Nonhomogeneous Dirichlet Problems with Unbounded Coefficient in the Principal Part

Abstract

The main result of the paper establishes the existence of a bounded weak solution for a nonlinear Dirichlet problem exhibiting full dependence on the solution u and its gradient $\nabla u$ in the reaction term, which is driven by a p-Laplacian-type operator with a coefficient $G\left(u\right)$ that can be unbounded. Through a special Moser iteration procedure, it is shown that the solution set is uniformly bounded. A truncated problem is formulated that drops that $G\left(u\right)$ be unbounded. The existence of a bounded weak solution to the truncated problem is proven via the theory of pseudomonotone operators. It is noted that the bound of the solution for the truncated problem coincides with the uniform bound of the original problem. This estimate allows us to deduce that for an appropriate choice of truncation, one actually resolves the original problem.

1. Introduction

In this paper, we study the following Dirichlet problem:

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

(1)

on a bounded domain $\Omega$ in ${\mathbb{R}}^N$ with a Lipschitz boundary $\partial \Omega$. In (1) we have a continuous function $G:\mathbb{R}\to [a_0,+\infty )$, with $a_0>0$, a number $p\in (1,+\infty )$ with $N>p$, and a Carathéodory function $F:\Omega \times \mathbb{R}\times {\mathbb{R}}^N\to \mathbb{R}$ (i.e., $F(·,t,\xi )$ is measurable on $\Omega$ for each $(t,\xi )\in \mathbb{R}\times {\mathbb{R}}^N$ and $F(x,·,·)$ is continuous on $\mathbb{R}\times {\mathbb{R}}^N$ for almost all $x\in \Omega$). The notation $\nabla u$ stands for the gradient of u in the distributional sense. It is seen that the driving operator in Equation (1) is the p-Laplacian with a coefficient $G\left(u\right)$ depending on the solution u. The notation $G\left(u\right)$ in Equation (1) means the composition of the functions $G:\mathbb{R}\to \mathbb{R}$ and $u:\Omega \to \mathbb{R}$, that is, $G\left(u\right)\left(x\right)=G\left(u\right(x\left)\right)$ for $x\in \Omega$. The main point is that $G\left(u\right)$ can be unbounded from above, which does not permit to apply any standard method. It is also worth mentioning that problem (1) is not in variational form.

The space underlying the Dirichlet problem (1) is the Banach space $W_0^{1,p}(\Omega )$ endowed with the norm

$$\parallel u\parallel :={\left({\int }_{\Omega }{|\nabla u\left(x\right)|}^{p}dx\right)}^{\frac{1}{p}},\forall u\in W_0^{1,p}(\Omega ).$$

The dual space of $W_0^{1,p}(\Omega )$ is denoted $W^{-1,p^{\prime }}(\Omega )$. Since it was supposed that $N>p$, the critical Sobolev exponent is $p^{*}=Np/(N-p)$. Refer to [1] for the background related to the space $W_0^{1,p}(\Omega )$.

The (negative) p-Laplacian is the nonlinear operator $-{\Delta }_p:W_0^{1,p}(\Omega )\to W^{-1,p^{\prime }}(\Omega )$ (linear for $p=2$) defined by

$$\begin{array}{c}\hfill ⟨-{\Delta }_p\left(u\right),v⟩:={\int }_{\Omega }{|\nabla u|}^{p-2}\nabla u\nabla vdx,\forall u,v\in W_0^{1,p}(\Omega ).\end{array}$$

(2)

Due to the unbounded function $G\left(u\right)$, one cannot build a definition as in (2) corresponding to the term $-\mathrm{div}(G\left(u\right)|\nabla u{|}^{p-2}\nabla u)$ in (1). A major tool in our arguments is the first eigenvalue ${\lambda }_1$ of $-{\Delta }_p$, which is positive and isolated in the spectrum of $-{\Delta }_p$, and is given by

$${\lambda }_1:=\underset{u\in W_0^{1,p}(\Omega ),u\ne 0}{inf}\frac{{\int }_{\Omega }{|\nabla u|}^{p}dx}{{\int }_{\Omega }{|u|}^{p}dx}.$$

(3)

For the the rest of the paper, in order to simplify the notation we make the notational convention that for any real number $r>1$ we denote $r^{\prime }:=r/(r-1)$ (the Hölder conjugate of r).

The Carathéodory function $F:\Omega \times \mathbb{R}\times {\mathbb{R}}^N\to \mathbb{R}$ determining the reaction term $F(x,u,\nabla u)$ is subject to the following hypotheses.

Hypothesis 1 (H1). 

There exist constants $c_1\ge 0$, $c_2\ge 0$, $c_3\ge 0$, and $r\in (p,p^{*})$ such that

$$\begin{array}{c}\hfill |F(x,t,\xi )|\le c_1{|\xi |}^{\frac{p}{r^{\prime }}}+c_2{|t|}^{r-1}+c_3\mathrm{for}\mathrm{a}.\mathrm{e}.x\in \Omega ,\forall t\in \mathbb{R},\forall \xi \in {\mathbb{R}}^N.\end{array}$$

Hypothesis 2 (H2). 

There exist constants $d_1\ge 0$ and $d_2\ge 0$ with $d_1+{\lambda }_1^{-1}{d}_2<a_0$, and a function $\sigma \in L^1(\Omega )$ such that

$$F(x,t,\xi )t\le d_1{|\xi |}^p+d_2{|t|}^p+\sigma \left(x\right)\mathrm{for}\mathrm{a}.\mathrm{e}.x\in \Omega ,\forall t\in \mathbb{R},\forall \xi \in {\mathbb{R}}^N,$$

where ${\lambda }_1$ denotes the first eigenvalue of $-{\Delta }_p$.

The main result of this paper is stated as follows.

Theorem 1. 

Assume that $G:\mathbb{R}\to [a_0,+\infty )$, with $a_0>0$, is a continuous function and $F:\Omega \times \mathbb{R}\times {\mathbb{R}}^N\to \mathbb{R}$ is a Carathéodory function satisfying the conditions (H1) and (H2). Then problem (1) has at least a bounded weak solution $u\in W_0^{1,p}(\Omega )$ in the following sense:

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

(4)

Under hypothesis (H1), the integrals in (4) exist. The proof of Theorem 1 is presented in Section 3. In order to see the effective applicability of Theorem 1, we provide an example.

Example 1. 

On a bounded domain Ω in ${\mathbb{R}}^N$ with a Lipschitz boundary $\partial \Omega$, we state the Dirichlet problem

$$\left\{\begin{array}{cc}-\mathrm{div}(e^{u^2}{|\nabla u|}^{p-2}\nabla u)=b_1{|u|}^{p-2}u+b_2\frac{u}{u^2+1}{|\nabla u|}^{\frac{p(r-1)}{r}}\hfill & in\Omega \hfill \\ u=0\hfill & on\partial \Omega ,\hfill \end{array}\right.$$

(5)

with constants $p\in (1,+\infty )$, $r\in (p,p^{*})$, $b_1\ge 0$, $b_2\ge 0$, provided that $N>p$ and $1>b_2+{\lambda }_1^{-1}{b}_1$, where ${\lambda }_1$ is given by (3). We readily check that (5) fits into the framework of problem (1) taking $G\left(t\right)=e^{t^2}$ for all $t\in \mathbb{R}$ and

$$\begin{array}{c}\hfill F(x,t,\xi )=b_1{|t|}^{p-2}t+b_2\frac{t}{t^2+1}{|\xi |}^{\frac{p(r-1)}{r}},\forall (x,t,\xi )\in \Omega \times \mathbb{R}\times \in {\mathbb{R}}^N.\end{array}$$

Indeed, one has $G\left(t\right)\ge a_0:=1$ for all $t\in \mathbb{R}$,

$$\begin{array}{c}\hfill |F(x,t,\xi )|\le b_1{(|t|}^{r-1}+1)+b_2{|\xi |}^{\frac{p}{r^{\prime }}},\forall (x,t,\xi )\in \Omega \times \mathbb{R}\times {\mathbb{R}}^N,\end{array}$$

$$\begin{array}{c}\hfill F(x,t,\xi )t\le b_1{|t|}^p+b_2{(|\xi |}^p+1),\forall (x,t,\xi )\in \Omega \times \mathbb{R}\times \in {\mathbb{R}}^N.\end{array}$$

Assumption (H1) is verified with $c_1=b_2$, $c_2=c_3=b_1$, while assumption (H2) holds with $d_1=b_2$, $d_2=b_1$, $\sigma \left(x\right)\equiv b_2$. Theorem 1 applies because $a_0>d_1+{\lambda }_1^{-1}{d}_2$.

The inspiration for the present work comes from the recent paper [2] that deals with the Dirichlet problem

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

(6)

for a positive $a\in L_{loc}^1(\Omega )$, a continuous function $g:[0,+\infty )\to [a_0,+\infty )$, with $a_0>0$, and a Carathéodory function $f:\Omega \times \mathbb{R}\times {\mathbb{R}}^N\to \mathbb{R}$. The standing point in that work was to use the theory of weighted Sobolev spaces in [3] (see also [4]) with the weight $a\in L_{loc}^1(\Omega )$ requiring the condition

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

If we consider our problem (1) as a particular case of (6) taking $a\left(x\right)\equiv 1$ on $\Omega$ and apply the result in [2], the issue is that one obtains a solution of (1) belonging to the space $W_0^{1,p_s}(\Omega )$ with

$$\begin{array}{c}\hfill p_s=\frac{ps}{s+1}.\end{array}$$

(7)

and not to the space $W_0^{1,p}(\Omega )$ as it would be natural according to the statement of (1). In this respect, by (7) we note that $p_s<p$, so $W_0^{1,p}(\Omega )$ is strictly contained in $W_0^{1,p_s}(\Omega )$. Moreover, the assumptions admitted therein for the reaction $f(x,u,\nabla u)$ in (6) are more restrictive than here because they are formulated in terms of $p_s$ corresponding to some s and not with p as in conditions (H1)–(H2) for $F(x,u,\nabla u)$. All of this shows that the treatment in [2] does not provide the right approach to obtain Theorem 1. For this reason, we develop a direct study for problem (1) relying just on the classical Sobolev space $W_0^{1,p}(\Omega )$. The present paper is the first work studying problem (1) with unbounded coefficient $G\left(u\right)$ in the Soboleev space $W_0^{1,p}(\Omega )$. Certainly, we use some previous ideas but with substantial modifications and in a different functional setting. The technique relies on truncation, which is needed because the coefficient $G\left(u\right)$ in the principal part of Equation (1) is unbounded. Other important tools in our study are a special version of Moser iteration and the surjectivity theorem for pseudomonotone operators.

We mention a few relevant works in the area of our paper. A large amount of results in the field is based on variational smooth or nonsmooth methods for which we refer to the recent publications [5,6,7]. They cannot be applied to problem (1) taking into account the lack of variational structure. Nonvariational problems with convection terms have been investigated in recent years through theoretic operator techniques, sub-supersolution and approximation (see, e.g., [8,9,10,11,12]). The main point in these works lies in the dependence of the reaction term with respect to the gradient of the solution without weakening the ellipticity condition of the driving operator. In this connection, we also cite papers dealing with the equations and inclusions driven by the $(p,q)$-Laplacian operators, such as, for instance [13,14]. As an extension of this setting, the paper [15] deals with degenerate $(p,q)$-Laplacian problems, but without dependence on the solution u in the principal part of the equation. An advance in this direction is ref. [2], where there is dependence on solution u in the principal part of the equation of type (6) subject to a weight $a\left(x\right)$. Here, we drop the dependence on weight $a\left(x\right)$ and allow to have a unbounded coefficient $G\left(u\right)$ in problem (1).

Regarding the rest of the paper, Section 2 focuses on the bounded solutions to problem (1), and Section 3 contains the proof of Theorem 1.

2. Bounded Solutions to Problem (1)

Our first goal is to estimate the solutions in $W_0^{1,p}(\Omega )$.

Lemma 1. 

Assume that condition (H2) holds. Then the set of solutions to problem (1) is bounded in $W_0^{1,p}(\Omega )$ with a bound that depends on function G only through the lower bound $a_0$ of G.

Proof. 

Let $u\in W_0^{1,p}(\Omega )$ be a solution of (1). Inserting $v=u$ in (4) yields

$${\int }_{\Omega }G\left(u\right){|\nabla u|}^{p}dx={\int }_{\Omega }F(x,u,\nabla u)udx.$$

Invoking hypothesis (H2) and (3), we arrive at

$$\begin{array}{c}\hfill a_0{\parallel u\parallel }^p\le (d_1+d_2{\lambda }_1^{-1}){\parallel u\parallel }^p+{\parallel \sigma \parallel }_{L^1(\Omega )}.\end{array}$$

Since by hypothesis $d_1+d_2{\lambda }_1^{-1}<a_0$, the stated result is true. □

We are now able to find a uniform bound for the solutions of (1).

Theorem 2. 

Assume that conditions (H1) and (H2) are satisfied. Then the solution set of problem (1) is uniformly bounded, that is, there exists a constant $C>0$ such that ${\parallel u\parallel }_{L^{\infty }(\Omega )}\le C$ for every weak solution $u\in W_0^{1,p}(\Omega )$ to problem (1). The dependence of the uniform bound C on the data in problem (1) and hypotheses (H1) and (H2) is indicated as $C=C(N,p,\Omega ,a_0,c_1,c_2,c_3,d_1,d_2,\parallel \sigma {\parallel }_{L^1(\Omega )})$. In particular, the uniform bound C depends on G only through its lower bound $a_0$.

Proof. 

Given a weak solution $u\in W_0^{1,p}(\Omega )$ to problem (1), we have the representation $u=u^{+}-u^{-}$ with $u^{+}=max\{u,0\}$ (the positive part of u) and $u^{-}=max\{-u,0\}$ (the negative part of u). We prove the uniform boundedness separately for $u^{+}$ and $u^{-}$. We only give the proof for $u^{+}$, noting that we can argue similarly in the case of $u^{-}$.

We proceed by using in (4) the test function $v=u^{+}{u}_h^{kp}\in W_0^{1,p}(\Omega )$, where $u_h:=min\{u^{+},h\}$ with arbitrary constants $h>0$ and $k>0$. The fact that $v\in W_0^{1,p}(\Omega )$ follows from $u\in L^p(\Omega )$ and $u_h$ is bounded, while the distributional partial derivatives

$$\begin{array}{c}\hfill \frac{\partial v}{\partial x_i}=u_h^{kp}\frac{\partial u^{+}}{\partial x_i}+kp{u}_h^{kp-1}{u}^{+}\frac{\partial u_h}{\partial x_i},\forall i=1,\cdots ,N,\end{array}$$

belong to $L^p(\Omega )$ because $u,\partial u_h/\partial x_i\in L^p(\Omega )$ and $u_h$ is bounded. This gives

$$\begin{array}{c}\hfill {\int }_{\Omega }G\left(u\right){|\nabla u|}^{p-2}\nabla u\nabla \left(u^{+}{u}_h^{kp}\right)dx={\int }_{\Omega }F(x,u,\nabla u)u^{+}{u}_h^{kp}dx.\end{array}$$

(8)

The left-hand side of (8) can be estimated as follows:

$$\begin{array}{ccc}\hfill & & {\int }_{\Omega }G\left(u\right){|\nabla u|}^{p-2}\nabla u\nabla \left(u^{+}{u}_h^{kp}\right)dx\hfill \\ & & ={\int }_{\Omega }G\left(u\right){|\nabla u|}^{p-2}\nabla u(u_h^{kp}\nabla \left(u^{+}\right)+kp{u}^{+}{u}_h^{kp-1}\nabla \left(u_h\right))dx\hfill \\ & & \ge a_0\left[{\int }_{\Omega }{u}_h^{kp}|\nabla \left(u^{+}\right){|}^{p}dx+kp{\int }_{\{0<u<h\}}{u}_h^{kp}{|\nabla \left(u^{+}\right)|}^{p}dx\right].\hfill \end{array}$$

(9)

For the right-hand side of (8), by hypothesis (H1), we obtain

$$\begin{array}{ccc}\hfill & & {\int }_{\Omega }F(x,u,\nabla u)u^{+}{u}_h^{kp}dx\hfill \\ & & \le c_1{\int }_{\Omega }{|\nabla u|}^{\frac{p}{r^{\prime }}}{u}_h^{kp}{u}^{+}dx+c_2{\int }_{\Omega }{|u|}^{r-1}{u}_h^{kp}{u}^{+}dx+c_3{\int }_{\Omega }{u}_h^{kp}{u}^{+}dx.\hfill \end{array}$$

(10)

By Young’s inequality, for each $\epsilon >0$ there is a constant $c(\epsilon )>0$ such that

$$\begin{array}{c}\hfill c_1{\int }_{\Omega }{|\nabla u|}^{\frac{p}{r^{\prime }}}{u}_h^{kp}{u}^{+}dx=c_1{\int }_{\Omega }(|\nabla (u^{+}){|}^{\frac{p}{r^{\prime }}}{u}_h^{\frac{kp}{r^{\prime }}})(u_h^{\frac{kp}{r}}{u}^{+})dx\\ \hfill \le \epsilon {\int }_{\Omega }{u}_h^{kp}{|\nabla (u^{+})|}^{p}dx+c(\epsilon ){\int }_{\Omega }{u}_h^{kp}{(u^{+})}^{r}dx.\end{array}$$

(11)

It is clear that

$$\begin{array}{c}\hfill {\int }_{\Omega }{|u|}^{r-1}{u}_h^{kp}{u}^{+}dx={\int }_{\Omega }{u}_h^{kp}{\left(u^{+}\right)}^{r}dx\end{array}$$

(12)

and, since $r>1$ and $u_h\le u^{+}$,

$$\begin{array}{ccc}\hfill & & {\int }_{\Omega }{u}_h^{kp}{u}^{+}dx={\int }_{\{u^{+}\ge 1\}}{u}_h^{kp}{u}^{+}dx+{\int }_{\{u^{+}<1\}}{u}_h^{kp}{u}^{+}dx\hfill \\ & & \le {\int }_{\Omega }{u}_h^{kp}{\left(u^{+}\right)}^{r}dx+|\Omega |,\hfill \end{array}$$

(13)

where $|\Omega |$ denotes the Lebesgue measure of $\Omega$.

If $\epsilon >0$ is sufficiently small, we deduce from (8), in conjunction with (9), (10), (11), (12), and (13) that

$$\begin{array}{ccc}\hfill & & {\int }_{\Omega }{u}_h^{kp}|\nabla \left(u^{+}\right){|}^{p}dx+kp{\int }_{\{0<u<h\}}{u}_h^{kp}{|\nabla \left(u^{+}\right)|}^{p}dx\hfill \\ & & \le b\left({\int }_{\Omega }{u}_h^{kp}{\left(u^{+}\right)}^{r}dx+1\right),\hfill \end{array}$$

(14)

with a constant $b>0$. The last integral exists because $r<p^{*}$.

On the other hand, by Bernoulli’s inequality and since $u_h=u^{+}$ on $\{0<u<h\}$, we derive

$$\begin{array}{ccc}\hfill & & {\int }_{\Omega }{u}_h^{kp}|\nabla \left(u^{+}\right){|}^{p}dx+kp{\int }_{\{0<u<h\}}{u}_h^{kp}{|\nabla \left(u^{+}\right)|}^{p}dx\hfill \\ & & ={\int }_{\{u\ge h\}}|\nabla \left(u_h^k{u}^{+}\right){|}^{p}dx+{\int }_{\{u<h\}}|\nabla \left(u_h^k{u}^{+}\right){|}^{p}dx+\frac{kp}{{(k+1)}^p}{\int }_{\{0<u<h\}}{|\nabla \left(u_h^k{u}^{+}\right)|}^{p}dx\hfill \\ & & ={\int }_{\{u\ge h\}}|\nabla \left(u_h^k{u}^{+}\right){|}^{p}dx+\frac{kp+1}{{(k+1)}^p}{\int }_{\{0<u<h\}}{|\nabla \left(u_h^k{u}^{+}\right)|}^{p}dx\hfill \\ & & \ge \frac{kp+1}{{(k+1)}^p}{\int }_{\Omega }{|\nabla \left(u_h^k{u}^{+}\right)|}^{p}dx.\hfill \end{array}$$

(15)

Combining (14) and (15) leads to

$$\begin{array}{c}\hfill \frac{kp+1}{{(k+1)}^p}{\int }_{\Omega }{|\nabla \left(u_h^k{u}^{+}\right)|}^{p}dx\le b\left({\int }_{\Omega }{u}_h^{kp}{\left(u^{+}\right)}^{r}dx+1\right).\end{array}$$

(16)

At this point, we choose $q\in (p,r)$ with

$$\begin{array}{c}\hfill \frac{(r-p)q}{q-p}<p^{*}.\end{array}$$

(17)

The validity of such a choice holds in view of $p<r<p^{*}$ as postulated in condition (H1). Then (17), the Sobolev embedding theorem, Hölder’s inequality with $(q-p)/q+p/q=1$, and Lemma 1 imply

$$\begin{array}{ccc}& & {\int }_{\Omega }{u}_h^{kp}{\left(u^{+}\right)}^{r}dx={\int }_{\Omega }{\left(u^{+}\right)}^{r-p}{\left(u_h^k{u}^{+}\right)}^{p}dx\hfill \\ & & \le {\left({\int }_{\Omega }{\left(u^{+}\right)}^{\frac{(r-p)q}{q-p}}dx\right)}^{\frac{q-p}{q}}{\left({\int }_{\Omega }{\left(u_h^k{u}^{+}\right)}^{q}dx\right)}^{\frac{p}{q}}\le K{\parallel u_h^k{u}^{+}\parallel }_{L^q(\Omega )}^p,\hfill \end{array}$$

with a constant $K>0$.

On the basis of the previous inequality and the Sobolev embedding theorem, we obtain from (16) that

$$\begin{array}{c}\hfill c_0\frac{kp+1}{{(k+1)}^p}{\parallel u_h^k{u}^{+}\parallel }_{L^{p^{*}}(\Omega )}^p\le b\left(\parallel {\left(u^{+}\right)}^{k+1}{\parallel }_{L^q(\Omega )}^p+1\right),\end{array}$$

with a constant $c_0>0$. Then Fatou’s lemma letting $h\to +\infty$ entails

$$\begin{array}{c}\hfill c_0\frac{kp+1}{{(k+1)}^p}{\parallel u^{+}\parallel }_{L^{p^{*}(k+1)}(\Omega )}^{p(k+1)}\le b\left(\parallel {\left(u^{+}\right)}^{k+1}{\parallel }_{L^q(\Omega )}^p+1\right).\end{array}$$

By some arrangements, we obtain for a constant $C_1>0$ the estimate

$$\begin{array}{c}\hfill \parallel u^{+}{\parallel }_{L^{(k+1)p^{*}}(\Omega )}\le C_1^{\frac{1}{k+1}}{(k+1)}^{\frac{1}{k+1}}{\left(\parallel {\left(u^{+}\right)}^{k+1}{\parallel }_{L^q(\Omega )}^p+1\right)}^{\frac{1}{(k+1)p}}.\end{array}$$

Noticing that the sequence ${(k+1)}^{\frac{1}{\sqrt{k+1}}}$ is bounded, we find a constant $C_0>0$ for which it holds

$$\begin{array}{c}\hfill \parallel u^{+}{\parallel }_{L^{(k+1)p^{*}}(\Omega )}\le C_0^{\frac{1}{\sqrt{k+1}}}{\left(\parallel u^{+}{\parallel }_{L^{(k+1)q}(\Omega )}^{(k+1)p}+1\right)}^{\frac{1}{(k+1)p}}.\end{array}$$

(18)

We claim that there exists a constant $C>0$ independent of the solution u to (1) such that

$$\begin{array}{c}\hfill \parallel u^{+}{\parallel }_{L^d(\Omega )}\le C,\forall d\ge 1.\end{array}$$

(19)

In the case where $\parallel u^{+}{\parallel }_{L^{(k+1)q}(\Omega )}\le 1$ for infinitely many k, it is straightforward to show the validity of the claim. Therefore, we may suppose that $\parallel u^{+}{\parallel }_{L^{(k+1)q}(\Omega )}>1$ for all $k\ge k_0$,

If $\parallel u^{+}{\parallel }_{L^{(k+1)q}(\Omega )}>1$ for all k, we see that (18) takes the form

$$\begin{array}{c}\hfill \parallel u^{+}{\parallel }_{L^{(k+1)p^{*}}(\Omega )}\le C_1^{\frac{1}{\sqrt{k+1}}}{\parallel u^{+}\parallel }_{L^{(k+1)q}(\Omega )},\end{array}$$

(20)

with a constant $C_1>0$. Through (20), we are able to carry on a Moser iteration, setting inductively $(k_n+1)q=(k_{n-1}+1)p^{*}$ with the initial step $(k_1+1)q=p^{*}$. Applying repeatedly (20), it turns out that

$$\begin{array}{c}\hfill \parallel u^{+}{\parallel }_{L^{(k_n+1)p^{*}}(\Omega )}\le C_1^{{\sum }_{1\le i\le n}\frac{1}{\sqrt{k_i+1}}}{\parallel u^{+}\parallel }_{L^{(k_1+1)q}(\Omega )},\forall n\ge 1.\end{array}$$

(21)

The series ${\sum }_{n\ge 1}\frac{1}{\sqrt{k_n+1}}$ converges because $q<p^{*}$ and $k_n\to +\infty$ as $n\to \infty$. Consequently, we can obtain (19) letting $n\to \infty$ in (21).

It remains to handle the case when the number $k_0$ is such that $\parallel u^{+}{\parallel }_{L^{(k_0+1)q}(\Omega )}\le 1$ and $\parallel u^{+}{\parallel }_{L^{(k+1)q}(\Omega )}>1$ for all $k>k_0$. In this case, the Moser iteration reads as $(k_n+1)q=(k_{n-1}+1)p^{*}$ with the initial step $(k_1+1)q=k_0$ if $k_0<p^{*}$ and $(k_1+1)q=p^{*}$ if $k_0\ge p^{*}$. In any case, we are led to (21) from which (19) can be established as before.

We can pass to the limit as $d\to \infty$ in (19) obtaining $\parallel u^{+}{\parallel }_{L^{\infty }(\Omega )}\le C$ for each weak solution $u\in W_0^{1,p}(\Omega )$ to problem (1). Analogously, we can prove that $\parallel u^{-}{\parallel }_{L^{\infty }(\Omega )}\le C$ for all weak solutions $u\in W_0^{1,p}(\Omega )$ to problem (1). Altogether, we have the uniform bound ${\parallel u\parallel }_{L^{\infty }(\Omega )}\le C$ for the solution set of problem (1).

A careful reading of the above proof reveals the dependence of the uniform bound C on the data in problem (1) and on the coefficients, entering assumptions (H1) and (H2). Precisely, we have to check how the constants b, q, K, $c_0$, $C_1$, and $C_0$ arising in the proof depend on the data given in (1), (H1), and (H2). Collecting all these renders the dependence $C=C(N,p,\Omega ,a_0,c_1,c_2,c_3,d_1,d_2,\parallel \sigma {\parallel }_{L^1(\Omega )})$. This completes the proof. □

3. Truncation Problem and Proof of Theorem 1

The method of proof relies on the truncation of the coefficient $G\left(u\right)$ of the p-Laplacian in problem (1) to drop its unboundedness. This idea was used in [2] in the context of the degenerate p-Laplacian. Specifically, for any number $R>0$, we introduce the truncation

$$G_R\left(t\right)=\left\{\begin{array}{cc}G\left(t\right)\hfill & \mathrm{if}|t|\le R\hfill \\ G\left(R\right)\hfill & \mathrm{if}t>R\hfill \\ G(-R)\hfill & \mathrm{if}t<-R.\hfill \end{array}\right.$$

(22)

By (22), we obtain a continuous function $G_R:\mathbb{R}\to [a_0,+\infty )$. We also consider the associated operator ${\mathcal{A}}_R:W_0^{1,p}(\Omega )\to W_0^{-1,p^{\prime }}(\Omega )$ given by

$$⟨{\mathcal{A}}_R\left(u\right),v⟩={\int }_{\Omega }{G}_R\left(u\right){|\nabla u|}^{p-2}\nabla u\nabla vdx,\forall u,v\in W_0^{1,p}(\Omega ).$$

(23)

The notation $G_R\left(u\right)|$ in Equation (23) means the composition of the functions $G_R:\mathbb{R}\to \mathbb{R}$ and $u:\Omega \to \mathbb{R}$, that is $G_R\left(u\right)\left(x\right)=G_R\left(u\left(x\right)\right)$ for $x\in \Omega$. The next proposition discusses the properties of ${\mathcal{A}}_R$.

Proposition 1. 

The nonlinear operator ${\mathcal{A}}_R$ in (23) is well defined, bounded (i.e., it maps bounded sets into bounded sets), continuous, and satisfies the $S_{+}$ property, that is, any sequence $\left\{u_n\right\}\subset W_0^{1,p}(\Omega )$ with $u_n\rightharpoonup u$ in $W_0^{1,p}(\Omega )$ and

$$\begin{array}{c}\hfill \underset{n\to \infty }{lim\; sup}⟨{\mathcal{A}}_R\left(u_n\right),u_n-u⟩\le 0\end{array}$$

(24)

fulfills $u_n\to u$ in $W_0^{1,p}(\Omega )$.

Proof. 

The continuity of the function G combined with (22), (23), and Hölder’s inequality ensures

$$\begin{array}{c}\hfill |⟨{\mathcal{A}}_R\left(u\right),v⟩|\le \underset{t\in [-R,R]}{max}{G\left(t\right)\parallel u\parallel }^{p-1}\parallel v\parallel .\end{array}$$

for all $u,v\in W_0^{1,p}(\Omega )$. It follows that the operator ${\mathcal{A}}_R$ is well-defined and bounded.

In order to show the continuity of ${\mathcal{A}}_R$ let $u_n\to u$ in $W_0^{1,p}(\Omega )$. By the continuity of G, (22), (23), Hölder’s inequality, and (2), we find

$$\begin{array}{ccc}& & \left|⟨{\mathcal{A}}_R\left(u_n\right)-{\mathcal{A}}_R\left(u\right),v⟩\right|\hfill \\ & & \le \left|{\int }_{\Omega }{G}_R\left(u_n\right)(|\nabla u_n{|}^{p-2}\nabla u_n-{|\nabla u|}^{p-2}\nabla u)\nabla vdx\right|\hfill \\ & & +\left|{\int }_{\Omega }(G_R\left(u_n\right)-G_R\left(u\right)){|\nabla u|}^{p-2}\nabla u\nabla vdx\right|\hfill \\ & & \le \underset{t\in [-R,R]}{max}G\left(t\right)|⟨-{\Delta }_p\left(u_n\right)-(-{\Delta }_p\left(u\right)),v⟩|\hfill \\ & & +{\left({\int }_{\Omega }|G_R\left(u_n\right)-G_R\left(u\right){|}^{\frac{p}{p-1}}{|\nabla u|}^{p}dx\right)}^{\frac{p-1}{p}}\parallel v\parallel \hfill \end{array}$$

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

$$\begin{array}{ccc}& & \parallel {\mathcal{A}}_R\left(u_n\right)-{\mathcal{A}}_R\left(u\right){\parallel }_{W^{-1,p^{\prime }}(\Omega )}\hfill \\ & & \le \underset{t\in [-R,R]}{max}G\left(t\right){\parallel -{\Delta }_p\left(u_n\right)-(-{\Delta }_p\left(u\right))\parallel }_{W^{-1,p^{\prime }}(\Omega )}\hfill \\ & & +{\left({\int }_{\Omega }|G_R\left(u_n\right)-G_R\left(u\right){|}^{\frac{p}{p-1}}{|\nabla u|}^{p}dx\right)}^{\frac{p-1}{p}}.\hfill \end{array}$$

The continuity of the p-Laplacian ${\Delta }_p$ implies that $-{\Delta }_p\left(u_n\right)\to -{\Delta }_p\left(u\right)$ in $W^{-1,p^{\prime }}(\Omega )$. By Lebesgue’s dominated convergence theorem, we derive

$$\begin{array}{c}\hfill \underset{n\to \infty }{lim}{\int }_{\Omega }|G_R\left(u_n\right)-G_R\left(u\right){|}^{\frac{p}{p-1}}{|\nabla u|}^{p}dx=0,\end{array}$$

whence ${\mathcal{A}}_R\left(u_n\right)\to {\mathcal{A}}_R\left(u\right)$ in $W^{-1,p^{\prime }}(\Omega )$, so the continuity of ${\mathcal{A}}_R$ is proven.

Now we show the $S_{+}$ property for the operator ${\mathcal{A}}_R$. Let a sequence $\left\{u_n\right\}$ satisfy $u_n\rightharpoonup u$ in $W^{1,p}(a,\Omega )$ and (24). It is seen that

$$\begin{array}{c}\hfill \underset{n\to \infty }{lim\; sup}⟨{\mathcal{A}}_R\left(u_n\right)-{\mathcal{A}}_R\left(u\right),u_n-u⟩\le 0.\end{array}$$

(25)

Taking into account (23) and the monotonicity of $-{\Delta }_p$, we have

$$\begin{array}{ccc}\hfill & & ⟨{\mathcal{A}}_R\left(u_n\right)-{\mathcal{A}}_R\left(u\right),u_n-u⟩\hfill \\ & =& {\int }_{\Omega }{G}_R\left(u_n\right)(|\nabla u_n{|}^{p-2}\nabla u_n-{|\nabla u|}^{p-2}\nabla u)\nabla (u_n-u)dx\hfill \\ & +& {\int }_{\Omega }(G_R\left(u_n\right)-G_R\left(u\right)){|\nabla u|}^{p-2}\nabla u\nabla (u_n-u)dx\hfill \\ & \ge & a_0⟨-{\Delta }_p\left(u_n\right)-(-{\Delta }_p\left(u\right)),u_n-u⟩\hfill \\ & +& {\int }_{\Omega }(G_R\left(u_n\right)-G_R\left(u\right)){|\nabla u|}^{p-2}\nabla u\nabla (u_n-u)dx.\hfill \end{array}$$

(26)

We claim that

$$\begin{array}{ccc}\hfill & & \underset{n\to \infty }{lim}{\int }_{\Omega }(G_R\left(u_n\right)-G_R\left(u\right)){|\nabla u|}^{p-2}\nabla u\nabla (u_n-u)dx=0.\hfill \end{array}$$

(27)

To this end, by Hölder’s inequality and since the sequence $\left\{u_n\right\}$ is bounded in $W_0^{1,p}(\Omega )$, we find a constant $C>0$ such that

$$\begin{array}{ccc}\hfill & & \left|{\int }_{\Omega }(G_R\left(u_n\right)-G_R\left(u\right)){|\nabla u|}^{p-2}\nabla u\nabla (u_n-u)dx\right|\hfill \\ & \le & C{\left({\int }_{\Omega }|G_R\left(u_n\right)-G_R\left(u\right){|}^{\frac{p}{p-1}}{|\nabla u|}^{p}dx\right)}^{\frac{p-1}{p}}.\hfill \end{array}$$

(28)

By Lebesgue’s dominated convergence theorem, it holds

$$\begin{array}{c}\hfill \underset{n\to \infty }{lim}{\int }_{\Omega }|G_R\left(u_n\right)-G_R\left(u\right){|}^{\frac{p}{p-1}}{|\nabla u|}^{p}dx=0.\end{array}$$

(29)

This is true because $G_R$ is continuous, $u_n\to u$ in $L^p(\Omega )$ and there is the domination

$$\begin{array}{c}\hfill |G_R\left(u_n\right)-G_R\left(u\right){|}^{\frac{p}{p-1}}{|\nabla u|}^p\le 2^{\frac{1}{p-1}}{\left(\underset{t\in [-R,R]}{max}G\left(t\right)\right)}^{\frac{p}{p-1}}{|\nabla u|}^p\in L^1(\Omega ).\end{array}$$

Then (25), (26), (27), (28), (29), and $u_n\rightharpoonup u$ in $W^{1,p}(\Omega )$ yield

$$\begin{array}{c}\hfill \underset{n\to \infty }{lim}⟨-{\Delta }_p\left(u_n\right),u_n-u⟩=0.\end{array}$$

(30)

Since it holds,

$$\begin{array}{ccc}& & \parallel u_n{\parallel }^p=⟨-{\Delta }_p\left(u_n\right),u⟩+⟨-{\Delta }_p\left(u_n\right),u_n-u⟩\hfill \\ & & \le \parallel u_n{\parallel }^{p-1}\parallel u\parallel +⟨-{\Delta }_p\left(u_n\right),u_n-u⟩,\hfill \end{array}$$

Equation (30) results in ${lim\; sup}_{n\to \infty }\parallel u_n\parallel \le \parallel u\parallel$. Recalling that space $W_0^{1,p}(\Omega )$ is uniformly convex, we conclude that $u_n\to u$ in $W_0^{1,p}(\Omega )$, which proves the $S_{+}$ property of the operator ${\mathcal{A}}_R$. The proof is thus complete. □

For any $R>0$ and the truncation $G_R$ in (22), let us consider the auxiliary problem

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

(31)

The solvability and a priori estimates for problem (31) are now studied.

Theorem 3. 

Assume that $G:[0,+\infty )\to [a_0,+\infty )$ is a continuous function with $a_0>0$, and that $F:\Omega \times \mathbb{R}\times {\mathbb{R}}^N\to \mathbb{R}$ is a Carathéodory function satisfying the conditions (H1) and (H2). Then, for every $R>0$, the auxiliary problem (31) has a weak solution $u_R\in W_0^{1,p}(\Omega )$ in the sense that

$${\int }_{\Omega }{G}_R\left(u_R\right){|\nabla u_R|}^{p-2}\nabla u_R\nabla vdx={\int }_{\Omega }F(x,u_R,\nabla u_R)vdx,\forall v\in W_0^{1,p}(\Omega ).$$

(32)

Moreover, the solution $u_R$ is uniformly bounded and fulfills the a priori estimate $\parallel u_R{\parallel }_{L^{\infty }(\Omega )}\le C$ with the constant $C=C(N,p,\Omega ,a_0,c_1,c_2,c_3,d_1,d_2,\parallel \sigma {\parallel }_{L^1(\Omega )})$ provided by Theorem 2.

Proof. 

Fix an $R>0$. In view of (23), equality (32) reads as

$$⟨{\mathcal{A}}_R\left(u_R\right),v⟩={\int }_{\Omega }F(x,u_R,\nabla u_R)vdx,\forall v\in W_0^{1,p}(\Omega ).$$

(33)

Through hypothesis (H1) and Hólder’s inequality, we find

$$\begin{array}{ccc}& & \left|{\int }_{\Omega }F(x,u,\nabla u)vdx\right|\le {\int }_{\Omega }|F(x,u,\nabla u)||v|dx\hfill \\ & & \le {\int }_{\Omega }(c_1{|\nabla u|}^{\frac{p}{r^{\prime }}}|v|+c_2{|u|}^{r-1}|v|+c_3|v|)dx\hfill \\ & & \le c_1{\parallel u\parallel }^{\frac{p}{r^{\prime }}}{\parallel v\parallel }_{L^r(\Omega )}+c_2{\parallel u\parallel }_{L^r(\Omega )}^{r-1}{\parallel v\parallel }_{L^r(\Omega )}+c_3{|\Omega |}^{\frac{1}{r^{\prime }}}{\parallel v\parallel }_{L^r(\Omega )}\hfill \end{array}$$

for all $u\in W_0^{1,p}(\Omega )$ and $v\in L^r(\Omega )$. We deduce that the mapping

$$\begin{array}{c}\hfill u\in W_0^{1,p}(\Omega )\mapsto F(·,u(·),\nabla u(·))\in L^{r^{\prime }}(\Omega )\end{array}$$

(34)

is well-defined and bounded. Furthermore, by Krasnoselskii’s theorem for Nemytskii operators, the mapping in (34) is continuous from $W_0^{1,p}(\Omega )$ to $L^{r^{\prime }}(\Omega )$, so continuous from $W_0^{1,p}(\Omega )$ to $W^{-1,p^{\prime }}(\Omega )$ due to the continuous embedding $W_0^{1,p}(\Omega )\subset L^r(\Omega )$.

Let us define the mapping ${\mathcal{B}}_R:W_0^{1,p}(\Omega )\to W^{-1,p^{\prime }}(\Omega )$ by

$$\begin{array}{c}\hfill {\mathcal{B}}_R\left(u\right)={\mathcal{A}}_R\left(u\right)-F(·,u(·),\nabla u(·)),\forall u\in W_0^{1,p}(\Omega ).\end{array}$$

(35)

On account of Proposition 1 and on what was said regarding the mapping in (34), we are entitled to assert that ${\mathcal{B}}_R:W_0^{1,p}(\Omega )\to W_0^{-1,p^{\prime }}(\Omega )$ introduced in (35) is well-defined, bounded and continuous.

The next step in the proof is to show that the mapping ${\mathcal{B}}_R:W_0^{1,p}(\Omega )\to W^{-1,p^{\prime }}(\Omega )$ is a pseodomonotone operator, which means that if $u_n\rightharpoonup u$ in $W_0^{1,p}(\Omega )$ and

$$\begin{array}{c}\hfill \underset{n\to \infty }{lim\; sup}⟨{\mathcal{B}}_R\left(u_n\right),u_n-u⟩\le 0,\end{array}$$

(36)

then

$$\begin{array}{c}\hfill ⟨{\mathcal{B}}_R\left(v\right),u-v⟩\le \underset{n\to \infty }{lim\; inf}⟨{\mathcal{B}}_R\left(u_n\right),u_n-v⟩\mathrm{for}\mathrm{all}v\in W_0^{1,p}(\Omega ).\end{array}$$

(37)

To this end, let $\left\{u_n\right\}$ be a sequence as above. By the Rellich–Kondrachov theorem, we derive from $u_n\rightharpoonup u$ in $W_0^{1,p}(\Omega )$ that $u_n\to u$ in $L^r(\Omega )$. As noted before, the sequence $\left\{F(·,u_n(·),\nabla u_n(·))\right\}$ is bounded in $L^{r^{\prime }}(\Omega )$. Therefore, we have

$$\begin{array}{c}\hfill \underset{n\to \infty }{lim}{\int }_{\Omega }F(x,u_n\left(x\right),\nabla u_n\left(x\right))(u_n\left(x\right)-u\left(x\right))dx=0.\end{array}$$

Then (36) entails that (24) holds true. As Proposition 1 guarantees that ${\mathcal{A}}_R$ has the $S_{+}$ property, we can conclude that $u_n\to u$ in $W_0^{1,p}(\Omega )$. From here, it can be readily shown (37) thanks to the continuity and boundedness properties stated in Proposition 1 and those related to (34). This amounts to saying that ${\mathcal{B}}_R$ is a pseudomonotone operator.

In the following, we prove that the operator ${\mathcal{B}}_R:W_0^{1,p}(\Omega )\to W_0^{-1,p^{\prime }}(\Omega )$ is coercive, that is

$$\underset{\parallel u\parallel \to \infty }{lim}\frac{⟨{\mathcal{B}}_R\left(u\right),u⟩}{\parallel u\parallel }=+\infty .$$

(38)

Toward this we infer from (35), (33), (22), (3), Hölder’s inequality and hypothesis (H2) that

$$\begin{array}{ccc}& & ⟨{\mathcal{B}}_R\left(u\right),u⟩={\int }_{\Omega }{G}_R\left(u\right){|\nabla u|}^{p}dx-{\int }_{\Omega }F(x,u,\nabla u)udx\hfill \\ & \ge & (a_0-d_1-d_2{\lambda }_1^{-1}){\parallel u\parallel }^p-{\parallel \sigma \parallel }_{L^1(\Omega )}\hfill \end{array}$$

for all $u\in W_0^{1,p}(\Omega )$. Since $p>1$ and $a_0-d_1-d_2{\lambda }_1^{-1}>0$ as known from hypothesis (H2), we confirm the validity of (38).

We showed on the reflexive Banach space $W_0^{1,p}(\Omega )$ that the operator ${\mathcal{B}}_R:W_0^{1,p}(\Omega )\to W_0^{-1,p^{\prime }}(\Omega )$ defined in (35) is bounded, pseudomonotone and coercive. According to the main theorem for pseudomonotone operators (see, for example, [16], Th. 2.99), we can conclude that the mapping ${\mathcal{B}}_R$ is surjective. So, in particular, there exists $u_R\in W_0^{1,p}(\Omega )$ such that ${\mathcal{B}}_R\left(u_R\right)=0$, which is exactly (32). Therefore $u_R$ is a weak solution of auxiliary problem (31).

Let us point out that the function G and its truncation $G_R$ take values in the same set $[a_0,+\infty )$, and function F is the same in both problems (1) and the (31). Consequently, Theorem 2 can be applied to the auxiliary problem (31) and provides the same uniform bound $C=C(N,p,\Omega ,a_0,c_1,c_2,c_3,d_1,d_2,\parallel \sigma {\parallel }_{L^1(\Omega )})$ of the solution set as for the original problem (1). This ensures that $\parallel u_R{\parallel }_{L^{\infty }(\Omega )}\le C$, which completes the proof. □

Relying on Theorem 3, we are now able to prove Theorem 1.

Proof of Theorem 1. 

It was established in Theorem 2 that the solution set of problem (1) is uniformly bounded by a constant $C=C(N,p,\Omega ,a_0,c_1,c_2,c_3,d_1,d_2,\parallel \sigma {\parallel }_{L^1(\Omega )})$, where $a_0$ is a lower bound of the function G. Since the truncated function $G_R$ has the lower bound $a_0$ too for all $R>0$ (see (22)) and the reaction term $F(x,t,\xi )$ is unchanged in problems (1) and (31) and is subject to the same hypotheses (H1)-(H2), Theorem 2 applies to the truncated problem (31) and provides the same bound C for its solution set whenever $R>0$. In particular, the solution $u_R\in W_0^{1,p}(\Omega )$ of problem (31) provided by Theorem 3 satisfies the estimate $\parallel u_R{\parallel }_{L^{\infty }(\Omega )}\le C$.

Owing to the crucial information that C is independent of $R>0$, we can choose $R\ge C$. Hence, the estimate $\parallel u_R{\parallel }_{L^{\infty }(\Omega )}\le C$ and (22) render that the functions $G_R$ and G coincide along the values $u_R\left(x\right)$ for all $x\in \Omega$. According to Theorem 3, $u_R\in W_0^{1,p}(\Omega )$ solves problem (31), and thus it becomes a bounded weak solution of the original problem (1). The conclusion of Theorem 1 is achieved. □