underline{u}(·),\nabla \underline{u}(·))\in L^{{\left(p_s^{*}\right)}^{\prime }}\left(\mathsf{\Omega }\right)$ and

$$\begin{array}{c}{\displaystyle {\int }_{\mathsf{\Omega }}a\left(x\right){|\nabla \underline{u}\left(x\right)|}^{p-2}\nabla \underline{u}\left(x\right)·\nabla v\left(x\right)dx\le {\int }_{\mathsf{\Omega }}f(x,\underline{u}\left(x\right),\nabla \underline{u}\left(x\right))v\left(x\right)dx}\hfill \end{array}$$

(3)

for all $v\in W_0^{1,p}(a,\mathsf{\Omega })$, $v\ge 0$ a.e. in $\mathsf{\Omega }$. Symmetrically, a function $\overline{u}\in W^{1,p}(a,\mathsf{\Omega })$ is called a supersolution for problem (P) if $\overline{u}\ge 0$ on $\partial \mathsf{\Omega }$ (in the sense of traces), $f(·,\overline{u}(·),\nabla \overline{u}(·))\in L^{{\left(p_s^{*}\right)}^{\prime }}\left(\mathsf{\Omega }\right)$ and

$$\begin{array}{c}{\displaystyle {\int }_{\mathsf{\Omega }}a\left(x\right){|\nabla \overline{u}\left(x\right)|}^{p-2}\nabla \overline{u}\left(x\right)·\nabla v\left(x\right)dx\ge {\int }_{\mathsf{\Omega }}f(x,\overline{u}\left(x\right),\nabla \overline{u}\left(x\right))v\left(x\right)dx}\hfill \end{array}$$

(4)

for all $v\in W_0^{1,p}(a,\mathsf{\Omega })$, $v\ge 0$ a.e. in $\mathsf{\Omega }$. Corresponding to a subsolution $\underline{u}$ and a supersolution $\overline{u}$ with $\underline{u}\le \overline{u}$ a.e. in $\mathsf{\Omega }$ we can consider the ordered interval

$$[\underline{u},\overline{u}]=\{w\in W^{1,p}(a,\mathsf{\Omega }):\underline{u}\le w\le \overline{u}\}.$$

The following hypothesis for $f(x,s,\xi )$ is adapted to an ordered sub-supersolution $\underline{u}\le \overline{u}$.

Hypothesis 1.

Given an ordered sub-supersolution $\underline{u}\le \overline{u}$ for problem (P), the Carathéodory function $f:\mathsf{\Omega }\times \mathbb{R}\times {\mathbb{R}}^N\to \mathbb{R}$ satisfies the growth condition

$$\left|f(x,t,\xi )\right|\le \sigma \left(x\right)+{b\left|\xi \right|}^r\mathrm{for}\mathrm{a}.\mathrm{e}.x\in \mathsf{\Omega },\mathrm{for}\mathrm{all}t\in [\underline{u}\left(x\right),\overline{u}\left(x\right)],\xi \in {\mathbb{R}}^N,$$

with a function $\sigma \in L^{\frac{p_s}{r}}\left(\mathsf{\Omega }\right)$ and constants $b>0$ and $r\in (0,\frac{p_s}{{\left(p_s^{*}\right)}^{\prime }})$.

According to Hypothesis 1 we have

$$f(x,u,\nabla u)\in L^{{\left(p_s^{*}\right)}^{\prime }}\left(\mathsf{\Omega }\right),\forall u\in [\underline{u},\overline{u}],$$

thus the integrals in the definitions above exist since

$$f(x,u,\nabla u)v\in L^1\left(\mathsf{\Omega }\right),\forall u\in [\underline{u},\overline{u}],v\in W_0^{1,p}(a,\mathsf{\Omega }).$$

Under Hypothesis 1, our main result establishes the existence of a weak solution to problem (P) with the additional location property $u\in [\underline{u},\overline{u}]$. We stress that this location property represents a significant qualitative information for the solution giving actually a priori estimates for it. As an application we prove the existence of a nontrivial nonnegative solution for a class of problems of type (P). The applicability of the stated result is demonstrated by an example.

2. Preliminary Material

The notation $\left|\mathsf{\Omega }\right|$ stands for the Lebesgue measure of the bounded domain $\mathsf{\Omega }$ in ${\mathbb{R}}^N$. In this section we discuss a few facts about the degenerated p-Laplacian entering problem (P). More details can be found in [1].

We note that (1) implies

$$a^{-\frac{1}{p-1}}\in L^1\left(\mathsf{\Omega }\right).$$

Indeed, it is seen that

$$\begin{array}{ccc}\hfill {\int }_{\mathsf{\Omega }}a{\left(x\right)}^{-\frac{1}{p-1}}dx& =& {\int }_{\left\{a\right(x)<1\}}a{\left(x\right)}^{-\frac{1}{p-1}}dx+{\int }_{\left\{a\right(x)\ge 1\}}a{\left(x\right)}^{-\frac{1}{p-1}}dx\hfill \\ & \le & {\int }_{\left\{a\right(x)<1\}}a{\left(x\right)}^{-s}dx+\left|\mathsf{\Omega }\right|<\infty \hfill \end{array}$$

since according to (1) one has $s\ge \frac{1}{p-1}$ and $a^{-s}\in L^1\left(\mathsf{\Omega }\right)$.

The weighted Sobolev space $W^{1,p}(a,\mathsf{\Omega })$ consists of all the functions $u\in L^p\left(\mathsf{\Omega }\right)$ for which $a^{\frac{1}{p}}|\nabla u|\in L^p\left(\mathsf{\Omega }\right)$. It is endowed with the norm

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

becoming a uniformly convex Banach space (due to the preceding property of the weight $a\left(x\right)$, see ([1], [Theorem 1.3])), thus reflexive, that contains $C_0^{\infty }\left(\mathsf{\Omega }\right)$. The space $W_0^{1,p}(a,\mathsf{\Omega })$ is the closure of $C_0^{\infty }\left(\mathsf{\Omega }\right)$ with respect to the norm ${\parallel ·\parallel }_{1,p,a}$.

There is an extensive literature devoted to the weighted Sobolev spaces including embeddings and traces related to different boundary value problems (see, e.g., [1,7,8]). The results depend strongly on what type of weight is used, generally attempting reduction to nonweighted spaces. As described below, under assumption (1), we can embed the space $W^{1,p}(a,\mathsf{\Omega })$ into the ordinary Sobolev space $W^{1,p_s}\left(\mathsf{\Omega }\right)$, hence automatically having the trace (note the boundary $\partial \mathsf{\Omega }$ is Lipschitz). This fact is needed in the definition of the sub-supersolution.

From (1) it is known that $s\ge \frac{1}{p-1}$, so one has $p_s\ge 1$ and the continuous embedding

$$W^{1,p}(a,\mathsf{\Omega })\hookrightarrow W^{1,p_s}\left(\mathsf{\Omega }\right),$$

(5)

which is relation (1.22) in [1]. More precisely, observing that $p>p_s$, through Holder’s inequality and (1) we get

$${\int }_{\mathsf{\Omega }}{|\nabla u|}^{p_s}dx={\int }_{\mathsf{\Omega }}{a}^{-\frac{p_s}{p}}{a}^{\frac{p_s}{p}}{|\nabla u|}^{p_s}dx\le {\left({\int }_{\mathsf{\Omega }}{a}^{-s}dx\right)}^{\frac{1}{s+1}}{\left({\int }_{\mathsf{\Omega }}a{|\nabla u|}^{p}dx\right)}^{\frac{p_s}{p}}$$

for all $u\in W^{1,p}(a,\mathsf{\Omega })$. As a consequence of the above inequality, we can endow $W_0^{1,p}(a,\mathsf{\Omega })$ with an equivalent norm

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

for which it holds

$${\parallel u\parallel }_{W_0^{1,p_s}\left(\mathsf{\Omega }\right)}\le \parallel a^{-s}{\parallel }_{L^1\left(\mathsf{\Omega }\right)}^{\frac{1}{ps}}\parallel u\parallel .$$

(6)

The Sobolev embedding theorem ensures the continuous embedding $W_0^{1,p_s}\left(\mathsf{\Omega }\right)\hookrightarrow L^{p_s^{*}}\left(\mathsf{\Omega }\right)$, with the critical exponent $p_s^{*}=\frac{N{p}_s}{N-p_s}$ (note that $1\le p_s<N$). Hence there exists a constant $T_0>0$ such that

$${\parallel u\parallel }_{L^{p_s^{*}}\left(\mathsf{\Omega }\right)}\le T_0{\parallel u\parallel }_{W_0^{1,p_s}\left(\mathsf{\Omega }\right)},\forall u\in W_0^{1,p_s}\left(\mathsf{\Omega }\right).$$

(7)

The best embedding constant $T_0$ has been estimated by Talenti [9] as follows

$$T_0\le {\pi }^{-\frac{1}{2}}{N}^{-\frac{1}{p_s}}{\left(\frac{p_s-1}{N-p_s}\right)}^{1-\frac{1}{p_s}}{\left(\frac{\mathsf{\Gamma }\left(1+\frac{N}{2}\right)\mathsf{\Gamma }\left(N\right)}{\mathsf{\Gamma }\left(\frac{N}{p_s}\right)\mathsf{\Gamma }\left(1+N-\frac{N}{p_s}\right)}\right)}^{\frac{1}{N}},$$

where $\mathsf{\Gamma }$ is the Euler function

$$\mathsf{\Gamma }\left(t\right)={\int }_0^{+\infty }{z}^{t-1}{e}^{-z}dz,\forall t>0.$$

Moreover, by the Rellich–Kondrachov compact embedding theorem, if $1\le r<p_s^{*}$ then the embedding $W_0^{1,p_s}\left(\mathsf{\Omega }\right)\hookrightarrow L^r\left(\mathsf{\Omega }\right)$ is compact.

By (7) and Hölder’s inequality we infer that

$${\parallel u\parallel }_{L^r\left(\mathsf{\Omega }\right)}\le T_0{\left|\mathsf{\Omega }\right|}^{\frac{p_s^{*}-r}{p_s^{*}}}{\parallel u\parallel }_{W_0^{1,p_s}\left(\mathsf{\Omega }\right)}$$

(8)

for every $u\in W_0^{1,p_s}\left(\mathsf{\Omega }\right)$ and $r\in [1,p_s^{*}]$. Combining (6) and (8) we arrive at

$${\parallel u\parallel }_{L^r\left(\mathsf{\Omega }\right)}\le {\kappa }_r\parallel u\parallel$$

(9)

for all $u\in W_0^{1,p}(a,\mathsf{\Omega })$ and $r\in [1,p_s^{*}]$, with the constant

$${\kappa }_r=T_0{\left|\mathsf{\Omega }\right|}^{\frac{p_s^{*}-r}{p_s^{*}}}{\parallel a^{-s}\parallel }_{L^1\left(\mathsf{\Omega }\right)}^{\frac{1}{ps}}.$$

The (negative) degenerated p-Laplacian with the weight $a\in L_{\mathrm{loc}}^1\left(\mathsf{\Omega }\right)$ satisfying condition (1) is the operator $A:W_0^{1,p}(a,\mathsf{\Omega })\to {\left(W_0^{1,p}(a,\mathsf{\Omega })\right)}^{*}$ defined by

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

(10)

We readily check that the operator A in (10) is well defined noticing by means of Hölder’s inequality that for all $u,v\in W_0^{1,p}(a,\mathsf{\Omega })$ it holds

$$\begin{array}{ccc}\hfill & & \left|{\int }_{\mathsf{\Omega }}a\left(x\right){|\nabla u|}^{p-2}\nabla u·\nabla vdx\right|\le {\int }_{\mathsf{\Omega }}a{\left(x\right)}^{\frac{p-1}{p}}{|\nabla u|}^{p-1}a{\left(x\right)}^{\frac{1}{p}}|\nabla v|dx\hfill \\ & \le & {\left({\int }_{\mathsf{\Omega }}a\left(x\right){|\nabla u|}^{p}dx\right)}^{\frac{p-1}{p}}{\left({\int }_{\mathsf{\Omega }}a\left(x\right){|\nabla v|}^{p}dx\right)}^{\frac{1}{p}}<\infty .\hfill \end{array}$$

(11)

Important properties of the operator A introduced in (10) are listed in the statement below.

Proposition 1.

Assume that the measurable function $a:\mathsf{\Omega }\to \mathbb{R}$ satisfies condition (1). Then the (negative) degenerated p-Laplacian $A:W_0^{1,p}(a,\mathsf{\Omega })\to {\left(W_0^{1,p}(a,\mathsf{\Omega })\right)}^{*}$ defined by (10) has the following properties:

(i) 

A is a bounded operator in the sense that it maps bounded sets to bounded sets;

(ii) 

A is a coercive operator, i.e.,

$$\underset{\parallel u\parallel \to \infty }{lim}\frac{\langle Au,u\rangle }{\parallel u\parallel }=+\infty ;$$

(iii) 

A is a strictly monotone operator, i.e.,

$$\langle Au-Av,u-v\rangle >0,u\ne v;$$

(iv) 

A has the $S_{+}$ property meaning that any sequence $\left\{u_n\right\}\subset W_0^{1,p}(a,\mathsf{\Omega })$ that satisfies $u_n\rightharpoonup u$ in $W_0^{1,p}(a,\mathsf{\Omega })$ and

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

(12)

is strongly convergent.

Proof.

$\left(i\right)$ From (10) and (11) we infer that

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

We obtain

$${\parallel Au\parallel }_{{\left(W_0^{1,p}(a,\mathsf{\Omega })\right)}^{*}}=\underset{v\in W_0^{1,p}(a,\mathsf{\Omega }),\parallel v\parallel \le 1}{sup}\left|\langle Au,v\rangle \right|\le {\parallel u\parallel }^{p-1},\forall u\in W_0^p(a,\mathsf{\Omega }),$$

whence A is bounded.

$\left(ii\right)$ By (10) we have that

$$\langle Au,u\rangle ={\int }_{\mathsf{\Omega }}{a\left(x\right)|\nabla u|}^{p}dx={\parallel u\parallel }^p,\forall u\in W_0^{1,p}(a,\mathsf{\Omega }).$$

Taking into account that $p>1$, it follows that the operator A is coercive.

$\left(iii\right)$ In view of the strict monotonicity of the mapping $\xi \mapsto {\left|\xi \right|}^{p-2}\xi$ on ${\mathbb{R}}^N$, it turns out

$$\langle Au-Av,u-v\rangle ={\int }_{\mathsf{\Omega }}a\left(x\right)\left({|\nabla u|}^{p-2}\nabla u-{|\nabla v|}^{p-2}\nabla v\right)·(\nabla u-\nabla v)dx>0,u\ne v,$$

so A is a strictly monotone operator.

$\left(iv\right)$ Let a sequence $\left\{u_n\right\}\subset W_0^{1,p}(a,\mathsf{\Omega })$ satisfy $u_n\rightharpoonup u$ in $W_0^{1,p}(a,\mathsf{\Omega })$ and (12). Using the monotonicity of the operator A and (12) we have

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

Through Hölder’s inequality we obtain

$$\begin{array}{ccc}& & {\displaystyle \langle A\left(u_n\right)-A\left(u\right),u_n-u\rangle ={\int }_{\mathsf{\Omega }}a\left(x\right)\left(|\nabla u_n{|}^{p-2}\nabla u_n-{|\nabla u|}^{p-2}\nabla u\right)·(\nabla u_n-\nabla u)dx}\hfill \\ =\hfill & & {\displaystyle {\int }_{\mathsf{\Omega }}a\left(x\right)|\nabla u_n{|}^{p}dx-{\int }_{\mathsf{\Omega }}a\left(x\right)|\nabla u_n{|}^{p-2}\nabla u_n·\nabla udx-{\int }_{\mathsf{\Omega }}{a\left(x\right)|\nabla u|}^{p-2}\nabla u·\nabla u_{n}dx+{\int }_{\mathsf{\Omega }}a\left(x\right){|\nabla u|}^{p}dx}\hfill \\ \ge \hfill & & {\displaystyle {\int }_{\mathsf{\Omega }}a\left(x\right){|\nabla u_n|}^{p}dx-{\left({\int }_{\mathsf{\Omega }}a\left(x\right){|\nabla u_n|}^{p}dx\right)}^{\frac{p-1}{p}}{\left({\int }_{\mathsf{\Omega }}a\left(x\right){|\nabla u|}^{p}dx\right)}^{\frac{1}{p}}}\hfill \\ & & {\displaystyle -{\left({\int }_{\mathsf{\Omega }}a\left(x\right){|\nabla u|}^{p}dx\right)}^{\frac{p-1}{p}}{\left({\int }_{\mathsf{\Omega }}a\left(x\right){|\nabla u_n|}^{p}dx\right)}^{\frac{1}{p}}+{\int }_{\mathsf{\Omega }}a\left(x\right){|\nabla u|}^{p}dx}\hfill \\ =\hfill & & {\displaystyle (\parallel u_n\parallel -\parallel u\parallel \left)\right(\parallel u_n{\parallel }^{p-1}-{\parallel u\parallel }^{p-1})\ge 0,}\hfill \end{array}$$

from which we find that ${lim}_{n\to +\infty }\parallel u_n\parallel =\parallel u\parallel$. Due to the uniform convexity of $W_0^{1,p}(a,\mathsf{\Omega })$ it follows that $u_n\to u$ in $W_0^{1,p}(a,\mathsf{\Omega })$, thus completing the proof. □

We also need the first eigenvalue ${\lambda }_1$ of the operator $A:W_0^{1,p}(a,\mathsf{\Omega })\to {\left(W_0^{1,p}(a,\mathsf{\Omega })\right)}^{*}$ in (10). Precisely, ${\lambda }_1>0$ is the least (positive) number for which the equation

$$\left\{\begin{array}{cc}-{\mathrm{div}\left(a\left(x\right)\right|\nabla u|}^{p-2}\nabla u)={\lambda }_1{\left|u\right|}^{p-2}u\hfill & \mathrm{in}\mathsf{\Omega }\hfill \\ \\ u=0\hfill & \mathrm{on}\partial \mathsf{\Omega }\hfill \end{array}\right.$$

(13)

admits a nontrivial solution called eigenfunction corresponding to the first eigenvalue ${\lambda }_1$. A solution to (13) is understood in the weak sense, i.e., $u\in W_0^{1,p}(a,\mathsf{\Omega })$ satisfying

$$\begin{array}{c}{\displaystyle {\int }_{\mathsf{\Omega }}{a\left(x\right)|\nabla u\left(x\right)|}^{p-2}\nabla u\left(x\right)·\nabla v\left(x\right)dx={\lambda }_1{\int }_{\mathsf{\Omega }}{\left|u\left(x\right)\right|}^{p-2}u\left(x\right)v\left(x\right)dx,\forall v\in W_0^{1,p}(a,\mathsf{\Omega }).}\hfill \end{array}$$

It is known that there exists an eigenfunction $u_1\in W_0^{1,p}(a,\mathsf{\Omega })$ corresponding to the first eigenvalue ${\lambda }_1$ such that $u_1\left(x\right)\ge 0$ for a.e. $x\in \mathsf{\Omega }$, $u_1\neg \equiv 0$, and $u_1\in L^{\infty }\left(\mathsf{\Omega }\right)$. For the proofs of these properties we refer to ([1], Chapter 3).

3. Main Results

Our main abstract result provides the existence of a solution to problem (P) and its location within the ordered interval determined by a sub-supersolution.

Theorem 1.

Let the weight $a\in L_{\mathrm{loc}}^1\left(\mathsf{\Omega }\right)$ fulfill the requirement (1) and assume that Hypothesis 1 for a subsolution $\underline{u}$ and a supersolution $\overline{u}$ with $\underline{u}\le \overline{u}$ a.e. is satisfied. Then problem (P) possesses at least a solution $u\in W_0^{1,p}(a,\mathsf{\Omega })$ with the location property $\underline{u}\le u\le \overline{u}$ for a.e. $x\in \mathsf{\Omega }$.

Proof. 

By means of the given sub-supersolution $\underline{u}\le \overline{u}$ for problem (P), we introduce some related mappings. The cut-off function for problem (P), we introduce some related mappings. The cut-off function $\pi :\mathsf{\Omega }\times \mathbb{R}\to \mathbb{R}$ is defined by

$$\pi (x,t)=\left\{\begin{array}{cc}-{(\underline{u}\left(x\right)-t)}^{\frac{r}{p_s-r}}\hfill & \mathrm{if}t<\underline{u}\left(x\right)\hfill \\ 0\hfill & \mathrm{if}\underline{u}\left(x\right)\le t\le \overline{u}\left(x\right)\hfill \\ {(t-\overline{u}\left(x\right))}^{\frac{r}{p_s-r}}\hfill & \mathrm{if}t>\overline{u}\left(x\right),\hfill \end{array}\right.$$

(14)

where s and r are the constants given in (1) and Hypothesis 1. Using (14) in conjunction with $\underline{u},\overline{u}\in L^{p_s^{*}}\left(\mathsf{\Omega }\right)$ enables us to find that

$$\left|\pi (x,t)\right|\le {c\left|t\right|}^{\frac{r}{p_s-r}}+\varrho \left(x\right)\mathrm{for}\mathrm{a}.\mathrm{e}.x\in \mathsf{\Omega },\mathrm{all}t\in \mathbb{R},$$

(15)

with a constant $c>0$ and a function $\varrho \in L^{\frac{p_s^{*}(p_s-r)}{r}}\left(\mathsf{\Omega }\right)$. Moreover, proceeding as in [4], we can establish that

$${\int }_{\mathsf{\Omega }}\pi (x,u\left(x\right))u\left(x\right)dx\ge b_1{\parallel u\parallel }_{L^{\frac{p_s}{p_s-r}}\left(\mathsf{\Omega }\right)}^{\frac{p_s}{p_s-r}}-b_2\mathrm{for}\mathrm{all}u\in W_0^{1,p}(a,\mathsf{\Omega }),$$

(16)

with positive constants $b_1$ and $b_2$.

In view of (15), the Nemytskij operator $u\mapsto \pi (·,u(·\left)\right)$ generated by $\pi$ maps continuously $L^{p_s^{*}}\left(\mathsf{\Omega }\right)$ to $L^{\frac{p_s^{*}(p_s-r)}{r}}\left(\mathsf{\Omega }\right)$. Therefore, the mapping $\Pi :W_0^{1,p}(a,\mathsf{\Omega })\to {\left(W_0^{1,p}(a,\mathsf{\Omega })\right)}^{*}$ defined by

$$\langle \Pi \left(u\right),v\rangle ={\int }_{\mathsf{\Omega }}\pi (x,u)vdx,\forall u,v\in W_0^{1,p}(a,\mathsf{\Omega })$$

is completely continuous. This is true because the inclusion $L^{\frac{p_s^{*}(p_s-r)}{r}}\left(\mathsf{\Omega }\right)\subset {\left(W_0^{1,p}(a,\mathsf{\Omega })\right)}^{*}$ is compact being the adjoint of the compact inclusion $W_0^{1,p}(a,\mathsf{\Omega })\subset L^{\frac{p_s^{*}(p_s-r)}{p_s^{*}(p_s-r)-r}}\left(\mathsf{\Omega }\right)$ (note that $\frac{p_s^{*}(p_s-r)}{p_s^{*}(p_s-r)-r}<p_s^{*}$ owing to the assumption $r\in (0,\frac{p_s}{{\left(p_s^{*}\right)}^{\prime }})$ in Hypothesis 1).

Hypothesis 1 and (5) imply that the Nemytskij operator $u\mapsto f(·,u(·),\nabla u(·\left)\right)$ maps continuously $[\underline{u},\overline{u}]\subset W^{1,p}(a,\mathsf{\Omega })$ to $L^{\frac{p_s}{r}}\left(\mathsf{\Omega }\right)$ with $r\in (0,\frac{p_s}{{\left(p_s^{*}\right)}^{\prime }})$. Composing the preceding Nemytskij operator with the inclusion $L^{\frac{p_s}{r}}\left(\mathsf{\Omega }\right)\subset {\left(W_0^{1,p}(a,\mathsf{\Omega })\right)}^{*}$, which is compact because it is the adjoint operator of the compact inclusion $W_0^{1,p}(a,\mathsf{\Omega })\subset L^{\frac{p_s}{p_s-r}}\left(\mathsf{\Omega }\right)$ (note that $\frac{p_s}{p_s-r}<p_s^{*}$ since $r\in (0,\frac{p_s}{{\left(p_s^{*}\right)}^{\prime }})$ in Hypothesis 1), we obtain a completely continuous mapping $N_f:[\underline{u},\overline{u}]\to {\left(W_0^{1,p}(a,\mathsf{\Omega })\right)}^{*}$ given by

$$\langle N_f\left(u\right),v\rangle ={\int }_{\mathsf{\Omega }}f(x,u\left(x\right),\nabla u\left(x\right))v\left(x\right)dx$$

for all $u\in [\underline{u},\overline{u}]$ and $v\in W_0^{1,p}(a,\mathsf{\Omega })$.

We also make use of the truncation operator $T:W_0^{1,p}(a,\mathsf{\Omega })\to W^{1,p}(a,\mathsf{\Omega })$ given by

$$\left(Tu\right)\left(x\right)=\left\{\begin{array}{cc}\underline{u}\left(x\right)\hfill & \mathrm{if}u\left(x\right)<\underline{u}\left(x\right)\hfill \\ u\left(x\right)\hfill & \mathrm{if}\underline{u}\left(x\right)\le u\left(x\right)\le \overline{u}\left(x\right)\hfill \\ \overline{u}\left(x\right)\hfill & \mathrm{if}u\left(x\right)>\overline{u}\left(x\right)\hfill \end{array}\right.$$

(17)

for all $u\in W_0^{1,p}(a,\mathsf{\Omega })$ and a.e. $x\in \mathsf{\Omega }$. It is a continuous and bounded mapping (in the sense that it maps bounded sets to bounded sets). Notice that its range lies in $[\underline{u},\overline{u}]$, so T can be composed with the operator $N_f$.

Now we consider for every $\lambda >0$ the operator $A_{\lambda }:W_0^{1,p}(a,\mathsf{\Omega })\to {\left(W_0^{1,p}(a,\mathsf{\Omega })\right)}^{*}$ defined by

$$\begin{array}{c}\hfill A_{\lambda }=A+\lambda \Pi -N_f\circ T.\end{array}$$

(18)

Explicitly, it reads as

$$\begin{array}{ccc}\hfill \langle A_{\lambda }\left(u\right),v\rangle & =& {\int }_{\mathsf{\Omega }}a\left(x\right){|\nabla u|}^{p-2}\nabla u·\nabla vdx+\lambda {\int }_{\mathsf{\Omega }}\pi (x,u)vdx\hfill \\ & -& {\int }_{\mathsf{\Omega }}f(x,Tu,\nabla \left(Tu\right))vdx\mathrm{for}\mathrm{all}u,v\in W_0^{1,p}(a,\mathsf{\Omega }).\hfill \end{array}$$

(19)

From Proposition 1$\left(i\right)$ it is known that the operator $A:W_0^{1,p}(a,\mathsf{\Omega })\to {\left(W_0^{1,p}(a,\mathsf{\Omega })\right)}^{*}$ is bounded, while the above comments demonstrate that the operators $\Pi$, $N_f$ and T are all of them bounded. Therefore from (18) we infer that the operator $A_{\lambda }:W_0^{1,p}(a,\mathsf{\Omega })\to {\left(W_0^{1,p}(a,\mathsf{\Omega })\right)}^{*}$ is bounded.

We claim that $A_{\lambda }:W_0^{1,p}(a,\mathsf{\Omega })\to {\left(W_0^{1,p}(a,\mathsf{\Omega })\right)}^{*}$ is a pseudomonotone operator. In this respect, let a sequence $\left\{u_n\right\}\subset W_0^{1,p}(a,\mathsf{\Omega })$ satisfy $u_n\rightharpoonup u$ in $W_0^{1,p}(a,\mathsf{\Omega })$ and

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

(20)

The sequence $\{\Pi \left(u_n\right)\}$ is bounded in $L^{\frac{p_s^{*}(p_s-r)}{r}}\left(\mathsf{\Omega }\right)$, while $u_n\to u$ in $L^{\frac{p_s^{*}(p_s-r)}{p_s^{*}(p_s-r)-r}}\left(\mathsf{\Omega }\right)$ by the compact embedding $W_0^{1,p}(a,\mathsf{\Omega })\subset L^{\frac{p_s^{*}(p_s-r)}{p_s^{*}(p_s-r)-r}}\left(\mathsf{\Omega }\right)$, thus

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

The sequence $\{N_f\circ T\left(u_n\right)\}$ is bounded in $L^{\frac{p_s}{r}}\left(\mathsf{\Omega }\right)$, while $u_n\to u$ in $L^{\frac{p_s}{p_s-r}}\left(\mathsf{\Omega }\right)$ by the compact embedding $W_0^{1,p}(a,\mathsf{\Omega })\subset L^{\frac{p_s}{p_s-r}}\left(\mathsf{\Omega }\right)$, producing

$$\underset{n\to \infty }{lim}\langle N_f\circ T\left(u_n\right),u_n-u\rangle =0.$$

Consequently, complying with (18), we see that (20) reduces to (12). This, in conjunction with the weak convergence $u_n\rightharpoonup u$, enables us to apply Proposition 1$\left(iv\right)$ ensuring that the strong convergence $u_n\to u$ in $W_0^{1,p}(a,\mathsf{\Omega })$ holds.

From the strong convergence $a{(·)}^{\frac{1}{p}}\nabla u_n(·)\to a{(·)}^{\frac{1}{p}}\nabla u(·)$ in ${\left(L^p\left(\mathsf{\Omega }\right)\right)}^N$ it follows the strong convergence $a{(·)}^{\frac{p-1}{p}}|\nabla u_n{(·)|}^{p-2}\nabla u_n(·)\to a{(·)}^{\frac{p-1}{p}}{|\nabla u(·)|}^{p-2}\nabla u(·)$ in ${\left(L^{\frac{p}{p-1}}\left(\mathsf{\Omega }\right)\right)}^N$. This amounts to saying that $A{u}_n\rightharpoonup Au$ in ${\left(W_0^{1,p}(a,\mathsf{\Omega })\right)}^{*}$ since

$${\displaystyle \langle A{u}_n,v\rangle ={\int }_{\mathsf{\Omega }}a\left(x\right)|\nabla u_n{|}^{p-2}\nabla u_n·\nabla vdx\to {\int }_{\mathsf{\Omega }}a\left(x\right){|\nabla u|}^{p-2}\nabla u·\nabla vdx=\langle Au,v\rangle ,\forall v\in W_0^{1,p}(a,\mathsf{\Omega }).}$$

Again, from the strong convergence $a{(·)}^{\frac{1}{p}}\nabla u_n(·)\to a{(·)}^{\frac{1}{p}}\nabla u(·)$ in ${\left(L^p\left(\mathsf{\Omega }\right)\right)}^N$ we infer that

$$\langle A{u}_n,u_n\rangle ={\int }_{\mathsf{\Omega }}a\left(x\right)|\nabla u_n{|}^{p}dx\to {\int }_{\mathsf{\Omega }}a\left(x\right){|\nabla u|}^{p}dx=\langle Au,u\rangle$$

as $n\to \infty$. Taking into account the continuity of the mappings $\Pi$ and $N_f\circ T$, we have

$$\langle A_{\lambda }{u}_n,v\rangle \to \langle A_{\lambda }u,v\rangle ,\forall v\in W_0^{1,p}(a,\mathsf{\Omega }),$$

and

$$\langle A_{\lambda }{u}_n,u_n\rangle \to \langle A_{\lambda }u,u\rangle$$

as $n\to \infty$, for every $\lambda >0$. We can conclude that $A_{\lambda }:W_0^{1,p}(a,\mathsf{\Omega })\to {\left(W_0^{1,p}(a,\mathsf{\Omega })\right)}^{*}$ is a pseudomonotone operator (see, e.g., ([2], Definition 2.97)).

The next step in the proof is to show that the operator $A_{\lambda }:W_0^{1,p}(a,\mathsf{\Omega })\to {\left(W_0^{1,p}(a,\mathsf{\Omega })\right)}^{*}$ is coercive provided $\lambda >0$ is large enough. Taking advantage of the fact that $Tu\in [\underline{u},\overline{u}]$ whenever $u\in W_0^{1,p}(a,\mathsf{\Omega })$, let us note by (16), (19) and Hypothesis 1 that

$$\begin{array}{ccc}\hfill & & \langle A_{\lambda }\left(u\right),u\rangle =\langle A\left(u\right),u\rangle +\lambda {\int }_{\mathsf{\Omega }}\pi (x,u)udx-{\int }_{\mathsf{\Omega }}f(x,Tu,\nabla \left(Tu\right))udx\hfill \\ \hfill \ge & & {\parallel u\parallel }^p+\lambda (b_1{\parallel u\parallel }_{L^{\frac{p_s}{p_s-r}}\left(\mathsf{\Omega }\right)}^{\frac{p_s}{p_s-r}}-b_2{)-\parallel \sigma \parallel }_{L^{\frac{p_s}{r}}\left(\mathsf{\Omega }\right)}{\parallel u\parallel }_{L^{\frac{p_s}{p_s-r}}\left(\mathsf{\Omega }\right)}-b{\int }_{\mathsf{\Omega }}{|\nabla \left(Tu\right)|}^r\left|u\right|dx\hfill \end{array}$$

(21)

for all $u\in W_0^{1,p}(a,\mathsf{\Omega })$. Now we estimate the last term in (21) based on the fact that by (5) we know that $\nabla u\in {\left(L^{p_s}\left(\mathsf{\Omega }\right)\right)}^N$, and so $\nabla \left(Tu\right)\in {\left(L^{p_s}\left(\mathsf{\Omega }\right)\right)}^N$. Using the definition of $Tu$ in (17), Hölder’s inequality and the continuous embedding in (9) it turns out that

$$\begin{array}{ccc}& & {\int }_{\mathsf{\Omega }}{|\nabla \left(Tu\right)|}^r\left|u\right|dx={\int }_{\{\underline{u}\le u\le \overline{u}\}}{|\nabla u|}^r\left|u\right|dx+{\int }_{\{u<\underline{u}\}}|\nabla \underline{u}{|}^r\left|u\right|dx+{\int }_{\{u>\overline{u}\}}|\nabla \overline{u}{|}^r\left|u\right|dx\hfill \\ \hfill \le & & {\int }_{\mathsf{\Omega }}{|\nabla u|}^r\left|u\right|dx+c_1\parallel u\parallel ,\forall u\in W_0^{1,p}(a,\mathsf{\Omega }),\hfill \end{array}$$

with a constant $c_1>0$. We can insert the preceding inequality in (21) to derive

$$\begin{array}{c}\hfill \langle A_{\lambda }\left(u\right),u\rangle \ge {\parallel u\parallel }^p+\lambda (b_1{\parallel u\parallel }_{L^{\frac{p_s}{p_s-r}}\left(\mathsf{\Omega }\right)}^{\frac{p_s}{p_s-r}}-b_2)-c_2\parallel u\parallel -b{\int }_{\mathsf{\Omega }}{|\nabla u|}^r\left|u\right|dx,\end{array}$$

(22)

with a constant $c_2>0$. The Hölder’s and Young’s inequalities in conjunction with embedding (5) imply

$$\begin{array}{ccc}& & {\int }_{\mathsf{\Omega }}{|\nabla u|}^r\left|u\right|dx\le {\parallel \nabla u\parallel }_{L^{p_s}\left(\mathsf{\Omega }\right)}^r{\parallel u\parallel }_{L^{\frac{p_s}{p_s-r}}\left(\mathsf{\Omega }\right)}\le c_3{\parallel u\parallel }^{p_s}+c_4{\parallel u\parallel }_{L^{\frac{p_s}{p_s-r}}\left(\mathsf{\Omega }\right)}^{\frac{p_s}{p_s-r}},\hfill \end{array}$$

with constants $c_3>0$ and $c_4>0$. Then (22) entails

$$\begin{array}{ccc}& & \langle A_{\lambda }\left(u\right),u\rangle \ge {\parallel u\parallel }^p+\lambda (b_1{\parallel u\parallel }_{L^{\frac{p_s}{p_s-r}}\left(\mathsf{\Omega }\right)}^{\frac{p_s}{p_s-r}}-b_2)-c_2\parallel u\parallel -b(c_3{\parallel u\parallel }^{p_s}+c_4{\parallel u\parallel }_{L^{\frac{p_s}{p_s-r}}\left(\mathsf{\Omega }\right)}^{\frac{p_s}{p_s-r}})\hfill \end{array}$$

(23)

for all $u\in W_0^{1,p}(a,\mathsf{\Omega })$. Recalling from (16) that $b_1>0$, we can choose $\lambda >0$ so large to have $\lambda b_1>b{c}_4$. Hence due to $p>p_s\ge 1$ (see (1)), (23) yields the coercivity of $A_{\lambda }$, i.e.,

$$\underset{\parallel u\parallel \to +\infty }{lim}\frac{\langle A_{\lambda }\left(u\right),u\rangle }{\parallel u\parallel }=+\infty .$$

We have shown that the nonlinear operator $A_{\lambda }:W_0^{1,p}(a,\mathsf{\Omega })\to {\left(W_0^{1,p}(a,\mathsf{\Omega })\right)}^{*}$ is bounded, pseudomonotone and coercive provided $\lambda >0$ is sufficiently large. Therefore, for such an $A_{\lambda }$ we can apply the main theorem of pseudomonotone operators (see, e.g., ([2], Theorem 2.99)) ensuring that there exists a solution $u\in W_0^{1,p}(a,\mathsf{\Omega })$ to the equation

$$\begin{array}{c}\hfill A_{\lambda }\left(u\right)=0.\end{array}$$

(24)

Fix an admissible $\lambda >0$ as pointed out above. We are going to prove that $u\in W_0^{1,p}(a,\mathsf{\Omega })$ resolving (24) is a weak solution of the original problem (P), which means that (2) is satisfied. To this end, notice that (19) and (24) yield

$$\begin{array}{cc}\hfill & {\int }_{\mathsf{\Omega }}a\left(x\right){|\nabla u\left(x\right)|}^{p-2}\nabla u\left(x\right)·\nabla v\left(x\right)dx+\lambda {\int }_{\mathsf{\Omega }}\pi (x,u)vdx\hfill \\ \hfill =& {\int }_{\mathsf{\Omega }}f(x,Tu,\nabla \left(Tu\right))vdx\mathrm{for}\mathrm{all}v\in W_0^{1,p}(a,\mathsf{\Omega }).\hfill \end{array}$$

(25)

We proceed by comparing u with the subsolution $\underline{u}$ and supersolution $\overline{u}$ postulated in Hypothesis 1. We claim that $u\le \overline{u}$ a.e. in $\mathsf{\Omega }$. Towards this, it can be readily checked that ${(u-\overline{u})}^{+}=max\{u-\overline{u},0\}\in W_0^{1,p}(a,\mathsf{\Omega })$, where the condition $\overline{u}\ge 0$ on $\partial \mathsf{\Omega }$ in the sense of traces is essentially used. Thus, we can insert $v={(u-\overline{u})}^{+}$ in (25) and (4) which gives

$$\begin{array}{cc}\hfill & {\int }_{\mathsf{\Omega }}a\left(x\right){|\nabla u\left(x\right)|}^{p-2}\nabla u\left(x\right)·\nabla {(u-\overline{u})}^{+}\left(x\right)dx+\lambda {\int }_{\mathsf{\Omega }}\pi (x,u\left(x\right)){(u-\overline{u})}^{+}\left(x\right)dx\hfill \\ \hfill & ={\int }_{\mathsf{\Omega }}f(x,Tu\left(x\right),\nabla \left(Tu\right)\left(x\right)){(u-\overline{u})}^{+}\left(x\right)dx\hfill \end{array}$$

(26)

and

$$\begin{array}{cc}\hfill & {\int }_{\mathsf{\Omega }}a\left(x\right){|\nabla \overline{u}\left(x\right)|}^{p-2}\nabla \overline{u}\left(x\right)·\nabla {(u-\overline{u})}^{+}\left(x\right)dx\ge {\int }_{\mathsf{\Omega }}f(x,\overline{u}\left(x\right),\nabla \overline{u}\left(x\right)){(u-\overline{u})}^{+}\left(x\right)dx.\hfill \end{array}$$

(27)

From (26) and (27), by subtraction we are led to

$$\begin{array}{cc}& {\displaystyle {\int }_{\mathsf{\Omega }}a\left(x\right)\left({|\nabla u\left(x\right)|}^{p-2}\nabla u\left(x\right)-{|\nabla \overline{u}\left(x\right)|}^{p-2}\nabla \overline{u}\left(x\right)\right)·\nabla {(u-\overline{u})}^{+}\left(x\right)dx+\lambda {\int }_{\mathsf{\Omega }}\pi (x,u\left(x\right)){(u-\overline{u})}^{+}\left(x\right)dx}\hfill \\ & {\displaystyle \le {\int }_{\mathsf{\Omega }}\left(f(x,Tu\left(x\right),\nabla \left(Tu\right)\left(x\right))-f(x,\overline{u}\left(x\right),\nabla \overline{u}\left(x\right))\right){(u-\overline{u})}^{+}\left(x\right)dx.}\hfill \end{array}$$

By (14), (17), and the preceding inequality we get

$$\begin{array}{cc}& {\int }_{\{u>\overline{u}\}}a\left(x\right)\left({|\nabla u\left(x\right)|}^{p-2}\nabla u\left(x\right)-{|\nabla \overline{u}\left(x\right)|}^{p-2}\nabla \overline{u}\left(x\right)\right)·\nabla (u-\overline{u})dx+\lambda {\int }_{\{u>\overline{u}\}}{(u\left(x\right)-\overline{u}\left(x\right))}^{\frac{p_s}{p_s-r}}dx\hfill \\ & \le {\int }_{\{u>\overline{u}\}}\left(f(x,Tu,\nabla \left(Tu\right))-f(x,\overline{u},\nabla \overline{u})\right)(u-\overline{u})dx=0.\hfill \end{array}$$

Since the function $a\left(x\right)$ is positive almost everywhere in $\mathsf{\Omega }$ and the mapping $\xi \mapsto {\left|\xi \right|}^{p-2}\xi$ on ${\mathbb{R}}^N$ is monotone, we arrive at

$${\int }_{\{u>\overline{u}\}}{(u\left(x\right)-\overline{u}\left(x\right))}^{\frac{p_s}{p_s-r}}dx\le 0.$$

Therefore, the Lebesgue measure of the set $\{u>\overline{u}\}$ is zero, i.e., $u\le \overline{u}$ a.e. in $\mathsf{\Omega }$.

Similarly, we can prove that $\underline{u}\le u$ a.e. in $\mathsf{\Omega }$. Specifically, relying on the condition $\underline{u}\le 0$ on $\partial \mathsf{\Omega }$ (in the sense of traces), it holds ${(\underline{u}-u)}^{+}=max\{\underline{u}-u,0\}\in W_0^{1,p}(a,\mathsf{\Omega })$, which allows us to test (25) and (3) with $v={(\underline{u}-u)}^{+}\in W_0^{1,p}(a,\mathsf{\Omega })$. This results in

$$\begin{array}{cc}\hfill & {\int }_{\mathsf{\Omega }}a\left(x\right){|\nabla u\left(x\right)|}^{p-2}\nabla u\left(x\right)·\nabla {(\underline{u}-u)}^{+}\left(x\right)dx+\lambda {\int }_{\mathsf{\Omega }}\pi (x,u\left(x\right)){(\underline{u}-u)}^{+}\left(x\right)dx\hfill \\ \hfill & ={\int }_{\mathsf{\Omega }}f(x,Tu\left(x\right),\nabla \left(Tu\right)\left(x\right)){(\underline{u}-u)}^{+}\left(x\right)dx\hfill \end{array}$$

(28)

and

$$\begin{array}{cc}\hfill & {\int }_{\mathsf{\Omega }}a\left(x\right){|\nabla \underline{u}\left(x\right)|}^{p-2}\nabla \underline{u}\left(x\right)·\nabla {(\underline{u}-u)}^{+}\left(x\right)dx\le {\int }_{\mathsf{\Omega }}f(x,\underline{u}\left(x\right),\nabla \underline{u}\left(x\right)){(\underline{u}-u)}^{+}\left(x\right)dx.\hfill \end{array}$$

(29)

Arguing as before, we deduce from (28), (29), (14), and (17) the following estimate

$$\begin{array}{cc}& {\displaystyle {\int }_{\{\underline{u}>u\}}a\left(x\right)\left(|\nabla \underline{u}{\left(x\right)|}^{p-2}\nabla \underline{u}\left(x\right)-{|\nabla u\left(x\right)|}^{p-2}\nabla u\left(x\right)\right)·\nabla (\underline{u}-u)dx+\lambda {\int }_{\{\underline{u}>u\}}{(\underline{u}\left(x\right)-u\left(x\right))}^{\frac{p_s}{p_s-r}}dx}\hfill \\ & {\displaystyle \le {\int }_{\mathsf{\Omega }}(f(x,\underline{u},\nabla \underline{u})-f(x,Tu,\nabla \left(Tu\right)){(\underline{u}-u)}^{+}dx}\hfill \\ & {\displaystyle ={\int }_{\{\underline{u}>u\}}(f(x,\underline{u},\nabla \underline{u})-f(x,Tu,\nabla \left(Tu\right))){(\underline{u}-u)}^{+}dx=0.}\hfill \end{array}$$

At this point, the positivity of the function $a\left(x\right)$ on $\mathsf{\Omega }$ and the monotonicity of the mapping $\xi \mapsto {\left|\xi \right|}^{p-2}\xi$ on ${\mathbb{R}}^N$ confirm that

$${\int }_{\{\underline{u}>u\}}{(\underline{u}\left(x\right)-u\left(x\right))}^{\frac{p_s}{p_s-r}}dx\le 0,$$

from which we can readily derive that $\underline{u}\le u$ a.e in $\mathsf{\Omega }$.

Based on the enclosure property $\underline{u}\le u\le \overline{u}$ a.e. in $\mathsf{\Omega }$, it follows through (17) that $T\left(u\right)=u$ and through (14) that $\Pi \left(u\right)=0$. As a result, (25) takes the form of (2), thus the proof is complete. □

Now we present an application of Theorem 1 describing how the existence of a nontrivial nonnegative solution can be established by effectively determining a sub-supersolution. In the sequel, by ${\lambda }_1$ we denote the first eigenvalue of problem (13) (see Section 2).

Theorem 2.

Let the weight $a\in L_{\mathrm{loc}}^1\left(\mathsf{\Omega }\right)$ fulfill the requirement (1). Assume that the Carathéodory function $f:\mathsf{\Omega }\times \mathbb{R}\times {\mathbb{R}}^N\to \mathbb{R}$ satisfies the conditions:

(j) 

there is a constant $\mu >0$ such that

$${\lambda }_1{t}^{p-1}\le f(x,t,\xi )\mathrm{for}\mathrm{a}.\mathrm{e}.x\in \mathsf{\Omega },\mathrm{all}t\in [0,\mu ],\xi \in {\mathbb{R}}^N;$$

(jj) 

there is a constant $C>0$ such that

$$f(x,C,0)\le 0\mathrm{for}\mathrm{a}.\mathrm{e}.x\in \mathsf{\Omega };$$

(jjj) 

there are a function $\sigma \in L^{\frac{p_s}{r}}\left(\mathsf{\Omega }\right)$ and constants $b>0$ and $r\in (0,\frac{p_s}{{\left(p_s^{*}\right)}^{\prime }})$ such that

$$\left|f(x,t,\xi )\right|\le \sigma \left(x\right)+{b\left|\xi \right|}^r\mathrm{for}\mathrm{a}.\mathrm{e}.x\in \mathsf{\Omega },\mathrm{all}t\in [0,C],\xi \in {\mathbb{R}}^N.$$

Then problem (P) has a nondegenerate, nonnegative and bounded weak solution $u\in W_0^{1,p}(a,\mathsf{\Omega })$ satisfying the estimate $u\le C$.

Proof. 

Our goal is to apply Theorem 1 by constructing an appropriate sub-supersolution. In order to determine a subsolution, we use an eigenfunction $u_1\in W_0^{1,p}(a,\mathsf{\Omega })$ corresponding to the first eigenvalue ${\lambda }_1$ of problem (13) with the properties $u_1\left(x\right)\ge 0$ for a.e. $x\in \mathsf{\Omega }$, $u_1\neg \equiv 0$, and $u_1\in L^{\infty }\left(\mathsf{\Omega }\right)$ as mentioned in Section 2. Then we choose an $\epsilon >0$ sufficiently small to verify

$$\begin{array}{c}\hfill \epsilon u_1\left(x\right)\le \mu \mathrm{for}\mathrm{a}.\mathrm{e}.x\in \mathsf{\Omega },\end{array}$$

(30)

where $\mu$ is the positive constant postulated in assumption $\left(j\right)$. Then assumption $\left(j\right)$ implies

$$\begin{array}{c}\hfill {\lambda }_1{\left(\epsilon u_1\right)}^{p-1}\le f(x,\epsilon u_1,\nabla \left(\epsilon u_1\right))\mathrm{for}\mathrm{a}.\mathrm{e}.x\in \mathsf{\Omega }.\end{array}$$

(31)

For a possibly smaller $\epsilon >0$ we can suppose

$$\begin{array}{c}\hfill \epsilon u_1\left(x\right)\le C\mathrm{for}\mathrm{a}.\mathrm{e}.x\in \mathsf{\Omega },\end{array}$$

(32)

with $C>0$ in assumption $\left(jj\right)$.

Let us fix an $\epsilon >0$ for which (30) and (32) are fulfilled. We claim that $\underline{u}=\epsilon u_1$ is a subsolution to problem (P). Indeed, by (13) with $u_1$ in place of u and (31) we note that

$$\begin{array}{cc}\hfill & {\int }_{\mathsf{\Omega }}a\left(x\right){|\nabla \underline{u}\left(x\right)|}^{p-2}\nabla \underline{u}\left(x\right)·\nabla v\left(x\right)dx={\epsilon }^{p-1}{\lambda }_1{\int }_{\mathsf{\Omega }}{u}_1{\left(x\right)}^{p-1}v\left(x\right)dx\hfill \\ \hfill & \le {\int }_{\mathsf{\Omega }}f(x,\epsilon u_1\left(x\right),\nabla \left(\epsilon u_1\right)\left(x\right))v\left(x\right)dx={\int }_{\mathsf{\Omega }}f(x,\underline{u}\left(x\right),\nabla \underline{u}\left(x\right))v\left(x\right)dx\hfill \end{array}$$

for all $v\in W_0^{1,p}(a,\mathsf{\Omega })$, $v\ge 0$ a.e. in $\mathsf{\Omega }$, thereby proving the claim.

Next we claim that the constant function $\overline{u}=C$, with $C>0$ in assumption $\left(jj\right)$, is a supersolution to problem (P). Accordingly, from assumption $\left(jj\right)$ we find that

$$\begin{array}{cc}& {\displaystyle {\int }_{\mathsf{\Omega }}a\left(x\right){|\nabla \overline{u}\left(x\right)|}^{p-2}\nabla \overline{u}\left(x\right)·\nabla v\left(x\right)dx=0\ge {\int }_{\mathsf{\Omega }}f(x,C,0)v\left(x\right)dx={\int }_{\mathsf{\Omega }}f(x,\overline{u}\left(x\right),\nabla \overline{u}\left(x\right))v\left(x\right)dx}\end{array}$$

for all $v\in W_0^{1,p}(a,\mathsf{\Omega })$, $v\ge 0$ a.e. in $\mathsf{\Omega }$, which proves the claim.

It is clear from (32) that $\underline{u}\left(x\right)\le \overline{u}\left(x\right)$ for a.e. in $\mathsf{\Omega }$. Assumption $\left(jjj\right)$ ensures that the growth condition required in Hypothesis 1 of Theorem 1 holds true. Therefore, all the hypotheses of Theorem 1 are verified, which permits the conclusion that there exists a solution $u\in W_0^{1,p}(a,\mathsf{\Omega })$ of problem (P) within the ordered interval $[\underline{u},\overline{u}]$. Since the function $\underline{u}=\epsilon u_1$ is nontrivial and nonnegative, and $u\ge \underline{u}$, we have that u is nontrivial and nonnegative, whereas $u\in [\underline{u},\overline{u}]$ renders the boundedness of u and the a priori estimate $u\le C$. The proof is complete. □

We end the paper with a simple example for which Theorem 2 applies.

Example 1.

Fix a positive weight $a\in L_{\mathrm{loc}}^1\left(\mathsf{\Omega }\right)$ with the property (1). Let the function $f:\mathsf{\Omega }\times \mathbb{R}\times {\mathbb{R}}^N\to \mathbb{R}$ be defined by

$$f(x,t,\xi )=\left\{\begin{array}{cc}0\hfill & \mathrm{if}t<0\hfill \\ t^{p-1}{(\rho \left(x\right)+|\xi |}^r)\hfill & \mathrm{if}0\le t\le 1\hfill \\ (2-t)(\rho \left(x\right)+|\xi {|}^r)\hfill & \mathrm{if}t>1,\hfill \end{array}\right.$$

with some $r\in [1,\frac{p_s}{{\left(p_s^{*}\right)}^{\prime }})$ and $\rho \in L^{\infty }\left(\mathsf{\Omega }\right)$ satisfying $\rho \left(x\right)\ge {\lambda }_1$ for a.e. $x\in \mathsf{\Omega }$. It follows that f is a Carathéodory function for which conditions $\left(j\right)-\left(jjj\right)$ in Theorem 2 are verified. Precisely, condition $\left(j\right)$ holds with $\mu =1$ because $\rho \left(x\right)\ge {\lambda }_1$, condition $\left(jj\right)$ holds with $C=2$, and condition $\left(jjj\right)$ is fulfilled with the given r. Hence Theorem 2 applies to problem (P) whose equation has the right-hand side expressed with the function $f(x,t,\xi )$ given above.