The Existence and Uniqueness of Nonlinear Elliptic Equations with General Growth in the Gradient

Abstract

In this paper, we prove the existence and uniqueness results for a weak solution to a class of Dirichlet boundary value problems whose prototype is $-{\Delta }_{p}u=\beta {|\nabla u|}^q+f \mathrm{i}\mathrm{n} \Omega , u=0 \mathrm{o}\mathrm{n} \partial \Omega ,$ where $\Omega$ is a bounded open subset of ${\mathbb{R}}^N$, $N\ge 2$, $1<p<N$, ${\Delta }_{p}u=\mathrm{div}\left({|\nabla u|}^{p-2}\nabla u\right)$, $p-1<q<p$, $\beta$ is a positive constant and f is a measurable function satisfying suitable summability conditions depending on q and a smallness condition.

1. Introduction

Let us consider the class of the homogeneous Dirichlet problems

$$\left\{\begin{array}{ccc}-\mathrm{d}\mathrm{i}\mathrm{v}\left(\mathbf{a}\left(x,\nabla u\right)\right)=H\left(x,\nabla u\right)+f\hfill & \hfill & \mathrm{in}\Omega \hfill \\ u=0\hfill & \hfill & \mathrm{on}\partial \Omega ,\hfill \end{array}\right.$$

(1)

where $\Omega$ is a bounded open subset of ${\mathbb{R}}^N$, $N\ge 2$. We assume that

$$\mathbf{a}:\Omega \times {\mathbb{R}}^N\to {\mathbb{R}}^N$$

and

$$H:\Omega \times {\mathbb{R}}^N\to \mathbb{R}$$

are Carathéodory functions which satisfy the ellipticity condition

$$\mathbf{a}\left(x,z\right)·z\ge {\left|z\right|}^p,$$

(2)

the monotonicity condition

$$\left(\mathbf{a}\left(x,z\right)-\mathbf{a}\left(x,z^{\prime }\right)\right)·\left(z-z^{\prime }\right)>0,z\ne z^{\prime },$$

(3)

and the growth conditions

$$\left|\mathbf{a}\left(x,z\right)\right|\le a_0{\left|z\right|}^{p-1}+a_1,a_0,a_1>0,$$

(4)

$$\left|H\left(x,z\right)\right|\le h{\left|z\right|}^q,h>0$$

(5)

with $1<p<N$, $p-1<q\le p$, for almost every $x\in {\mathbb{R}}^N,$ for every $z,z^{\prime }\in {\mathbb{R}}^N$, and f is in a suitable Lorentz space.

Existence of solutions to problem (1) have been extensively studied in the literature; we quote only some contributions and refer to the references therein: [1,2,3]. Existence results have been proved under suitable assumptions of the summability of the datum f and a smallness condition on its norm. Usually, one has to distinguish three intervals for q, i.e.,

$$p-1<q<{\displaystyle \frac{N(p-1)}{N-1}},$$

(6)

$${\displaystyle \frac{N(p-1)}{N-1}}\le q<p-1+{\displaystyle \frac{p}{N}},$$

(7)

$$p-1+{\displaystyle \frac{p}{N}}\le q\le p.$$

(8)

Depending on these intervals, the notion of the solution to problem (1) has to be specified. Indeed, a solution u to problem (1) is the standard weak solution when the datum f is an element of the dual space $W^{-1,p^{\prime }}(\Omega )$, such as, for example, when q satisfies (8), but the notion of a weak solution does not fit the cases when q satisfies (6) or (7). In [1], for example, the notion of “solution obtained as limit of approximations” is used ([4]; see also [5]), which is based on a delicate procedure of passage to the limit. Other equivalent notions of solutions are available in the literature, such as the renormalized solution ([6,7,8]) or entropy solution ([9]).

When dealing with the question of uniqueness, one has to consider the following well-known counterexample (see, for example, [10]) for the model problem ($p=2$)

$$\left\{\begin{array}{ccc}-\Delta u={|\nabla u|}^q\hfill & \hfill & \mathrm{in}{B}_1\left(0\right),\hfill \\ \hfill & \hfill & \\ u=0\hfill & \hfill & \mathrm{on}\partial B_1\left(0\right),\hfill \end{array}\right.$$

(9)

where $B_1\left(0\right)$ is the unit ball. It shows that uniqueness does not hold in $W_0^q\left(B_1\left(0\right)\right)$, with $q>{\displaystyle \frac{N}{N-1}}$ since, in addition to the trivial solution $u=0$, the function

$$u\left(x\right)={\mathcal{C}}_{\alpha }{\left(\right|x|}^{-\alpha }-1),\alpha ={\displaystyle \frac{2-q}{q-1}},{\mathcal{C}}_{\alpha }={\displaystyle \frac{{(N-\alpha -2)}^{\frac{1}{q-1}}}{\alpha }},$$

(10)

solves problem (9) when $N>2$. Moreover, when $1+2/N<q<2$, $u\in H_0^1\left(B_1\left(0\right)\right)$. Therefore, the uniqueness fails also in $H_0^1\left(B_1\left(0\right)\right)$.

In analogous way, it is easy to verify that uniqueness fails for problem (9) with $q={\displaystyle \frac{N}{N-1}}$, since, in addition to the trivial solution $u=0$, such a problem also has the following solution:

$$u_R\left(\right|x\left|\right)={(N-1)}^{N-1}{\int }_{\left|x\right|}^1{\displaystyle \frac{dt}{t^{N-1}{\left[log\left(\frac{R}{t}\right)\right]}^{N-1}}},$$

(11)

for every fixed $R>R_{\Omega }=1$.

In order to prove uniqueness results, elliptic operators that satisfy further standard structural conditions were considered. Monotonicity condition (3) is usually replaced by the following “strong monotonicity” condition:

$$\left(\mathbf{a}\left(x,z\right)-\mathbf{a}\left(x,z^{\prime }\right)\right)·\left(z-z^{\prime }\right)\ge \alpha {(\epsilon +\left|z\right|+\left|z^{\prime }\right|)}^{p-2}{\left|z-z^{\prime }\right|}^2,$$

(12)

for some $\alpha >0$, with $\epsilon$ non-negative and strictly positive if $p>2$. Moreover, the following local Lipschitz assumption on H is made

$$\left|H\left(x,z\right)-H\left(x,z^{\prime }\right)\right|\le \beta {\left(\left|z\right|+\left|z^{\prime }\right|\right)}^{q-1}\left|z-z^{\prime }\right|,$$

(13)

where $\beta >0$.

Uniqueness results for problem (1) have been obtained under various structural assumptions on the operators. A relatively complete account on state-of-the-art approaches can be found in [11] (see also [12,13,14,15,16,17,18,19,20,21] and references therein).

Our main interest is to investigate the uniqueness issue for solutions to problem (1) in cases that, to our knowledge, do not appear in literature. Namely, we consider the case $p\le 2$ in Section 2 under general assumptions (12) and (13) on a and H, respectively. Uniqueness is proven under sharp smallness assumptions of the data.

We also discuss (see Section 3) the case where the divergence part of the operator is the laplacian ($p=2$). Classical regularity theory proves the existence of a weak solution with a higher summability of the gradient under a suitable smallness assumption of data. Such a regularity of the solution proves the uniqueness of weak solutions.

We finally observe (see Section 4) that the solution to the problem (1) can be obtained as the limit of solutions of approximated problems where a term of the type $\epsilon \Delta$ is added. The uniqueness of solutions to approximated problems is also proven.

2. Uniqueness Result for Weak Solutions in the Case $p\le \mathbf{2}$

Let us denote

$$Qu\equiv -\mathrm{d}\mathrm{i}\mathrm{v}\left(\mathbf{a}\left(x,\nabla u\right)\right)-H\left(x,\nabla u\right),u\in W_0^{1,p}(\Omega ).$$

(14)

Using a comparison principle, we mean that if $u,v$ are weak solutions to the following respective Dirichlet problems,

$$Qu=f\mathrm{in}\Omega ,u=0\mathrm{on}\partial \Omega$$

(15)

$$Qv=g\mathrm{in}\Omega ,v=0\mathrm{on}\partial \Omega$$

(16)

with $f,g\in L^{{\left(p^{*}\right)}^{\prime }}(\Omega )$, where $p^{*}={\displaystyle \frac{Np}{N-p}}$ denotes the Sobolev conjugate exponent, and

$$f\le g\mathrm{in}{\mathcal{D}}^{\prime }(\Omega ),$$

(17)

then,

$$u\le v\mathrm{a}.\mathrm{e}.\mathrm{in}\Omega .$$

(18)

Observe that (15)–(17) and the definition of the weak solution imply

$${\int }_{\Omega }[\mathbf{a}(x,\nabla u)-\mathbf{a}(x,\nabla v)]·\nabla \phi dx-{\int }_{\Omega }[H(x,\nabla u)-H(x,\nabla v)]\phi dx\le 0$$

(19)

for all non-negative $\phi \in W_0^{1,p}(\Omega )$.

We begin by proving the following result:

Theorem 1.

Let $N\ge 2$ and p such that

$$\left\{\begin{array}{cc}{\displaystyle 1\le p<2,}\hfill & \mathrm{if}N=2\hfill \\ \\ {\displaystyle \frac{2N}{N+2}\le p\le 2,}\hfill & \mathrm{if}N\ge 3.\hfill \end{array}\right.$$

(20)

Assume (2), (4), (5), (12) and (13) with

$$p-1<q\le p-1+{\displaystyle \frac{p}{N}}.$$

(21)

Denote $u,v\in W_0^{1,p}(\Omega )$ as weak solutions to the problems (15), and (16), respectively, with $f,g\in L^{{\left(p^{*}\right)}^{\prime }}(\Omega )$, which satisfy (17) and

$${\parallel f\parallel }_{L^{{\left(p^{*}\right)}^{\prime }}(\Omega )}<K_2,{\parallel g\parallel }_{L^{{\left(p^{*}\right)}^{\prime }}(\Omega )}<K_2,$$

(22)

where

$$K_2={\displaystyle \frac{q-p+1}{q}}{\left({\omega }_N^{1/N}{\displaystyle \frac{N(q-p+1)-q}{q-p+1}}\right)}^{{\displaystyle \frac{q}{q-p+1}}}{\left({\displaystyle \frac{p-1}{\beta q}}\right)}^{{\displaystyle \frac{N(p-1)}{N(q-p+1)}}}$$

is the sharp constant. Then, (18) holds. In particular, problem (15) has an unique weak solution.

Remark 1.

We explicitly observe that ${\displaystyle \frac{N(q-p+1)}{q}}={\left(p^{*}\right)}^{\prime }$. The existence of a weak solution to problems (15) and (16) is assured by the results contained in [1], where the sharpness of the constant $K_2$ is proven.

Remark 2.

We explicitly recall that Theorem 1 improves the uniqueness result proved in [16] and [21] since it gives a larger interval of values of p for which uniqueness holds true, but its proof is essentially contained in [15].

Proof of Theorem 1.

Let us denote

$$w={(u-v)}_{+}$$

and

$$D=\left\{x\in \Omega :w\left(x\right)>0\right\}.$$

Assume that D has positive measure. Let us fix $t\in \left[0,supw\right[.$ We denote

$$w_t=\left\{\begin{array}{ccc}w-t& & \mathrm{if}w>t\\ \\ 0& & \mathrm{otherwise}\end{array}\right.$$

and we use $w_t$ as test function in (19) obtaining

$$\begin{array}{c}{\int }_{E_t}\left[\mathbf{a}(x,\nabla u)-\mathbf{a}(x,\nabla v)\right]\nabla wdx\le {\int }_{E_t}\left[H(x,\nabla u)-H(x,\nabla v)\right]w_{t}dx,\end{array}$$

(23)

where

$$E_t=\left\{x\in D:t<w\left(x\right)<supw\right\}.$$

Observe that $\nabla w=0$ a.e. on $\left\{x\in D:w\left(x\right)=supw\right\}$. By assumptions (12) and (13), we have

$$\alpha {\int }_{E_t}{\displaystyle \frac{{\left|\nabla w_t\right|}^2}{{(\left|\nabla u\right|+\left|\nabla v\right|)}^{2-p}}}dx\le \beta {\int }_{E_t}{\left(\left|\nabla u\right|+\left|\nabla v\right|\right)}^{q-1}\left|\nabla w_t\right|w_{t}dx.$$

(24)

Let us estimate the integral on the right-hand side of (24) by using Hölder inequality and Sobolev inequality. Since $p\ge {\displaystyle \frac{2N}{N+2}}$ and $q\le p-1+{\displaystyle \frac{p}{N}}$, we obtain

$$\begin{array}{cc}{\displaystyle {\int }_{E_t}}\hfill & {\displaystyle {\left(\left|\nabla u\right|+\left|\nabla v\right|\right)}^{q-1}\left|\nabla w_t\right|w_{t}dx\le {\left({\int }_{E_t}\frac{{\left|\nabla w_t\right|}^2}{{(\left|\nabla u\right|+\left|\nabla v\right|)}^{2-p}}dx\right)}^{1/2}}\hfill \\ \hfill & \\ \hfill & {\displaystyle \times {\left({\int }_{E_t}{\left(\left|\nabla u\right|+\left|\nabla v\right|\right)}^{p}dx\right)}^{\frac{2q-p}{2p}}{\left({\int }_{E_t}{\left|w_t\right|}^{p^{*}}dx\right)}^{\frac{1}{p^{*}}}{|\Omega |}^{\frac{1}{\theta }}}\hfill \\ \hfill & \\ \hfill & {\displaystyle \le C_{N,p}{\left({\int }_{E_t}\frac{{\left|\nabla w_t\right|}^2}{{(\left|\nabla u\right|+\left|\nabla v\right|)}^{2-p}}dx\right)}^{1/2}}\hfill \\ \hfill & \\ \hfill & {\displaystyle \times {\left({\int }_{E_t}{\left(\left|\nabla u\right|+\left|\nabla v\right|\right)}^{p}dx\right)}^{\frac{2q-p}{2p}}{\left({\int }_{E_t}{|\nabla w_t|}^{p}dx\right)}^{\frac{1}{p}}{|\Omega |}^{\frac{1}{\theta }},}\hfill \end{array}$$

(25)

where ${\displaystyle \frac{1}{2}}+{\displaystyle \frac{2q-p}{p}}+{\displaystyle \frac{1}{p^{*}}}+{\displaystyle \frac{1}{\theta }}=1$ and $C_{N,p}$ is the best constant in Sobolev embedding $W_0^{1,p}(\Omega )\subset L^{p^{*}}(\Omega )$. From (24) and (25) we obtain

$$\begin{array}{c}{\left({\int }_{E_t}{\displaystyle \frac{{\left|\nabla w_t\right|}^2}{{(\left|\nabla u\right|+\left|\nabla v\right|)}^{2-p}}}dx\right)}^{{\displaystyle \frac{1}{2}}}\le \\ \le C_{N,p}{\displaystyle \frac{\beta }{\alpha }}{\left({\int }_{E_t}{\left(\left|\nabla u\right|+\left|\nabla v\right|\right)}^{p}dx\right)}^{{\displaystyle \frac{2q-p}{2p}}}{\left({\int }_{E_t}{|\nabla w_t|}^{p}dx\right)}^{{\displaystyle \frac{1}{p}}}{|\Omega |}^{{\displaystyle \frac{1}{\theta }}}.\end{array}$$

(26)

On the other hand, since $p\le 2,$ if $N\ge 3$ or $p<2$ if $N=2$, Hölder inequality gives

$$\begin{array}{c}\hfill {\int }_{E_t}{\left|\nabla w_t\right|}^{p}dx\le {\int }_{E_t}{\displaystyle \frac{{\left|\nabla w_t\right|}^p}{{(\left|\nabla u\right|+\left|\nabla v\right|)}^{\left(2-p\right)\frac{p}{2}}}}{\left(\left|\nabla u\right|+\left|\nabla v\right|\right)}^{\left(2-p\right){\displaystyle \frac{p}{2}}}dx\\ \hfill \le {\left({\int }_{E_t}{\displaystyle \frac{{\left|\nabla w_t\right|}^2}{{(\left|\nabla u\right|+\left|\nabla v\right|)}^{2-p}}}dx\right)}^{{\displaystyle \frac{p}{2}}}{\left({\int }_{E_t}{\left(\left|\nabla u\right|+\left|\nabla v\right|\right)}^{p}dx\right)}^{{\displaystyle \frac{2-p}{2}}}.\end{array}$$

Then, by using (26) we obtain

$$1\le {\displaystyle \frac{{\beta }^p{C}_{N,p}^p}{{\alpha }^p}}{|\Omega |}^{{\displaystyle \frac{p}{\theta }}}{\left({\int }_{E_t}{\left(\left|\nabla u\right|+\left|\nabla v\right|\right)}^{p}dx\right)}^{q-p+1}.$$

Letting $t\to supw$ on the right-hand side gives a value of zero; this creates a contradiction. Therefore, we conclude that $\left|D\right|=0$ and we obtain the assert. □

Remark 3.

Let us explicitly observe that in the case where $p=N=2$, the proof can be easily adapted and the interval of values of q for which uniqueness holds true coincides with the whole interval $(1,2)$.

Remark 4.

Let us explicitly observe that this approach does not prove a comparison result when $p>2$ and q satisfies $q>p-1+{\displaystyle \frac{p}{N}}$ since it would require a higher summability of $|\nabla u|$, which is not natural for a weak solution u. The same also occurs in [21].

3. Uniqueness for Weak Solutions in the Case of Laplacian Operator

Let us consider the class of homogeneous Dirichlet problems

$$\left\{\begin{array}{ccc}-\Delta u=H\left(x,\nabla u\right)+f\hfill & \hfill & \mathrm{in}\Omega \hfill \\ u=0\hfill & \hfill & \mathrm{on}\partial \Omega ,\hfill \end{array}\right.$$

(27)

where the function $H\left(x,z\right)$ satisfies condition (5) with $N>2$, $1+{\displaystyle \frac{2}{N}}\le q<2$ and

$$f\in L^{{\displaystyle \frac{N(q-1)}{q}}}(\Omega ).$$

In this section, we assume that $\Omega$ is a $C^{1,1}$ domain. It is well known that, under a suitable smallness assumption on the datum f, the existence of a weak solution belonging to $H_0^1(\Omega )$ has been proved (see [1,3]).

3.1. Existence

We begin by proving the existence of a weak solution belonging to the Sobolev space $W_0^{1,N(q-1)}(\Omega )$, which is smaller than $H_0^1(\Omega )$, and therefore has a more regular gradient.

The following existing result holds true:

Theorem 2.

Let $\Omega \subset {\mathbb{R}}^N$, $N\ge 3$, be a $C^{1,1}$ domain. Assume (5) with

$$1+{\displaystyle \frac{2}{N}}\le q<2.$$

(28)

If $f\in L^{{\displaystyle \frac{N(q-1)}{q}}}(\Omega )$ satisfies the following smallness condition

$${\parallel f\parallel }_{L^{{\displaystyle \frac{N(q-1)}{q}}}(\Omega )}\le {\displaystyle \frac{q-1}{h^{\frac{1}{q-1}}{q}^{\frac{q}{q-1}}}}{\left({\displaystyle \frac{1}{S_{N,q}\sqrt{2}}}{\left({\displaystyle \frac{2}{N}}\right)}^{{\displaystyle \frac{1}{N(q-1)}}}\right)}^{{\displaystyle \frac{q}{q-1}}},$$

(29)

where $S_{N,q}$ is the best constant in the Sobolev embedding $W_0^{1,{\left(N(q-1)\right)}^{\prime }}(\Omega )$ in $L^{{\left[{\left(N(q-1)\right)}^{\prime }\right]}^{*}}(\Omega )$, then at least a weak solution to problem (27) u belonging to the Sobolev space $W_0^{1,N(p-1)}(\Omega )$ exists such that

$${\parallel \nabla u\parallel }_{L^{N(q-1)}(\Omega )}<C,$$

(30)

where C is a positive constant depending only on q, N, h and ${\parallel f\parallel }_{L^{{\displaystyle \frac{N(q-1)}{q}}}(\Omega )}$.

3.2. A Priori Estimates

We consider the following sequence of approximated problems

$$\left\{\begin{array}{ccc}-\Delta u_n=T_n\left(H\left(x,\nabla u_n\right)\right)+f\hfill & \hfill & \mathrm{in}\Omega \hfill \\ u_n=0\hfill & \hfill & \mathrm{on}\partial \Omega ,\hfill \end{array}\right.$$

(31)

Here, for any $s\in \mathbb{R}$ and any $k>0$, we define

$$T_k\left(s\right)=max\{-k,min\{s,k\}\}.$$

(32)

Classical results assure the existence of a weak solution $u_n\in H_0^1(\Omega )\cap L^{\infty }(\Omega )$, i.e.,

$${\int }_{\Omega }\nabla u_n·\nabla \phi dx={\int }_{\Omega }{T}_n\left(H(x,\nabla u_n)\right)\phi dx+{\int }_{\Omega }f\phi dx,$$

(33)

for any $\phi \in H_0^1(\Omega )$.

Moreover, since $T_n\left(H(x,\nabla u)\right)\in L^{\infty }(\Omega )$ and $f\in L^{{\displaystyle \frac{N(q-1)}{q}}}(\Omega )$, then the weak solution u belongs to the Sobolev space $W^{2,{\displaystyle \frac{N(q-1)}{q}}}(\Omega )$ (see, for example, [22]) and, by Sobolev embedding theorem, in $W^{1,N(q-1)}(\Omega )$, where $N(q-1)={\left({\displaystyle \frac{N(q-1)}{q}}\right)}^{*}$.

Now, we prove the following a priori estimates

Lemma 1.

Under the assumptions of Theorem 2, the following a priori estimates holds true

$$\parallel \nabla u_n{\parallel }_{L^{N(q-1)}(\Omega )}\le C$$

(34)

for any $n\in \mathbb{N}$, where C is a positive constant depending only on q, N, h and ${\parallel f\parallel }_{L^{{\displaystyle \frac{N(q-1)}{q}}}(\Omega )}$.

Proof of Lemma 1.

Since $u_n$ satisfies (33), by density arguments, the following equality holds true:

$${\int }_{\Omega }\nabla u_n·\nabla \phi dx={\int }_{\Omega }{T}_n\left(H(x,\nabla u_n)\right)\phi dx+{\int }_{\Omega }f\phi dx,$$

(35)

for any $\phi \in W_0^{1,{\left(N(q-1)\right)}^{\prime }}(\Omega )$.

Fix $k>0$ and denote

$$\omega =\{x\in \Omega :|u_n\left(x\right)|>k\}.$$

Consider a test function $\phi \in C_0^{\infty }\left(\omega \right)$ such that ${\parallel \phi \parallel }_{W_0^{1,{\left(N(q-1)\right)}^{\prime }}\left(\omega \right)}\le 1$.

By (35) the following equality holds true:

$${\int }_{|u_n\left(x\right)|>k}\nabla u_n·\nabla \phi dx={\int }_{|u_n\left(x\right)|>k}{T}_n\left(H(x,\nabla u_n)\right)\phi dx+{\int }_{|u_n\left(x\right)|>k}f\phi dx,$$

for any $\phi \in W_0^{1,{\left(N(q-1)\right)}^{\prime }}\left(\omega \right)$.

Taking into account the fact that $\nabla u_n\in {\left(L^{N(p-1)}\left(\omega \right)\right)}^N$, we can consider the functional

$$T\left(\phi \right)={\int }_{\omega }\nabla u_n·\nabla \phi dx,\phi \in W_0^{1,{\left(N(q-1)\right)}^{\prime }}\left(\omega \right).$$

Recall that

$$\underset{{\parallel \phi \parallel }_{W_0^{1,{\left(N(q-1)\right)}^{\prime }}\left(\omega \right)}\le 1}{sup}{\int }_{\omega }\nabla u_n·\nabla \phi dx=\underset{1\le i\le N}{max}{\parallel {\left(u_n\right)}_{x_i}\parallel }_{L^{N(q-1)}\left(\omega \right)}.$$

(36)

It is easy to verify that

$$\underset{1\le i\le N}{max}\parallel {\left(u_n\right)}_{x_i}{\parallel }_{L^{N(q-1)}\left(\omega \right)}\ge c(N,q){\parallel \nabla u_n\parallel }_{L^{N(q-1)}\left(\omega \right)},$$

(37)

where

$$c(N,q)={\displaystyle \frac{1}{\sqrt{2}}}{\left({\displaystyle \frac{2}{N}}\right)}^{{\displaystyle \frac{1}{N(q-1)}}}.$$

Moreover, using Hölder inequality, since

$${\displaystyle \frac{q}{N(q-1)}}+{\displaystyle \frac{1}{{\left[{\left(N(q-1)\right)}^{\prime }\right]}^{*}}}=1,$$

$\phi \in W_0^{1,{\left(N(q-1)\right)}^{\prime }}\left(\omega \right)\subset L^{{\left[{\left(N(q-1)\right)}^{\prime }\right]}^{*}}\left(\omega \right)$ and ${\parallel \phi \parallel }_{W_0^{1,{\left(N(q-1)\right)}^{\prime }}\left(\omega \right)}\le 1$, we obtain

$$\begin{array}{cc}{\displaystyle {\int }_{|u_n\left(x\right)|>k}{T}_n\left(H(x,\nabla u_n)\right)\left|\phi \right|dx}\hfill & {\displaystyle \le h{\left({\int }_{|u_n\left(x\right)|>k}{|\nabla u_n|}^{N(q-1)}dx\right)}^{\frac{q}{N(q-1)}}\times }\hfill \\ \hfill & \\ \hfill & \times {\left({\displaystyle {\int }_{|u_n\left(x\right)|>k}{\left|\phi \right|}^{\frac{N(q-1)}{N(q-1)-q}}dx}\right)}^{{\displaystyle \frac{N(q-1)-q}{N(q-1)}}}\hfill \\ \hfill & \\ \hfill & {\displaystyle \le h{S}_{N,q}{\left({\int }_{|u_n\left(x\right)|>k}{|\nabla u_n|}^{N(q-1)}dx\right)}^{\frac{q}{N(q-1)}},}\hfill \end{array}$$

(38)

where $S_{N,q}$ is the best constant of the Sobolev embedding $W_0^{1,{\left(N(q-1)\right)}^{\prime }}(\Omega )\subset L^{{\left({\left(N(q-1)\right)}^{\prime }\right)}^{*}}(\Omega )$. In analogous way, we have

$$\left|{\int }_{|u_n\left(x\right)|>k}f\phi dx\right|\le {\int }_{\Omega }\left|f\right|\left|\phi \right|dx\le S_{N,q}{\left({\int }_{\Omega }{\left|f\right|}^{{\displaystyle \frac{N(q-1)}{q}}}dx\right)}^{{\displaystyle \frac{q}{N(q-1)}}}.$$

(39)

Collecting (35)–(39), we obtain

$$c(N,q)\parallel \nabla u_n{\parallel }_{L^{N(q-1)}\left(\right|u_n\left(x\right)|>k)}\le h{S}_{N,q}\parallel \nabla u_n{\parallel }_{L^{N(q-1)}\left(\right|u_n\left(x\right)|>k)}^q+S_{N,q}{\parallel f\parallel }_{L^{{\displaystyle \frac{N(q-1)}{q}}}(\Omega )}.$$

(40)

Since ${\parallel f\parallel }_{L^{{\displaystyle \frac{N(q-1)}{q}}}(\Omega )}$ satisfies the smallness condition (29), the inequality of (40) implies

$$\forall k>0,\parallel \nabla u_n{\parallel }_{L^{N(q-1)}\left(\right|u_n\left(x\right)|>k)}\le Z_1,\mathrm{or}{\parallel \nabla u_n\parallel }_{L^{N(q-1)}\left(\right|u_n\left(x\right)|>k)}\ge Z_2,$$

(41)

where $Z_1<Z_2$ are the two positive zeros of the function

$$F\left(\sigma \right)=h{S}_{N,q}{\sigma }^q-c(N,q)\sigma +S_{N,q}{\parallel f\parallel }_{L^{{\displaystyle \frac{N(q-1)}{q}}}(\Omega )},\sigma >0.$$

Now, when k goes to $+\infty$, $\parallel \nabla u_n{\parallel }_{L^{N(q-1)}(u_n\left(x\right)>k)}$ tends to zero; therefore, by the continuity of the function

$$k\to \parallel \nabla u_n{\parallel }_{L^{N(q-1)}\left(\right\{|u_n\left(x\right)|>k\})},$$

we conclude that for any k,

$$\parallel \nabla u_n{\parallel }_{L^{N(q-1)}\left(\right\{|u_n\left(x\right)|>k\})}<Z_1.$$

(42)

By choosing $k=0$, (34) is obtained. □

3.3. Proof of Theorem 2

The proof is based on a well-known procedure of passage to the limit. We repeat it here for the sake of completeness. We consider a weak solution $u_n\in W_0^{1,2}(\Omega )\cap L^{\infty }(\Omega )$ to the approximate problem (27). By the a priori estimates obtained in Lemma 1 we deduce that $|\nabla u_n{|}^q$ is bounded in $L^{{\displaystyle \frac{N(q-1)}{q}}}(\Omega )$. Therefore by growth assumption (5) on H we deduce that $T_n\left(H(x,\nabla u_n)\right)$ is bounded in $L^{{\displaystyle \frac{N(q-1)}{q}}}(\Omega )$. Moreover, for every fixed $k>0$, $T_k\left(u_n\right)$ can be used as test function in the usual weak formulation of (31) and we get

$${\int }_{\Omega }{|\nabla T_k\left(u_n\right)|}^{2}dx\le k{\int }_{\Omega }[T_n\left(H(x,\nabla u_n)\right)+f]dx.$$

(43)

This implies that $T_k\left(u_n\right)$ is bounded in $W_0^{1,2}(\Omega )$, for every $k>0$. Since the right-hand side in (31) is bounded in $L^1(\Omega )$, we can apply a well-known compactness result (see [23]), which implies that a function u exists such that, up to extracting a subsequence,

$$u_n\to u\mathrm{and}\nabla u_n\to \nabla u\mathrm{a}.\mathrm{e}.\mathrm{in}\Omega$$

(44)

with $u\in L^{{\left(N(q-1)\right)}^{*}}(\Omega )$ and $|\nabla u|\in L^2(\Omega )$.

We deduce that $T_n\left(H(x,\nabla u_n)\right)+f$ converges pointwise to $H(x,\nabla u)+f$. Using Vitali’s theorem, we can pass to the limit in the weak formulation of the approximate problem (31), i.e.,

$${\int }_{\Omega }\nabla u_n·\nabla \varphi dx={\int }_{\Omega }{T}_n\left(H(x,\nabla u_n)\right)\varphi dx+{\int }_{\Omega }f\left(x\right)\varphi dx,$$

for any $\varphi \in W_0^{1,p}(\Omega )$ and we find that u is a weak solution to problem (27), i.e.,

$${\int }_{\Omega }\nabla u·\nabla \varphi dx={\int }_{\Omega }H(x,\nabla u))\varphi dx+{\int }_{\Omega }f\left(x\right)\varphi dx.$$

□

3.4. Uniqueness

In this section, we prove n uniqueness result for problem (27). The following uniqueness result holds true:

Theorem 3.

Under the assumption of Theorem 2, there exists a unique weak solution to problem (27) u belonging to the Sobolev space $W^{1,N(q-1)}(\Omega )$, such that (30) holds true.

Proof of  Theorem 3.

The proof proceeds by adapting the proof of Theorem 1. Let us denote

$$w={(u-v)}_{+}$$

and

$$D=\left\{x\in \Omega :w\left(x\right)>0\right\}.$$

Assume that D has a positive measure. Let us fix $t\in \left[0,supw\right[.$ We denote

$$w_t=\left\{\begin{array}{ccc}w-t& & \mathrm{if}w>t\\ \\ 0& & \mathrm{otherwise}\end{array}\right.$$

and we use $w_t$ as test function in (51) obtaining

$$\begin{array}{c}{\int }_{E_t}(\nabla u-\nabla v)\nabla wdx\le {\int }_{E_t}\left[H(x,\nabla u)-H(x,\nabla v)\right]w_{t}dx,\end{array}$$

(45)

where

$$E_t=\left\{x\in D:t<w\left(x\right)<supw\right\}.$$

Observe that $\nabla w=0$ a.e. on $\left\{x\in D:w\left(x\right)=supw\right\}$. By assumption (13), we have

$${\int }_{E_t}{|\nabla w_t|}^{2}dx\le \beta {\int }_{E_t}{\left(\left|\nabla u\right|+\left|\nabla v\right|\right)}^{q-1}\left|\nabla w_t\right|w_{t}dx.$$

(46)

Let us estimate the integral on the right-hand side of (46) by using Hölder inequality and Sobolev inequality. Since ${\displaystyle \frac{1}{2^{*}}}+{\displaystyle \frac{1}{N}}+{\displaystyle \frac{1}{2}}=1$, we obtain

$$\begin{array}{cc}{\displaystyle {\int }_{E_t}}\hfill & {\displaystyle {\left(\left|\nabla u\right|+\left|\nabla v\right|\right)}^{q-1}\left|\nabla w_t\right|w_{t}dx}\hfill \\ \hfill & {\displaystyle \le {\left({\displaystyle {\int }_{E_t}{\left|\nabla w_t\right|}^{2}dx}\right)}^{1/2}{\left({\int }_{E_t}{\left(\left|\nabla u\right|+\left|\nabla v\right|\right)}^{N(q-1)}dx\right)}^{\frac{1}{N}}{\left({\int }_{E_t}{\left|w_t\right|}^{2^{*}}dx\right)}^{\frac{1}{2^{*}}}}\hfill \\ \hfill & \\ \hfill & {\displaystyle \le C_{N,2}{\left({\int }_{E_t}{\left(\left|\nabla u\right|+\left|\nabla v\right|\right)}^{N(q-1)}dx\right)}^{\frac{1}{N}}{\int }_{E_t}{\left|\nabla w_t\right|}^{2}dx,}\hfill \end{array}$$

(47)

where $C_{N,2}$ is the best constant in Sobolev embedding $W_0^{1,2}(\Omega )\subset L^{2^{*}}(\Omega )$. Therefore, by (46) and (47), we obtain

$$1\le {\displaystyle \frac{\beta C_{N,p}}{\alpha }}{\left({\int }_{E_t}{\left(\left|\nabla u\right|+\left|\nabla v\right|\right)}^{N(q-1)}dx\right)}^{{\displaystyle \frac{1}{N}}}.$$

Letting $t\to supw$, the left-hand side goes to zero; this gives a contradiction. Hence we conclude that $\left|D\right|=0$ and we get the assert. □

4. Comparison Principle for Weak Solutions in the Case $p\ge \mathbf{2}$

For fixed $\epsilon >0$, let us denote

$${\tilde{Q}}_{\epsilon }u\equiv -\epsilon \Delta u_{\epsilon }-\mathrm{div}\left(\mathbf{a}\left(x,\nabla u_{\epsilon }\right)\right)-H\left(x,\nabla u_{\epsilon }\right),u_{\epsilon }\in W_0^{1,p}(\Omega ).$$

(48)

Assume that $u_{\epsilon },v_{\epsilon }$ are weak solutions to the following Dirichlet problems, respectively.

$${\tilde{Q}}_{\epsilon }{u}_{\epsilon }=f\mathrm{in}\Omega ,u_{\epsilon }=0\mathrm{on}\partial \Omega$$

(49)

$${\tilde{Q}}_{\epsilon }{v}_{\epsilon }=g\mathrm{in}\Omega ,v_{\epsilon }=0\mathrm{on}\partial \Omega$$

(50)

with $f,g\in L^{{\left(p^{*}\right)}^{\prime }}(\Omega )$, which satisfy (17).

Observe that (49), (50) and (17) and definition of weak solution, imply

$$\epsilon {\int }_{\Omega }(\nabla u_{\epsilon }-\nabla v_{\epsilon })·\nabla \phi dx+{\int }_{\Omega }[\mathbf{a}(x,\nabla u_{\epsilon })-\mathbf{a}(x,\nabla v_{\epsilon })]·\nabla \phi dx-{\int }_{\Omega }[H(x,\nabla u_{\epsilon })-H(x,\nabla v_{\epsilon })]\phi dx\le 0$$

(51)

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

Let us explicitly remark that $W_0^{1,p}(\Omega )\subset W_0^{1,2}(\Omega )$, since $p\ge 2$. Moreover, ${|\nabla v|}^q\in L^{{\displaystyle \frac{p}{q}}}(\Omega )$ with $q\le p-1+{\displaystyle \frac{p}{N}}$ and therefore $H(x,\nabla v)\in L^{{\left(p^{*}\right)}^{\prime }}(\Omega )$. Hence every term in (51) has a meaning.

We begin by proving the following result

Theorem 4.

Let $N\ge 3$ and p such that

$${\displaystyle 2\le p\le \frac{2N}{N-1}.}$$

(52)

Assume (2), (3), (4), (5) and (13) with

$$p-1<q\le 1+{\displaystyle \frac{p}{N}}(\le p-1+{\displaystyle \frac{p}{N}}.)$$

(53)

Denote $u,v\in W_0^{1,p}(\Omega )$ weak solutions to the problems (49), (50) respectively with $f,g\in L^{{\left(p^{*}\right)}^{\prime }}(\Omega )$ which satisfy

$$f\le g\mathrm{in}{\mathcal{D}}^{\prime }(\Omega ),$$

and

$${\parallel f\parallel }_{L^{{\displaystyle \frac{N(q-p+1)}{q}}}(\Omega )}<K,{\parallel g\parallel }_{L^{{\displaystyle \frac{N(q-p+1)}{q}}}(\Omega )}<K,$$

(54)

for a suitable constant $K>0$. Then

$$u_{\epsilon }\le v_{\epsilon }\mathrm{a}.\mathrm{e}.\mathrm{in}\Omega .$$

(55)

In particular, problem (49) has a unique weak solution.

Remark 5.

Let us observe that Theorem 4 holds true under the assumption that the operator $\mathbf{a}(x,z)$ is a monotone operator and not a strong monotone operator.

Proof of Theorem 4.

Let us denote

$$w_{\epsilon }={(u_{\epsilon }-v_{\epsilon })}_{+}$$

and

$$D=\left\{x\in \Omega :w_{\epsilon }\left(x\right)>0\right\}.$$

Assume that D has a positive measure. Let us fix $t\in \left[0,sup{w}_{\epsilon }\right[.$ We denote

$$w_{\epsilon ,t}=\left\{\begin{array}{ccc}{w}_{\epsilon }-t& & \mathrm{if}{w}_{\epsilon }>t\\ \\ 0& & \mathrm{otherwise}\end{array}\right.$$

and

$$E_{\epsilon ,t}=\left\{x\in D:w_{\epsilon }\left(x\right)>t\right\}.$$

Using $w_{\epsilon ,t}$ as a test function in (51), we obtain

$$\begin{array}{c}\epsilon {\int }_{E_{\epsilon ,t}}{|\nabla w_{\epsilon }|}^{2}dx+{\int }_{E_{\epsilon ,t}}\left[\mathbf{a}(x,\nabla u_{\epsilon })-\mathbf{a}(x,\nabla v_{\epsilon })\right]\nabla w_{\epsilon }dx\le {\int }_{E_{\epsilon ,t}}\left[H(x,\nabla u_{\epsilon })-H(x,\nabla v_{\epsilon })\right]w_{\epsilon }dx.\end{array}$$

(56)

Using assumptions (3) and (13), we have

$$\epsilon {\int }_{E_{\epsilon ,t}}{|\nabla w_{\epsilon }|}^{2}dx\le \beta {\int }_{E_{\epsilon ,t}}{\left(\left|\nabla u_{\epsilon }\right|+\left|\nabla v_{\epsilon }\right|\right)}^{q-1}\left|\nabla w_{\epsilon }\right|w_{\epsilon }dx.$$

(57)

By using Hölder inequality, Sobolev inequality, and the bounds on q, since $N>2$, we obtain

$$\begin{array}{cc}{\displaystyle \epsilon {\int }_{E_{\epsilon ,t}}}\hfill & {\displaystyle {\left|\nabla w_{\epsilon }\right|}^{2}dx\le {\left({\displaystyle {\int }_{E_{\epsilon ,t}}{\left|\nabla w_{\epsilon }\right|}^{2}dx}\right)}^{\frac{1}{2}}}\hfill \\ \hfill & \\ \hfill & {\displaystyle \times {\left({\int }_{E_{\epsilon ,t}}{\left(\left|\nabla u_{\epsilon }\right|+\left|\nabla v_{\epsilon }\right|\right)}^{p}dx\right)}^{\frac{q-1}{p}}{\left({\int }_{E_{\epsilon ,t}}{w}_{\epsilon }^{2^{*}}dx\right)}^{\frac{1}{2^{*}}}{|\Omega |}^{\frac{1}{\theta }}}\hfill \\ \hfill & \\ \hfill & {\displaystyle \le C_{N,2}{\left({\int }_{E_{\epsilon ,t}}{\left(\left|\nabla u_{\epsilon }\right|+\left|\nabla v_{\epsilon }\right|\right)}^{p}dx\right)}^{\frac{q-1}{p}}{\int }_{E_{\epsilon ,t}}{\left|\nabla w_{\epsilon ,t}\right|}^{2}dx{|\Omega |}^{\frac{1}{\theta }}}\hfill \end{array}$$

(58)

where $C_{N,2}$ is the best constant in Sobolev embedding $W_0^{1,2}(\Omega )\subset L^{2^{*}}(\Omega )$. Therefore, we obtain

$$1\le {\displaystyle \frac{\beta C_{N,2}}{\epsilon }}{\left({\int }_{E_{\epsilon ,t}}{\left(\left|\nabla u_{\epsilon }\right|+\left|\nabla v_{\epsilon }\right|\right)}^{p}dx\right)}^{{\displaystyle \frac{q-1}{p}}}.$$

Letting $t\to sup{w}_{\epsilon }$ the right-hand side goes to zero; this gives a contradiction. Therefore, we conclude that $\left|D\right|=0$ and we obtain the assert. □

We conclude this section by observing that a weak solution u to problem (1) can be obtained as the limit of a sequence of solutions $u_{\epsilon }$ to problems (49).

Let us consider the approximate problems:

$$\left\{\begin{array}{ccc}-\epsilon \Delta u_{\epsilon ,n}-div\left(\mathbf{a}(x,\nabla u_{\epsilon ,n})\right)=T_n\left(H(x,\nabla u_{\epsilon ,n})\right)+T_n\left(f\right)\hfill & \hfill & \mathrm{in}\Omega ,\hfill \\ \hfill & \hfill & \\ u_{\epsilon ,n}=0\hfill & \hfill & \mathrm{on}\partial \Omega .\hfill \end{array}\right.$$

(59)

This problem has at least a weak solution $u_{\epsilon ,n}\in W_0^{1,p}(\Omega )\cap L^{\infty }(\Omega )$ ([24]). Proceeding as in [1], we consider the function

$$\phi \left(x\right)=\mathrm{sign}\left(u_{\epsilon ,n}\left(x\right)\right){\int }_0^{|u_{\epsilon ,n}\left(x\right)|}{\displaystyle \frac{1}{{\left[{\mu }_{\epsilon ,n}\left(t\right)\right]}^{\alpha }}}dt,$$

(60)

where ${\mu }_{\epsilon ,n}\left(t\right)$ denotes the distribution function of $u_{\epsilon ,n}$, i.e.,

$${\mu }_{\epsilon ,n}\left(t\right)=\left|\left\{x\in \Omega :\left|u_{\epsilon ,n}\left(x\right)\right|\ge t\right\}\right|,t\ge 0$$

and

$$0<\alpha <1-{\displaystyle \frac{p}{N(q-p+1)}}.$$

Since $u_{\epsilon ,n}\in W_0^{1,p}(\Omega )\cap L^{\infty }(\Omega )$, $\phi \in W_0^{1,p}(\Omega )\cap L^{\infty }(\Omega )$, and therefore it is a test function for the problem (59). Using the ellipticity condition (2) and the growth condition on H (5) we obtain

$$\epsilon {\int }_{\Omega }{\displaystyle \frac{|\nabla u_{\epsilon ,n}{|}^2}{[{\mu }_{\epsilon ,n}\left(\right|u_{\epsilon ,n}{\left(x\right)\left|\right)]}^{\alpha }}}dx+{\int }_{\Omega }{\displaystyle \frac{|\nabla u_{\epsilon ,n}{|}^p}{[{\mu }_{\epsilon ,n}\left(\right|u_{\epsilon ,n}{\left(x\right)\left|\right)]}^{\alpha }}}dx\le h{\int }_{\Omega }|\nabla u_{\epsilon ,n}{|}^q\left|\phi \right|dx+{\int }_{\Omega }\left|T_n\left(f\right)\phi \right|dx$$

(61)

and therefore

$${\int }_{\Omega }{\displaystyle \frac{|\nabla u_{\epsilon ,n}{|}^p}{[{\mu }_{\epsilon ,n}\left(\right|u_{\epsilon ,n}{\left(x\right)\left|\right)]}^{\alpha }}}dx\le h{\int }_{\Omega }|\nabla u_{\epsilon ,n}{|}^q\left|\phi \right|dx+{\int }_{\Omega }\left|T_n\left(f\right)\phi \right|dx.$$

(62)

By using the approach in [1], under the smallness assumptions (54), we obtain the a priori estimates

$$\parallel u_{\epsilon ,n}{\parallel }_{M^{{\displaystyle \frac{N(q-p+1)}{p-q}}}(\Omega )}\le C,{\parallel \nabla u_{\epsilon ,n}\parallel }_{L^p(\Omega )}\le C,$$

where $M^{{\displaystyle \frac{N(q-p+1)}{p-q}}}(\Omega )$ is a Marcinkiewicz space (We recall that, by definition, $u\in M^p(\Omega )$, $p>1$, if $\underset{t}{sup}t\mu {\left(t\right)}^{{\displaystyle \frac{1}{p}}}<+\infty$, where $\mu \left(t\right)$ denotes the distribution function of $u\left(x\right)$.), C is a positive constant depending only on q, N, $|\Omega |$, h and ${\parallel f\parallel }_{L^{{\displaystyle \frac{N(q-p+1)}{q}}}(\Omega )}$.

By usual procedure of passage to the limit, for $n\to \infty$, we can find that $u_{\epsilon }\in W_0^{1,p}(\Omega )$ is a weak solution to problem (49) and

$$\parallel \nabla u_{\epsilon }{\parallel }_{L^p(\Omega )}\le C,$$

(63)

for any $\epsilon >0$. Therefore, we deduce the existence of a function $u\in W_0^{1,p}(\Omega )\subset W_0^{1,2}(\Omega )$ such that, for a subsequence which we denote again $u_{\epsilon }$,

$$\begin{array}{cc}\hfill & u_{\epsilon }\rightharpoonup u,\mathrm{weakly}\mathrm{in}{W}_0^{1,p}(\Omega )\hfill \\ \hfill & u_{\epsilon }\to u,\mathrm{strongly}\mathrm{in}{L}^r(\Omega ),r<p^{*}\hfill \\ \hfill & u_{\epsilon }\to u,\mathrm{a}.\mathrm{e}.\mathrm{in}\Omega .\hfill \end{array}$$

This function u is a solution in the sense of distribution of problem (1), that is

$${\int }_{\Omega }\mathbf{a}(x,\nabla u)·\nabla \phi dx={\int }_{\Omega }H(x,\nabla u)\phi dx+{\int }_{\Omega }f\phi dx,$$

for any $\phi \in C_0^{\infty }(\Omega )$.

Remark 6.

We explicitly observe that, if the whole sequence $u_{\epsilon }$ converges to u as $\epsilon \to 0$, the solution u to problem (1) obtained via the approximated problems (14) is unique.