Existence, Regularity, and Uniqueness of Solutions to Some Noncoercive Nonlinear Elliptic Equations in Unbounded Domains

Abstract

In this paper, we study a noncoercive nonlinear elliptic operator with a drift term in an unbounded domain. The singular first-order term grows like $\left|E\right(x\left)\right||\nabla u|$, where $E\left(x\right)$ is a vector field belonging to a suitable Morrey-type space. Our operator arises as a stationary equation of diffusion–advection problems. We prove existence, regularity, and uniqueness theorems for a Dirichlet problem. To obtain our main results, we use the weak maximum principle and the same a priori estimates.

1. Introduction

In this paper, we study a Dirichlet boundary value problem for a noncoercive operator whose model appears in stationary diffusion–advection problems. Precisely, we consider the following Dirichlet problem:

$$\left\{\begin{array}{c}-\mathrm{div}\left(b\right(x)\nabla u)+\mu u=H(x,\nabla u)+f\left(x\right)\mathrm{in}\Omega ,\hfill \\ u\in W_0^{1,2}(\Omega ),\hfill \end{array}\right.$$

(1)

where $\Omega$ is an open unbounded subset of ${\mathbb{R}}^N$, $N>2$, $b:\Omega \to {\mathbb{R}}^{N^2}$ is a measurable matrix field such that almost every $x\in \Omega$, and for some $\alpha$, $\beta \in {\mathbb{R}}_{+}$.

$$\begin{array}{c}{\alpha \left|\xi \right|}^2\le b\left(x\right)\xi \xi ,\left|b\left(x\right)\right|\le \beta ,\forall \xi \in {\mathbb{R}}^N,\hfill \end{array}$$

(2)

$$\mu >0.$$

(3)

The drift term $H:\Omega \times {\mathbb{R}}^N\to \mathbb{R}$ is a Carathéodory function verifying

$$\left|H(x,\xi )\right|\le \left|E\left(x\right)\right|\left|\xi \right|,\forall x\in \Omega ,\forall \xi \in {\mathbb{R}}^N,$$

(4)

where $E:\Omega \to {\mathbb{R}}^N$ is a vector field such that

$$\left|E\right|\in L^2(\Omega )\cap M_0^{s,N-s}(\Omega ),\mathrm{for}\mathrm{some}s\in ]2,N].$$

(5)

In the datum, we assume that

$$f\in L^{{\left(2^{\ast }\right)}^{\prime }}(\Omega ),\mathrm{where}{\left(2^{\ast }\right)}^{\prime }=\frac{2N}{N+2}.$$

(6)

In this paper, we show the existence and regularity of a solution of problem (1) by using the spaces of a Morrey-type introduced in [1], denoted by $M^{p,\lambda }(\Omega )$. These spaces were introduced to overcome some difficulties encountered when studying variational elliptic problems in unbounded domains. More precisely, the natural decreasing inclusions among Lebesgue spaces and compactness results do not hold, and the norm in $W_0^{1,2}(\Omega )$ is not equivalent to the $L^2(\Omega )$ norm of a gradient. In Section 2, the decreasing inclusion relation and the compactness result for Morrey-type spaces are given. For the last statement, we make hypothesis (3). In a recent paper [2], problem (1) was studied assuming that

$$\left|E\right|\in L^2(\Omega )\cap M_0^N(\Omega ),$$

where $M_0^N(\Omega )$ is a generalization of Lebesgue spaces in unbounded domains. The space $M^{p,\lambda }(\Omega )$ is smaller than the class of the space $M^p(\Omega )$ of [3] and larger than the class of the spaces $L^{p,\lambda }\left({\mathbb{R}}^N\right)$ studied in [4] when $\Omega ={\mathbb{R}}^N$. Noting that, for any $0<s\le N$, it results in $M^N(\Omega )\subset M^{s,N-s}(\Omega )$, the need to study the problem in the setting of Morrey-type spaces arises quite naturally.

The uniqueness of the solution is proved if the following holds:

$$|H(x,\xi )-H(x,\eta )|\le \left|E\left(x\right)\right||\xi -\eta |\forall \xi ,\eta \in {\mathbb{R}}^N.$$

(7)

We observe that the problem is not coercive unless ${\parallel E\parallel }_{M_0^{s,N-s}(\Omega )}$ is small enough.

Our results are obtained following some nonlinear methods. The main difficulty that arises is obtaining uniform (with respect to n) a priori estimates on the solutions $u_n$ of approximating problems. Our strategy is as follows: first, we prove the existence of weak solutions of coercive nonlinear approximating problems (18). Then we show an a priori estimate using the weak maximum principle, since the method of test functions cannot be applied. Indeed, as shown in [5] (see also [6]), one cannot find a real Lipschitz continuous function $\phi$ such that using $\phi \left(u_n\right)$ as a test function provides uniform a priori estimates for the solutions of (18).

Finally, passing the limit, we obtain the existence of a weak solution of problem (1), contained in Theorem 5. We achieve the Stampacchia-type regularity results [7], which can be found in Theorem 6. The uniqueness result of the solution is given in Theorem 7.

Our problem recovers the case

$$\left\{\begin{array}{c}-\Delta u+\mu u=E\left(x\right)\nabla u+f\left(x\right)\mathrm{in}\Omega ,\hfill \\ u\in W_0^{1,2}(\Omega )\hfill \end{array}\right.$$

(8)

which appears in the stationary diffusion–advection problems. The study of (8) reflects on its dual problem.

$$\left\{\begin{array}{c}-\Delta v+\mu v=\mathrm{div}\left(E\right(x)\nabla v)+g\left(x\right)\mathrm{in}\Omega ,\hfill \\ v\in W_0^{1,2}(\Omega ).\hfill \end{array}\right.$$

(9)

The dual problem can be seen as the stationary counterpart of the Fokker–Planck equation, which appears in several contexts, such as the description of the Brownian motion of a particle (for instance, in a fluid). See also [8] for applications to mean field games. On the other hand, since we study problems dealing with diffusion–advection phenomena, it is natural to consider cases where $\Omega$ can be unbounded. If we think of a diffusing gas, ideally in applications, the gas diffuses into containers that are extremely large. In view of this, it could be useful to work in the whole ${\mathbb{R}}^N$ or in a domain that is unbounded at least along one direction. Although (8) and (9) are dual problems with different features, one can use the results on one model to deduce results for the other model, thanks to linear theory. When we consider nonlinear generalizations of such models, the approaches for these two types could be quite different. In general, results in unbounded domains are applied to studies in different fields, such as population dynamics, phase transition theory (see [9]), pseudoplastic fluids in [10], and ecology models in [11]. For the reader’s complete understanding of this topic, see also [12].

Existence and uniqueness results for problem (8) in a coercive case were studied in [13]. The dual problem of (8) in a noncoercive case was considered in [14] (see also [15]). Existence and regularity results for a noncoercive nonlinear operator that behaves like the p-Laplacian were considered in [16,17] (see also [18,19]). A related obstacle problem can be found in [20]. Regularity results for noncoercive problems in divergence form in a linear case were obtained from [21]. We refer to [22,23,24] for the study of the Dirichlet problems in an unbounded domain under various assumptions.

The existence and regularity of the solutions for a Dirichlet problem in an bounded domain have been largely studied. Here, we recall the classical works [7,25,26] regarding the study of the coercive linear problem (8), where $\left|E\right(x\left)\right|$ is assumed to belong to $L^N(\Omega )$ with a sufficiently small $L^N$-norm.

In [27], existence and regularity results were obtained for the noncoercive linear problem (8), assuming $\mu =0$ and $E\in L^N(\Omega )$.

We also quote [28], where a weaker datum f was considered. Analogous results in the nonlinear case were obtained in [6]. We refer to [29,30,31] for existence results.

2. Spaces of Morrey-Type in Unbounded Domains

We recall the definitions of the function spaces we deal with that are suitable for the study of variational problems in unbounded domains.

For this aim, let $\Omega$ be an unbounded open subset of ${\mathbb{R}}^N$, $N>2$, and $\Sigma (\Omega )$ the $\sigma$-algebra of Lebesgue measurable subsets of $\Omega$. Let $O\in \Sigma (\Omega )$, ${\chi }_O$ the characteristic function of O, $O(x,r)$ the intersection $O\cap B(x,r)$ ($x\in {\mathbb{R}}^N,r\in {\mathbb{R}}_{+}$), and $\left|O\right|$ the Lebesgue measure of O.

The class $\mathcal{D}\left(\overline{\Omega }\right)$ contains the restrictions to $\overline{\Omega }$ of functions $\zeta \in C_0^{\infty }\left({\mathbb{R}}^N\right)$, while $L_{loc}^t\left(\overline{\Omega }\right)$ denotes the class of functions $g:\Omega \to \mathbb{R}$ such that $\zeta g\in L^t(\Omega )$ for any $\zeta \in$$\mathcal{D}\left(\overline{\Omega }\right)$.

The space of Morrey-type $M^{p,\lambda }(\Omega )$, $0\le \lambda <n$ and $1\le p<+\infty$, is the set of all the functions $g\in L^p(\Omega \cap B_r)$ for each $r\in {\mathbb{R}}_{+}$ such that

$${\left|\right|g\left|\right|}_{M^{p,\lambda }(\Omega )}:=\underset{\genfrac{}{}{0pt}{}{x\in \Omega }{0<r\le 1}}{\sup }{\left|r^{-\lambda }g\right|}_{p,\Omega \cap B(x,r)}<+\infty ,$$

(10)

equipped with the norm defined above.

These functional spaces generalize the classical notion of Morrey spaces to the case of unbounded domains and are introduced and largely studied in [1]. For these spaces, different from Lebesgue spaces defined on unbounded domains, one has the decreasing inclusion

$$M^{p_0,{\lambda }_0}(\Omega )\to M^{p,\lambda }(\Omega ),\mathrm{if}p\le p_0\mathrm{and}\frac{\lambda -N}{p}\le \frac{{\lambda }_0-N}{p_0},$$

and the following inclusion holds:

$$L^{\infty }(\Omega )\subset M^{p,\lambda }(\Omega ).$$

Moreover, as observed in [32], for any $0<s\le N$,

$$M^N(\Omega )\subset M^{s,N-s}(\Omega ).$$

Indeed, if we consider

$$g\left(x\right)=\frac{1}{\left|x\right|},$$

we have $g\in M^{s,N-s}\left({\mathbb{R}}^n\right),0<s<N$, but $g\notin L_{loc}^N\left({\mathbb{R}}^n\right)$ and so $g\notin M^N\left({\mathbb{R}}^n\right).$

The space $M_0^{p,\lambda }(\Omega )$ denotes the subspace of $M^{p,\lambda }(\Omega )$ of the functions $g\in M^{p,\lambda }(\Omega )$ such that

$$\forall ϵ\in {\mathbb{R}}_{+}\exists {\nu }_{ϵ}{\ni }^{\prime }E\in \Sigma (\Omega ),|E(0,{\sigma }_{ϵ})|\le {\nu }_{ϵ}\Rightarrow {\parallel g{\chi }_E\parallel }_{M^{p,\lambda }(\Omega )}\le ϵ.$$

(11)

In the sequel, we will use the following embedding result from [33]:

Theorem 1.

If $g\in M^{p,\lambda }(\Omega )$, with $p>2$ and $\lambda =0$ if $N=2$, and $p\in ]2,N]$ and $\lambda =N-p$ if $N>2$, then the operator

$$u\in W_0^{1,2}(\Omega )\to gu\in L^2(\Omega )$$

(12)

is bounded and there exists a constant $c\in {\mathbb{R}}_{+}$ such that

$${\left|\right|gu\left|\right|}_{L^2(\Omega )}\le {c\left|\right|g\left|\right|}_{M^{p,\lambda }(\Omega )}{\left|\right|u\left|\right|}_{W^{1,2}(\Omega )},\forall u\in W_0^{1,2}(\Omega ),$$

(13)

with $c=c(N,p)$. Moreover, if $g\in M_0^{p,\lambda }(\Omega )$, then the operator in (12) is also compact.

3. Weak Maximum Principle

This section is dedicated to the weak maximum principle, which is an indispensable tool for obtaining our results. It extends the well-known one contained in [25] (see also [31]).

Theorem 2.

Assume (2)–(5). If $w\in W_0^{1,2}(\Omega )$ is such that

$${\displaystyle {\int }_{\Omega }b\left(x\right)\nabla w\nabla \phi dx+\mu {\int }_{\Omega }w\phi dx\le {\int }_{\Omega }\left|E\left(x\right)\right||\nabla w|\left|\phi \right|dx,\forall \phi \in W_0^{1,2}(\Omega ),}$$

(14)

then $w\le 0$.

Proof. 

For $k>0$, we choose $w_k={(w-k)}^{+}$ as a test function in (14), and by (3), we have

$${\displaystyle {\int }_{\Omega }b\left(x\right)\nabla w\nabla w_{k}dx+\mu {\int }_{\Omega }(w-k)w_{k}dx\le {\int }_{\Omega }\left|E\left(x\right)\right||\nabla w|w_{k}dx.}$$

Now, observe that $\nabla w=\nabla w_k$, if $w>k$; thus by (2), Hölder’s inequality and Theorem 1, we have

$$min(\alpha ,\mu ){\parallel w_k\parallel }_{W^{1,2}(\Omega )}^2\le C{\parallel E\parallel }_{M^{s,N-s}\left(E_k\right)}{\parallel w_k\parallel }_{W^{1,2}(\Omega )}^2,$$

(15)

where $E_k=\{x\in \Omega :w\left(x\right)>k,|\nabla w\left(x\right)|>0\}$ and $C=C(N,s)$ is a positive constant.

Therefore,

$$(min(\alpha ,\mu )-C{\parallel E\parallel }_{M^{s,N-s}\left(E_k\right)}){\parallel w_k\parallel }_{W^{1,2}(\Omega )}^2\le 0.$$

(16)

Now, by contradiction, let us suppose that $supw>0$ and set $M=supw$. If $M=+\infty$, then

$$\underset{k\to M}{\lim }meas\left(E_k\right)=0.$$

(17)

If M is finite, since $w\in W_0^{1,2}(\Omega )$, by the known properties of Sobolev functions (see, e.g., ref. [25]), we have $|\nabla w(x\left)\right|=0$ a.e. on $\{x\in \Omega :w(x)=M\}$; hence, we still deduce (17).

By (11) and (17), we have ${\lim }_{k\to M}{\parallel E\parallel }_{M^{s,N-s}\left(E_k\right)}=0$; hence, there exists $k_0<M$ such that ${\parallel E\parallel }_{M^{s,N-s}\left(E_k\right)}<\frac{min(\alpha ,\mu )}{C}$, for $k\ge k_0$. This, together with (16) written for $k=k_0$, implies that ${\parallel w_{k_0}\parallel }_{W^{1,2}(\Omega )}=0$ and then $w_{k_0}=0$ a.e. in $\Omega$, that is, $w\le k_0$ a.e. in $\Omega$. With $supw\le k_0<M,$ the contradiction follows. □

4. Existence Result

In order to prove the existence of a weak solution of problem (1), we start to study the existence of solutions to some approximating problems.

We consider the following coercive nonlinear approximating problems of (1):

$$\left\{\begin{array}{c}{\displaystyle -\mathrm{div}(b\left(x\right)\nabla u_n)+\mu u_n}\hfill \\ \\ {\displaystyle =\frac{H(x,\nabla u_n)}{\left(1+\frac{1}{n}|\nabla u_n|\right)\left(1+\frac{1}{n}\left|E\right|\right)}+\frac{f}{1+\frac{1}{n}\left|f\right|},}\hfill \\ \\ u_n\in W_0^{1,2}(\Omega ).\hfill \end{array}\right.$$

(18)

As usual, we first prove the existence of a solution sequence to (18) using the next surjectivity theorem (see [34]). Then, applying the weak maximum principle, we archive uniform a priori estimates for $\left\{u_n\right\}$, and finally, using the method of passing the limit, we establish the existence result for problem (1).

Theorem 3 (Surjectivity).

Let V be a reflexive and separable Banach space. If the operator $A:V\to V^{\prime }$ is

1. 

coercive, i.e.,

$$\frac{<A\left(u\right),u>}{\parallel u\parallel }\to +\infty ,\parallel u\parallel \to +\infty ;$$

2. 

pseudomonotone, i.e.,

(i) 

A is bounded (it transforms bounded sets of V in bounded sets of $V^{\prime }$);

(ii) 

if $u_n\rightharpoonup u$ weakly in V and $\underset{n\to +\infty }{\lim \sup }<A\left(u_n\right),u_n-u>\le 0$, then

${\displaystyle \underset{n\to +\infty }{\lim \inf }<A\left(u_n\right),u_n-w>\ge <A\left(u\right),u-w>}$ for all w in V.

Then A is surjective; i.e., for every f in $V^{\prime }$, there exists u in V such that $A\left(u\right)=f$.

We will use the following lemma shown in Theorem 2.1 of [34] in the case of an bounded domain, but the result holds also for unbounded domains.

Lemma 1.

Let $p>1$, $\left\{f_n\right\}$ be a sequence of functions in $L^p(\Omega )$, and f be a function in $L^p(\Omega )$. Assume that

1. 

$\left\{f_n\right\}$ is bounded in $L^p(\Omega )$;

2. 

$f_n\to f$ a.e. in Ω.

Then $f_n\rightharpoonup f$ weakly in $L^p(\Omega )$.

We also need the following lemma (see [35]).

Lemma 2.

Let $u_n,u\in W_0^{1,2}(\Omega )$. Under hypotheses (2) and (3) and if

$${\displaystyle b\left(x\right){\left(\nabla u_n-\nabla u\right)}^2+\mu {\left(u_n-u\right)}^2\to 0a.e. in \Omega ,}$$

then

$$u_n\to ua.e. in \Omega ,$$

(19)

and

$$\nabla u_n\to \nabla ua.e. in \Omega .$$

(20)

Theorem 4.

Assume (2)–(6). Then, for any $n\in \mathbb{N}$, there exists a weak solution $u_n$ of problem (18).

Proof. 

The proof is obtained for $n=1$ using the surjectivity Theorem 3. For this aim, we define the operator A as follows:

$$\begin{array}{c}{\displaystyle A:u\in W_0^{1,2}(\Omega )\to -\mathrm{div}\left(b\left(x\right)\nabla u\right)+\mu u}\\ \\ {\displaystyle -\frac{H(x,\nabla u)}{(1+|\nabla u\left|\right)(1+|E\left|\right)}-\frac{f}{1+\left|f\right|}\in W^{-1,2}(\Omega ).}\end{array}$$

First, we show that A is coercive.

Indeed, from (2)–(6), Hölder’s and Sobolev inequality, we have for every $u\in W_0^{1,2}(\Omega )$

$$\begin{array}{ll}{\displaystyle <A\left(u\right),u>}& {\displaystyle \ge \alpha {\int }_{\Omega }{|\nabla u|}^{2}dx+\mu {\int }_{\Omega }{\left|u\right|}^{2}dx-{\int }_{\Omega }\frac{\left|E\right(x\left)\right||\nabla u|}{(1+|\nabla u\left|\right)(1+|E\left|\right)}\left|u\right|dx-{\int }_{\Omega }\frac{\left|f\right|\left|u\right|}{1+\left|f\right|}dx}\\ & {\displaystyle \ge C(\parallel u{\parallel }_{W^{1,2}(\Omega )}^2-\parallel E{\parallel }_{L^2(\Omega )}\parallel u{\parallel }_{W^{1,2}(\Omega )}-\parallel f{\parallel }_{L^{{\left(2^{\ast }\right)}^{\prime }}(\Omega )}\parallel u{\parallel }_{W^{1,2}(\Omega )})}\\ & {\displaystyle \hspace{1em}\hspace{1em}=C\left({\parallel u\parallel }_{W^{1,2}(\Omega )}-{\parallel E\parallel }_{L^2(\Omega )}-{\parallel f\parallel }_{L^{{\left(2^{\ast }\right)}^{\prime }}(\Omega )}\right){\parallel u\parallel }_{W^{1,2}(\Omega )},}\end{array}$$

where the constant $C=C(N,\alpha ,\mu ).$

Now, we prove that A is pseudomonotone.

We start to show that A is bounded. Indeed, for every $u,v\in W_0^{1,2}(\Omega )$ also by the Sobolev inequality, one has

$${\displaystyle \begin{array}{c}<A\left(u\right),v>\\ \le C\left({\parallel \nabla u\parallel }_{L^2(\Omega )}{\parallel \nabla v\parallel }_{L^2(\Omega )}+{\parallel u\parallel }_{L^2(\Omega )}{\parallel v\parallel }_{L^2(\Omega )}\right.\\ \left.{\displaystyle +{\parallel E\parallel }_{L^2(\Omega )}{\parallel v\parallel }_{L^2(\Omega )}+{\parallel f\parallel }_{L^{{\left(2^{\ast }\right)}^{\prime }}(\Omega )}{\parallel v\parallel }_{L^{2^{\ast }}(\Omega )}}\right)\\ {\displaystyle \le C\left({\parallel u\parallel }_{W^{1,2}(\Omega )}+{\parallel E\parallel }_{L^2(\Omega )}+{\parallel f\parallel }_{L^{{\left(2^{\ast }\right)}^{\prime }}(\Omega )}\right){\parallel v\parallel }_{W^{1,2}(\Omega )},}\end{array}}$$

where $C=C(N,\beta ,\mu )$.

Now, we have to prove that if

$$u_n\rightharpoonup u\mathrm{weakly}\mathrm{in}{W}_0^{1,2}(\Omega )$$

(21)

and

$$\underset{n\to +\infty }{\lim \sup }<A\left(u_n\right),u_n-u>\le 0,$$

(22)

then

$$\underset{n\to +\infty }{\lim \inf }<A\left(u_n\right),u_n-w>\ge <A\left(u\right),u-w>,\forall w\in W_0^{1,2}(\Omega ).$$

(23)

Step 1. We start to archive that

$$b\left(x\right){\left(\nabla u_n-\nabla u\right)}^2+\mu {\left(u_n-u\right)}^2\to 0 \mathrm{strongly}\mathrm{in} L^1(\Omega ).$$

(24)

From (21), we have

$$\underset{n\to +\infty }{\lim }\left[{\int }_{\Omega }b\left(x\right)\nabla u\nabla (u_n-u)dx+\mu {\int }_{\Omega }u(u_n-u)dx\right]=0,$$

(25)

and so, we deduce that

$$\begin{array}{c}{\displaystyle \underset{n\to +\infty }{\lim \sup }<A\left(u_n\right),u_n-u>}\hfill \\ \\ {\displaystyle =\underset{n\to +\infty }{\lim \sup }[{\int }_{\Omega }b\left(x\right){(\nabla u_n-\nabla u)}^{2}dx+\mu {\int }_{\Omega }{(u_n-u)}^{2}dx}\hfill \\ \\ {\displaystyle -{\int }_{\Omega }\frac{H(x,\nabla u_n)}{(1+|\nabla u_n\left|\right)(1+\left|E\left(x\right)\right|)}(u_n-u)dx-{\int }_{\Omega }\frac{f}{1+\left|f\right|}(u_n-u)dx].}\hfill \end{array}$$

(26)

Combining hypothesis (5), convergence (21), and the compactness result of Theorem 1, we obtain, up to a subsequence, that

$$\left|E\left(x\right)\right|u_n\to \left|E\left(x\right)\right|u\mathrm{strongly}\mathrm{in}{L}^2(\Omega ).$$

(27)

Thus, (4) and (21), Hölder’s inequality, and (27) give

$$\begin{array}{c}{\displaystyle \underset{n\to +\infty }{\lim }{\int }_{\Omega }\frac{\left|H\right(x,\nabla u_n\left)\right|}{(1+|\nabla u_n\left|\right)(1+\left|E\left(x\right)\right|)}|u_n-u|dx}\\ \\ {\displaystyle \le \underset{n\to +\infty }{\lim }\parallel E\left(x\right)(u_n-u){\parallel }_{L^2(\Omega )}{\parallel \nabla u_n\parallel }_{L^2(\Omega )}=0.}\end{array}$$

(28)

Then it results in

$$\underset{n\to +\infty }{\lim }{\int }_{\Omega }\frac{H(x,\nabla u_n)}{(1+|\nabla u_n\left|\right)(1+\left|E\left(x\right)\right|)}(u_n-u)dx=0.$$

(29)

In view of (21), one has

$${\displaystyle \underset{n\to +\infty }{\lim }{\int }_{\Omega }\frac{f}{1+\left|f\right|}(u_n-u)dx=0.}$$

(30)

Now, combining (22), (26), (29), and (30), we deduce

$${\displaystyle \underset{n\to +\infty }{\lim \sup }\left({\int }_{\Omega }b\left(x\right){(\nabla u_n-\nabla u)}^{2}dx+\mu {\int }_{\Omega }{(u_n-u)}^{2}dx\right)\le 0.}$$

(31)

Putting together (2), (3), and (31), we have

$$\underset{n\to +\infty }{\lim }\left[{\int }_{\Omega }b\left(x\right){\left(\nabla u_n-\nabla u\right)}^{2}dx+\mu {\int }_{\Omega }{\left(u_n-u\right)}^{2}dx\right]=0,$$

(32)

and so we obtain (24).

• Step 2. Let us now prove (23).

Let $u_{n_k}$ be the subsequence of $u_n$ such that

$$\underset{n\to +\infty }{\lim \inf }<A\left(u_n\right),u_n-w>=\underset{k\to +\infty }{\lim }<A(u_{n_k}),u_{n_k}-w>.$$

(33)

In view of (24), we deduce that there exists $u_{n_{k_m}}$ such that

$$b\left(x\right){\left(\nabla u_{n_{k_m}}-\nabla u\right)}^2{(u_{n_{k_m}}-u)}^2\to 0\mathrm{a}.\mathrm{e}. \mathrm{i}\mathrm{n} \Omega .$$

Thus, from Lemma 2, we deduce that

$$\begin{array}{c}{u}_{n_{k_m}}\to u\mathrm{a}.\mathrm{e}. \mathrm{i}\mathrm{n} \Omega \hfill \\ \\ \nabla u_{n_{k_m}}\to \nabla u\mathrm{a}.\mathrm{e}.\mathrm{i}\mathrm{n} \Omega .\hfill \end{array}$$

(34)

Using (33) and the definition of A, we have

$$\underset{n\to +\infty }{\mathrm{lim\; inf}}<A\left(u_n\right),u_n-w>=\underset{m\to +\infty }{\lim }({\int }_{\Omega }b\left(x\right)|\nabla u_{n_{k_m}}{|}^{2}dx+\mu {\int }_{\Omega }{|u_{n_{k_m}}|}^{2}dx$$

$$-{\int }_{\Omega }\frac{H(x,\nabla u_{n_{k_m}})}{(1+|\nabla u_{n_{k_m}}|)(1+|E\left(x\right)|)}{u}_{n_{k_m}}dx+{\int }_{\Omega }\frac{H(x,\nabla u_{n_{k_m}})}{(1+|\nabla u_{n_{k_m}}|)(1+|E\left(x\right)|)}wdx$$

$$-{\int }_{\Omega }b\left(x\right)\nabla u_{n_{k_m}}\nabla wdx-\mu {\int }_{\Omega }{u}_{n_{k_m}}wdx-{\int }_{\Omega }\frac{f}{1+|f|}{u}_{n_{k_m}}dx+{\int }_{\Omega }\frac{f}{1+|f|}wdx).$$

Now, we pass the limit as $m\to +\infty$ on the right-hand side of the previous inequality. Concerning the first two integrals, thanks to (2), (3), and (34), we can apply Fatou’s lemma. The third and fourth integrals can be estimated using (21) and (34) and Lemma 1. For the fifth integral, observing that $H(x,\nabla u)$ is a Carathéodory function and using the second convergence in (34), we obtain

$$\frac{H(x,\nabla u_{n_{k_m}})}{(1+|\nabla u_{n_{k_m}}|)(1+|E\left(x\right)|)}\to \frac{H(x,\nabla u)}{(1+|\nabla u|)(1+|E\left(x\right)|)}\mathrm{q}.\mathrm{o}.\mathrm{in}\Omega .$$

(35)

Moreover, with

$$\frac{H(x,\nabla u_{n_{k_m}})}{(1+|\nabla u_{n_{k_m}}|)(1+|E\left(x\right)|)}\le \frac{|E\left(x\right)||\nabla u_{n_{k_m}}|}{(1+|\nabla u_{n_{k_m}}|)(1+|E\left(x\right)|)}\le |E\left(x\right)|\in L^2(\Omega ),$$

from (35) and the Lebesgue Theorem, we obtain

$$\frac{H(x,\nabla u_{n_{k_m}})}{(1+|\nabla u_{n_{k_m}}|)(1+|E\left(x\right)|)}\to \frac{H(x,\nabla u)}{(1+|\nabla u|)(1+|E\left(x\right)|)}\mathrm{strongly}\mathrm{in}{L}^2(\Omega ).$$

(36)

Hence, by (21),

$${\int }_{\Omega }\frac{H(x,\nabla u_{n_{k_m}})}{(1+|\nabla u_{n_{k_m}}|)(1+|E\left(x\right)|)}{u}_{n_{k_m}}dx\to {\int }_{\Omega }\frac{H(x,\nabla u)}{(1+|\nabla u|)(1+|E\left(x\right)|)}udx.$$

From (21) and (36), we deduce the convergences of the other integrals. It follows that

$$\begin{array}{c}\underset{n\to +\infty }{\lim \inf }<A\left(u_n\right),u_n-w>\ge {\int }_{\Omega }b\left(x\right){|\nabla u|}^{2}dx+\mu {\int }_{\Omega }{|u|}^{2}dx\\ -{\int }_{\Omega }\frac{H(x,\nabla u)}{(1+|\nabla u|)(1+|E\left(x\right)|)}udx-{\int }_{\Omega }\frac{f}{1+|f|}udx-{\int }_{\Omega }b\left(x\right)\nabla u\nabla wdx-\mu {\int }_{\Omega }uwdx\\ +{\int }_{\Omega }\frac{H(x,\nabla u)}{(1+|\nabla u|)(1+|E\left(x\right)|)}wdx+{\int }_{\Omega }\frac{f}{1+|f|}wdx=<A\left(u\right),u-w>,\end{array}$$

and the theorem is proved. □

Lemma 3.

Assume (2)–(6) and let $\left\{u_n\right\}$ be a solution sequence to problem (18). Then, there exists a constant C, independent of n, such that

$$||u_n{||}_{W^{1,2}(\Omega )}\le C.$$

(37)

Proof. 

Reasoning by contradiction, we assume that

$$\parallel u_n{\parallel }_{W^{1,2}(\Omega )}\to +\infty .$$

(38)

Putting

$$w_n=\frac{u_n}{\parallel u_n{\parallel }_{W^{1,2}(\Omega )}},$$

(39)

it follows that $\parallel w_n{\parallel }_{W^{1,2}(\Omega )}=1$, and that there exists $\overline{w}\in W_0^{1,2}(\Omega )$ such that

$$w_n\rightharpoonup \overline{w}\mathrm{weakly}\mathrm{in}{W}_0^{1,2}(\Omega ),$$

(40)

up to a subsequence not relabeled.

Dividing the variational formulation of (18) by $\parallel u_n{\parallel }_{W^{1,2}(\Omega )}$ and using (4), it results in

$$\begin{array}{c}{\displaystyle {\int }_{\Omega }b\left(x\right)\nabla w_n\nabla \phi dx+\mu {\int }_{\Omega }{w}_n\phi dx}\\ \\ {\displaystyle \le {\int }_{\Omega }\frac{|E\left(x\right)||\nabla w_n||\phi |}{(1+\frac{1}{n}|\nabla u_n|)\left(1+\frac{1}{n}|E|\right)}dx+\frac{1}{\parallel u_n{\parallel }_{W^{1,2}(\Omega )}}{\int }_{\Omega }\frac{f}{\left(1+\frac{1}{n}|f|\right)}\phi dx}\\ \\ {\displaystyle \le {\int }_{\Omega }|E\left(x\right)||\nabla w_n||\phi |dx+\frac{1}{\parallel u_n{\parallel }_{W^{1,2}(\Omega )}}{\int }_{\Omega }\frac{f}{\left(1+\frac{1}{n}|f|\right)}\phi dx,\forall \phi \in W_0^{1,2}(\Omega ).}\end{array}$$

(41)

Putting $\phi =w_n-\overline{w}$ in (41), we obtain

$$\begin{array}{c}{\displaystyle {\int }_{\Omega }b\left(x\right)\nabla w_n\nabla (w_n-\overline{w})dx+\mu {\int }_{\Omega }{w}_n(w_n-\overline{w})dx}\\ \\ {\displaystyle \le {\int }_{\Omega }|E\left(x\right)||\nabla w_n||w_n-\overline{w}|dx+\frac{1}{||u_n{||}_{W^{1,2}(\Omega )}}{\int }_{\Omega }\frac{f}{\left(1+\frac{1}{n}|f|\right)}(w_n-\overline{w})dx.}\end{array}$$

(42)

Thus,

$$\begin{array}{c}{\displaystyle {\int }_{\Omega }b\left(x\right){(\nabla w_n-\nabla \overline{w})}^{2}dx+\mu {\int }_{\Omega }{(w_n-\overline{w})}^{2}dx}\\ \\ {\displaystyle \le {\int }_{\Omega }|E\left(x\right)||\nabla w_n||w_n-\overline{w}|dx+\frac{1}{||u_n{||}_{W^{1,2}(\Omega )}}{\int }_{\Omega }\frac{f}{\left(1+\frac{1}{n}|f|\right)}(w_n-\overline{w})dx}\\ \\ {\displaystyle -\left[{\int }_{\Omega }b\left(x\right)\nabla \overline{w}\nabla (w_n-\overline{w})dx+\mu {\int }_{\Omega }\overline{w}(w_n-\overline{w})dx\right].}\end{array}$$

(43)

Repeating the reasoning for obtaining (32), we have

$${\displaystyle \underset{n\to +\infty }{\lim }{\int }_{\Omega }b\left(x\right){(\nabla w_n-\nabla \overline{w})}^{2}dx+\mu {\int }_{\Omega }{(w_n-\overline{w})}^{2}dx=0,}$$

(44)

and we deduce, up to a subsequence not relabeled, that

$$b\left(x\right){\left(\nabla w_n-\nabla w\right)}^2+\mu {\left(w_n-w\right)}^2\to 0\mathrm{a}.\mathrm{e}. \mathrm{i}\mathrm{n} \Omega .$$

(45)

Then, by Lemma 2, we obtain

$$w_n\to \overline{w}\mathrm{a}.\mathrm{e}. \mathrm{i}\mathrm{n} \Omega$$

(46)

and

$$\nabla w_n\to \nabla \overline{w}\mathrm{a}.\mathrm{e}. \mathrm{i}\mathrm{n} \Omega .$$

(47)

As a consequence of (46) and (47) and the boundedness of the $W^{1,2}$-norm of $w_n$, applying the Lebesgue Theorem and (38), we can pass the limit in (41), obtaining

$$\begin{array}{c}{\displaystyle {\int }_{\Omega }b\left(x\right)\nabla \overline{w}\nabla \phi dx+\mu {\int }_{\Omega }\overline{w}\phi dx}\\ \\ {\displaystyle \le {\int }_{\Omega }|E\left(x\right)||\nabla \overline{w}||\phi |dx,\forall \phi \in W_0^{1,2}(\Omega ).}\end{array}$$

(48)

From the previous inequality, we deduce that $\overline{w}$ satisfies the hypothesis of Proposition 2, and so $\overline{w}\le 0$. Reasoning in a similar way for $-\overline{w}$, we have $\overline{w}\ge 0$, and hence, $\overline{w}=0$. Thus, we deduce that

$$w_n\rightharpoonup 0\mathrm{weakly}\mathrm{in}{W}_0^{1,2}(\Omega ).$$

(49)

Now, putting $\phi =w_n$ in (41), from (2) and (3), we have

$$\begin{array}{c}{\displaystyle min\{\alpha ,\mu \}\parallel w_n{\parallel }_{W^{1,2}(\Omega )}^2}\\ \\ {\displaystyle \le {\int }_{\Omega }|E\left(x\right)||\nabla w_n|w_{n}dx+\frac{1}{\parallel u_n{\parallel }_{W^{1,2}(\Omega )}}{\int }_{\Omega }\frac{f}{1+\frac{1}{n}|f|}{w}_{n}dx.}\\ {\displaystyle }\end{array}$$

(50)

Thanks to (38) and (49) and the compactness result of Theorem 1, we pass the limit in (50), obtaining that $w_n\to 0\mathrm{strongly}\mathrm{in}{W}_0^{1,2}(\Omega )$. This is a contradiction since $\parallel w_n{\parallel }_{W^{1,2}(\Omega )}=1$. □

Corollary 1.

Assume (2)–(6). Let $\left\{u_n\right\}$ be a solution sequence to problem (18). Then, for any $\epsilon >0$, there exists $k_{\epsilon }$, independent of n, such that

$${\displaystyle |{\Omega }_{n,k}|\le \epsilon ,\forall k>k_{\epsilon },}$$

(51)

where ${\Omega }_{n,k}$ is defined by

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

(52)

Now, we can prove the existence result for our problem.

Theorem 5.

Assume (2)–(6). Then, there exists a $u\in W_0^{1,2}(\Omega )$ weak solution to (1).

Proof. 

Let $\left\{u_n\right\}$ be a solution sequence to (18), that is,

$$\begin{array}{c}{\displaystyle {\int }_{\Omega }b\left(x\right)\nabla u_n\nabla vdx+\mu {\int }_{\Omega }{u}_{n}vdx}\\ \\ {\displaystyle ={\int }_{\Omega }\frac{H(x,\nabla u_n)v}{(1+\frac{1}{n}|\nabla u_n|)\left(1+\frac{1}{n}|E|\right)}dx+{\int }_{\Omega }\frac{f}{1+\frac{1}{n}|f|}vdx,}\end{array}$$

(53)

for every $v\in W_0^{1,2}(\Omega )$.

• From (37), we deduce that there exists $u\in W_0^{1,2}(\Omega )$ such that, unless passing a subsequence not relabeled,

$$u_n\rightharpoonup u\mathrm{weakly}\mathrm{in}{W}_0^{1,2}(\Omega ).$$

(54)

Arguing as shown (46) and (47), we have

$$u_n\to u\mathrm{a}.\mathrm{e}. \mathrm{i}\mathrm{n} \Omega$$

(55)

and

$$\nabla u_n\to \nabla u \mathrm{a}.\mathrm{e}. \mathrm{i}\mathrm{n} \Omega .$$

(56)

By (54)–(56), the property of $H(x,\xi )$ and the Lebesgue Theorem, we can take the limit $n\to +\infty$ with respect to (53), obtaining the thesis. □

5. Regularity

The following lemma contained in [13] generalizes a well-known result proved in [7] to the case of unbounded domains.

Lemma 4.

Let G be a uniformly Lipschitz function satisfying $G\left(0\right)=0$ and $u\in W_0^{1,2}(\Omega )$. Then $G\circ u\in W_0^{1,2}(\Omega )$.

For $k\in {\mathbb{R}}_{+}$, we consider the truncation function

$$T_k\left(s\right)=\left\{\begin{array}{cc}s,\hfill & \mathrm{if}|s|\le k,\hfill \\ k\frac{s}{|s|},\hfill & \mathrm{if}|s|>k\hfill \end{array}\right.$$

(57)

and we set

$$G_k\left(s\right)=s-T_k\left(s\right).$$

(58)

For every $u\in W_0^{1,2}(\Omega )$, we define

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

(59)

As a consequence of Lemma 4, we have

Lemma 5.

Let $u\in W_0^{1,2}(\Omega )$ and $k\in {\mathbb{R}}_{+}$. The following properties hold:

$$G_k\left(u\right)=G_k\circ u\in W_0^{1,2}(\Omega ),$$

(60)

$$|G_k\left(u\right)|\le |u|,a.e. in \Omega ,$$

(61)

$$|u|\le |G_k\left(u\right)|+k,a.e. in \Omega ,$$

(62)

$$\nabla u\nabla G_k\left(u\right)={|\nabla G_k\left(u\right)|}^2, a.e. in \Omega ,$$

(63)

$$u{G}_k\left(u\right)\ge {|G_k\left(u\right)|}^2, a.e. in \Omega ,$$

(64)

$$\mathrm{supp}{G}_k\left(u\right)\subseteq {\overline{\Omega }}_k,$$

(65)

$${\left(G_k\left(u\right)\right)}_{x_i}=\left\{\begin{array}{c}{\displaystyle u_{x_i}a.e. in {\Omega }_k,}\hfill \\ 0a.e. in \Omega \setminus {\Omega }_k,i=1\dots N,\hfill \end{array}\right.$$

(66)

$$T_k\left(u\right)=T_k\circ u\in W_0^{1,2}(\Omega ),$$

(67)

$$\nabla u\nabla T_k\left(u\right)={|\nabla T_k\left(u\right)|}^2, a.e. in \Omega ,$$

(68)

$$u{T}_k\left(u\right)\ge {|T_k\left(u\right)|}^2, a.e. in \Omega ,$$

(69)

$$u\nabla T_k\left(u\right)=T_k\left(u\right)\nabla T_k\left(u\right),a.e. in \Omega .$$

(70)

To prove the boundedness result, we need the following lemma due to G. Stampacchia (see Lemma 4.1 of [7]).

Lemma 6.

Let $k_0>0$ and $\phi : [k_0,+\infty [\to {\mathbb{R}}_{+}$ be a nonincreasing function such that

$$\phi \left(h\right)\le \frac{C}{{(h-k)}^{\gamma }}{\left[\phi \left(k\right)\right]}^{\delta }\forall h>k\ge k_0,$$

(71)

where C, γ, and δ are positive constants, with $\delta >1$. For

$$d=2^{\frac{\delta }{\delta -1}}{C}^{1/\gamma }{\left[\phi \left(k_0\right)\right]}^{\frac{\delta -1}{\gamma }},$$

(72)

one has

$$\phi (k_0+d)=0.$$

(73)

Lemma 7.

Assume (2)–(5) and $f\in L^{{\left(2^{\ast }\right)}^{\prime }}(\Omega )\cap L^r(\Omega )$, with $r>\frac{N}{2}$. Let $\left\{u_n\right\}$ be a solution sequence of problem (18). Then, there exists a positive constant $C_r$, independent of n, such that

$$\parallel u_n{\parallel }_{L^{\infty }(\Omega )}\le C_r\left({\alpha ,\mu ,N,s,\parallel E\parallel }_{M^{s,N-s}(\Omega )},{\parallel f\parallel }_{L^r(\Omega )}\right).$$

(74)

Proof. 

From (60), we consider $G_k\left(u_n\right)$ as test function in the variational formulations of (18). Using (2), (3), (4), (52), and (63)–(65) and Young’s inequality, we obtain

$${\displaystyle \alpha {\int }_{\Omega }|\nabla G_k\left(u_n\right){|}^{2}dx+\mu {\int }_{\Omega }{|G_k\left(u_n\right)|}^{2}dx}$$

$${\displaystyle \le \frac{\alpha }{2}{\int }_{{\Omega }_{n,k}}|\nabla G_k\left(u_n\right){|}^{2}dx+\frac{1}{2\alpha }{\int }_{{\Omega }_{n,k}}|E\left(x\right){|}^2{|G_k\left(u_n\right)|}^{2}dx}$$

$$+{\int }_{{\Omega }_{n,k}}|f||G_k\left(u_n\right)|dx.$$

Therefore, from Theorem 1 and Hölder’s inequality, we have

$$\begin{array}{c}{\displaystyle \frac{\alpha }{2}{\int }_{\Omega }|\nabla G_k\left(u_n\right){|}^{2}dx+\mu {\int }_{\Omega }{|G_k\left(u_n\right)|}^{2}dx}\hfill \end{array}$$

$${\displaystyle \le \frac{1}{2\alpha }{\int }_{{\Omega }_{n,k}}{|E\left(x\right)|}^2|G_k\left(u_n\right){|}^{2}dx+{\int }_{{\Omega }_{n,k}}|f||G_k\left(u_n\right)|dx}$$

$${\displaystyle \le {C\parallel E\parallel }_{M^{s,N-s}\left({\Omega }_{n,k}\right)}^2\parallel G_k\left(u_n\right){\parallel }_{W^{1,2}(\Omega )}^2+{\parallel f\parallel }_{L^{{\left(2^{\ast }\right)}^{\prime }}\left({\Omega }_{n,k}\right)}{\parallel G_k\left(u_n\right)\parallel }_{L^{2^{\ast }}\left({\Omega }_{n,k}\right)},}$$

where $C=C(\alpha ,N,s)$.

Then, from (3),

$$\min \left\{\frac{\alpha }{2},\mu \right\}{\parallel G_k\left(u_n\right)\parallel }_{W_0^{1,2}(\Omega )}^2$$

$$\le {C\parallel E\parallel }_{M^{s,N-s}\left({\Omega }_{n,k}\right)}^2\parallel G_k\left(u_n\right){\parallel }_{W^{1,2}(\Omega )}^2+{\parallel f\parallel }_{L^{{\left(2^{\ast }\right)}^{\prime }}\left({\Omega }_{n,k}\right)}{\parallel G_k\left(u_n\right)\parallel }_{L^{2^{\ast }}\left({\Omega }_{n,k}\right)}.$$

(75)

Using (11) and Corollary 1, there exists $k_0\in {\mathbb{R}}_{+}$, independent of n, such that

$${\parallel E\parallel }_{M^{s,N-s}\left({\Omega }_{n,k}\right)}^2<\frac{min\{\frac{\alpha }{2},\mu \}}{C}\forall k\ge k_0.$$

(76)

From previous inequalities using the Sobolev inequality, we obtain

$$\parallel G_k\left(u_n\right){\parallel }_{L^{2^{\ast }}(\Omega )}^2\le {C\parallel f\parallel }_{L^{{\left(2^{\ast }\right)}^{\prime }}\left({\Omega }_{n,k}\right)}{\parallel G_k\left(u_n\right)\parallel }_{L^{2^{\ast }}(\Omega )},$$

where $C=C(\alpha ,\mu ,N,s,\parallel E{\parallel }_{M^{s,N-s}(\Omega )})$.

Applying Hölder’s inequality, we deduce

$$\begin{array}{c}\parallel G_k\left(u_n\right){\parallel }_{L^{2^{\ast }}(\Omega )}\le {C\parallel f\parallel }_{L^{{\left(2^{\ast }\right)}^{\prime }}\left({\Omega }_{n,k}\right)}\le {C\parallel f\parallel }_{L^r(\Omega )}{|{\Omega }_{n,k}|}^{\frac{1}{{\left(2^{\ast }\right)}^{\prime }}-\frac{1}{r}}.\hfill \end{array}$$

(77)

On the other hand, from (52) and (62), for every $h>0$, we obtain

$$\begin{array}{c}{\displaystyle h|{\Omega }_{n,h}{|}^{\frac{1}{2^{\ast }}}={\left({\int }_{{\Omega }_{n,h}}{|h|}^{2^{\ast }}dx\right)}^{\frac{1}{2^{\ast }}}\le \parallel u_n{\parallel }_{L^{2^{\ast }}\left({\Omega }_{n,h}\right)}\le \parallel G_k\left(u_n\right){\parallel }_{L^{2^{\ast }}\left({\Omega }_{n,h}\right)}+k{|{\Omega }_{n,h}|}^{\frac{1}{2^{\ast }}},}\hfill \end{array}$$

thus

$${\displaystyle (h-k)|{\Omega }_{n,h}{|}^{\frac{1}{2^{\ast }}}\le {\parallel G_k\left(u_n\right)\parallel }_{L^{2^{\ast }}\left({\Omega }_{n,h}\right)},\forall h>k.}$$

(78)

Combining (77) and (78), we have

$$|{\Omega }_{n,h}|\le C\frac{|{\Omega }_{n,k}{|}^{2^{\ast }\left(\frac{1}{{\left(2^{\ast }\right)}^{\prime }}-\frac{1}{r}\right)}}{{(h-k)}^{2^{\ast }}},\forall h>k\ge k_0,$$

with $C=C(\alpha ,\mu ,N,s,\parallel E{\parallel }_{M^{s,N-s}(\Omega )},\parallel f{\parallel }_{L^r(\Omega )})$. Now, since $r>\frac{N}{2}$, one has $2^{\ast }\left(\frac{1}{{\left(2^{\ast }\right)}^{\prime }}-\frac{1}{r}\right)>1$. Thus, from Lemma 6, we deduce that there exists $d\in {\mathbb{R}}_{+}$ such that $|{\Omega }_{k_0+d}|=0$. The (74) follows. □

The following lemma gives a further regularity result.

Lemma 8.

Assume (2)–(5) and $f\in L^1(\Omega )\cap L^m(\Omega )$. If ${\left(2^{\ast }\right)}^{\prime }<m<\frac{N}{2}$, then there exists a positive constant $C=C(\alpha ,\mu ,N,m,||E|{|}_{M^{s,N-s}(\Omega )},$ ${||E||}_{L^2(\Omega )}{,||f||}_{L^m(\Omega )}),$ independent of n, such that

$$\parallel u_n{\parallel }_{L^{m^{\ast \ast }}(\Omega )}\le C.$$

(79)

Proof. 

We prove (79) by means of three steps. For $k\in {\mathbb{R}}_{+}$, being $u_n=T_k\left(u_n\right)+G_k\left(u_n\right)$ for obtaining the estimate, we first show that the sequence $\left\{T_k\left(u_n\right)\right\}$ is bounded in $L^{m^{\ast \ast }}$, and then we prove that there exists $k_0>0$ such that the sequence $\left\{G_k\left(u_n\right)\right\}$ is bounded in $L^{m^{\ast \ast }}$, for every $k\ge k_0$. Combining these results, we conclude the proof.

• Step 1. If $|t|\le M$, for some $M>0$ and for every $\lambda >1$, the function ${|t|}^{2(\lambda -1)}t$ satisfies the hypotheses of Lemma 4. Then we can choose ${\displaystyle \frac{|T_k\left(u_n\right){|}^{2(\lambda -1)}{T}_k\left(u_n\right)}{2\lambda -1},}$ with $\lambda =\frac{m^{\ast \ast }}{2^{\ast }}$, as a test function in the variational formulation of problem (18).
• Then, by (2), (4), (52), (68), and (70) and Young’s inequality, we have

$$\begin{array}{c}{\displaystyle \alpha {\int }_{\Omega }|T_k\left(u_n\right){|}^{2(\lambda -1)}|\nabla T_k\left(u_n\right){|}^{2}dx+\frac{\mu }{2\lambda -1}{\int }_{\Omega }{|T_k\left(u_n\right)|}^{2\lambda }dx}\hfill \\ \\ {\displaystyle \le \frac{1}{2\lambda -1}{\int }_{\Omega }|T_k\left(u_n\right){|}^{2\lambda -1}|E\left(x\right)||\nabla u_n|dx+\frac{1}{2\lambda -1}{\int }_{\Omega }|f||T_k\left(u_n\right){|}^{2\lambda -1}dx}\hfill \\ \\ {\displaystyle \le \frac{1}{2\lambda -1}{\int }_{\Omega \setminus {\Omega }_{n,k}}|T_k\left(u_n\right){|}^{2\lambda -1}|E\left(x\right)||\nabla T_k\left(u_n\right)|dx}\hfill \\ \\ {\displaystyle +\frac{1}{2\lambda -1}{\int }_{{\Omega }_{n,k}}|T_k\left(u_n\right){|}^{2\lambda -1}|E\left(x\right)||\nabla G_k\left(u_n\right)|dx}\hfill \\ \\ {\displaystyle +\frac{k^{2\lambda -1}}{2\lambda -1}{\int }_{\Omega }|f|dx.}\hfill \end{array}$$

(80)

From Young’s inequality, we obtain

$$\begin{array}{c}{\displaystyle {\int }_{\Omega \setminus {\Omega }_{n,k}}|T_k\left(u_n\right){|}^{2\lambda -1}|E\left(x\right)||\nabla T_k\left(u_n\right)|dx}\hfill \\ {\displaystyle \le ϵ{\int }_{\Omega \setminus {\Omega }_{n,k}}|T_k\left(u_n\right){|}^{2(\lambda -1)}|\nabla T_k\left(u_n\right){|}^{2}dx+C\left(ϵ\right){\int }_{\Omega \setminus {\Omega }_{n,k}}|E\left(x\right){|}^2{|T_k\left(u_n\right)|}^{2\lambda }dx,}\hfill \end{array}$$

(81)

with $\epsilon =\epsilon (\alpha ,\lambda )$.

Combining (3), (80), and (81), we obtain

$$\begin{array}{c}{\displaystyle {\int }_{\Omega }|T_k\left(u_n\right){|}^{2(\lambda -1)}{|\nabla T_k\left(u_n\right)|}^{2}dx}\hfill \\ \\ {\displaystyle \le C\left\{k^{2\lambda }{\int }_{\Omega \setminus {\Omega }_{n,k}}{|E\left(x\right)|}^{2}dx+{\int }_{{\Omega }_{n,k}}|T_k\left(u_n\right){|}^{2\lambda -1}|E\left(x\right)||\nabla G_k\left(u_n\right)|dx+k^{2\lambda -1}{\int }_{\Omega }|f|dx\right\},}\hfill \end{array}$$

(82)

with $C=C(\alpha ,\lambda )$. Using the Sobolev inequality, we deduce

$$\begin{array}{c}{\displaystyle {\left({\int }_{\Omega }{|T_k\left(u_n\right)|}^{\lambda 2^{\ast }}dx\right)}^{\frac{2}{2^{\ast }}}\le C_S{\int }_{\Omega }|\nabla (|T_k\left(u_n\right){|}^{\lambda }{)|}^{2}dx}\hfill \\ {\displaystyle =C_S{\int }_{\Omega }|T_k\left(u_n\right){|}^{2(\lambda -1)}{|\nabla T_k\left(u_n\right)|}^{2}dx,}\hfill \end{array}$$

(83)

where $C_S$ denotes the Sobolev constant. Putting together (82) and (83), by Hölder’s inequality and (37), we obtain

$$\begin{array}{c}{\displaystyle {\left({\int }_{\Omega }{|T_k\left(u_n\right)|}^{m^{\ast \ast }}dx\right)}^{\frac{2}{2^{\ast }}}\le C\left(k^{2\lambda }{\int }_{\Omega }{|E\left(x\right)|}^{2}dx+k^{2\lambda -1}{\parallel E\parallel }_{L^2(\Omega )}+k^{2\lambda -1}{\int }_{\Omega }|f|dx\right),}\hfill \end{array}$$

(84)

with a C positive constant independent of n.

• Step 2. For Lemma 7 and Lemma 4, ${\displaystyle \frac{|G_k\left(u_n\right){|}^{2(\lambda -1)}{G}_k\left(u_n\right)}{2\lambda -1}sign\left(\left(u_n\right)\right)}$ can be considered as test function in the variational formulation of problem (18). Combining (2), (4), and (63), we have

$$\begin{array}{c}{\displaystyle \alpha {\int }_{\Omega }|G_k\left(u_n\right){|}^{2(\lambda -1)}|\nabla G_k\left(u_n\right){|}^{2}dx+\frac{\mu }{2\lambda -1}{\int }_{\Omega }{|G_k\left(u_n\right)|}^{2\lambda }dx}\hfill \\ \\ {\displaystyle \le \frac{1}{2\lambda -1}({\int }_{\Omega }|E\left(x\right))|\nabla G_k\left(u_n\right)||G_k\left(u_n\right){|}^{2\lambda -1}dx}\hfill \\ \\ {\displaystyle +{\int }_{\Omega }|f||G_k\left(u_n\right){|}^{2\lambda -1}dx).}\hfill \end{array}$$

(85)

By Young’s inequality, we have

$$\begin{array}{c}{\displaystyle \alpha {\int }_{\Omega }|G_k\left(u_n\right){|}^{2(\lambda -1)}|\nabla G_k\left(u_n\right){|}^{2}dx+\frac{\mu }{2\lambda -1}{\int }_{\Omega }{|G_k\left(u_n\right)|}^{2\lambda }dx}\\ \\ {\displaystyle \le \frac{1}{2\lambda -1}(ϵ{\int }_{\Omega }|\nabla G_k\left(u_n\right){|}^2{|G_k\left(u_n\right)|}^{2(\lambda -1)}dx}\\ \\ {\displaystyle +C\left(ϵ\right){\int }_{{\Omega }_{k,n}}{|E\left(x\right)|}^2|G_k\left(u_n\right){|}^{2\lambda }dx+{\int }_{\Omega }|f||G_k\left(u_n\right){|}^{2\lambda -1}dx),}\end{array}$$

(86)

with $\epsilon =\epsilon (\alpha ,\lambda )$.

From (3), Theorem 1 and Hölder’s inequality, we have

$$\begin{array}{c}{\displaystyle \parallel |G_k\left(u_n\right){{|}^{\lambda }\parallel }_{W^{1,2}(\Omega )}^2}\\ \\ \le C({\parallel E\parallel }_{M^{s,N-s}\left({\Omega }_{n,k}\right)}^2\parallel |G_k\left(u_n\right){{|}^{\lambda }\parallel }_{W^{1,2}(\Omega )}^2\\ \\ {\displaystyle +{\parallel f\parallel }_{L^m(\Omega )}\left.{\left({\int }_{\Omega }{|G_k\left(u_n\right)|}^{(2\lambda -1)m^{\prime }}dx\right)}^{\frac{1}{m^{\prime }}}\right),}\end{array}$$

(87)

$C=C(\alpha ,\mu ,N,\lambda )$.

• Therefore, from the same argument as in Lemma 7, using (11) and Corollary 1, we observe that there exists $k_0\in {\mathbb{R}}_{+}$, independent of n, such that

$$\begin{array}{c}{\displaystyle \parallel |G_k\left(u_n\right){|}^{\lambda }{\parallel }_{W^{1,2}(\Omega )}^2\le C{\parallel f\parallel }_{L^m(\Omega )}{\left({\int }_{\Omega }{|G_k\left(u_n\right)|}^{(2\lambda -1)m^{\prime }}dx\right)}^{\frac{1}{m^{\prime }}},\forall k\ge k_0,}\end{array}$$

(88)

with $C=C(\alpha ,\mu ,N,\lambda ,||E|{|}_{M^{s,N-s}(\Omega )})$.

By the Sobolev inequality, since $2^{\ast }\lambda =(2\lambda -1)m^{\prime }=m^{\ast \ast }$, we have

$$\begin{array}{c}{\displaystyle {\left({\int }_{\Omega }{|G_k\left(u_n\right)|}^{m^{\ast \ast }}dx\right)}^{\frac{2}{2^{\ast }}}\le C{\parallel f\parallel }_{L^m(\Omega )}{\left({\int }_{\Omega }{|G_k\left(u_n\right)|}^{m^{\ast \ast }}dx\right)}^{\frac{1}{m^{\prime }},}\forall k\ge k_0,}\end{array}$$

(89)

with $C=C(\alpha ,\mu ,N,\lambda ,||E|{|}_{M^{s,N-s}(\Omega )})$.

Since $m<\frac{N}{2}$, one has $\frac{2}{2^{\ast }}-\frac{1}{m^{\prime }}>0$, and it follows that

$${\left({\int }_{\Omega }{|G_k\left(u_n\right)|}^{m^{\ast \ast }}dx\right)}^{\frac{2}{2^{\ast }}-\frac{1}{m^{\prime }}}\le C{\parallel f\parallel }_{L^m(\Omega )},\forall k\ge k_0.$$

(90)

Step 3. Combining (84) with (90), we have

$$\begin{array}{c}{\displaystyle {\int }_{\Omega }{|u_n|}^{m^{\ast \ast }}dx}\\ \\ {\displaystyle \le C\left[{\parallel f\parallel }_{L^m(\Omega )}^{\frac{2^{\ast }{m}^{\prime }}{2m^{\prime }-2^{\ast }}}+{\left(k_0^{2\lambda }{\int }_{\Omega }{|E\left(x\right)|}^{2}dx+k_0^{2\lambda -1}{\parallel E\parallel }_{L^2(\Omega )}+k_0^{2\lambda -1}{\int }_{\Omega }|f|dx\right)}^{\frac{2^{\ast }}{2}}\right],}\end{array}$$

(91)

with C independent of n, and the theorem is proved. □

In line with [18] and the references therein, we show the exponential summability of $u_n$.

Lemma 9.

Assume (2)–(5) and $f\in L^1(\Omega )\cap L^{N/2}(\Omega )$. Then, for any $\lambda >0$, there exists a positive constant C, independent of n, such that

$${\displaystyle {\int }_{\Omega }{\left(e^{\lambda |u_n|}-1\right)}^{2^{\ast }}dx\le C.}$$

(92)

Proof. 

We obtain the result arguing as in the proof of Lemma 4.6 of [2]. We give the proof in three steps.

• Step 1. First, we prove that, for any $\lambda >0$, there exist $k_0>0$ and a positive constant C, independent of n, such that one has

$${\displaystyle {\int }_{\Omega }{\left(e^{\lambda |G_{k_0}\left(u_n\right)|}-1\right)}^{2^{\ast }}dx\le C.}$$

(93)

By Lemma 4 and Lemma 5, we can take $\left(e^{2\lambda |G_k\left(u_n\right)|}-1\right)\mathrm{sgn}\left(G_k\left(u_n\right)\right)$ as a test function in the variational formulation of problem (18), and using (2) and (4), we have

$$2\lambda \alpha {\int }_{\Omega }{|\nabla G_k\left(u_n\right)|}^2{e}^{2\lambda |G_k\left(u_n\right)|}dx+\mu {\int }_{\Omega }{u}_n\left(e^{2\lambda |G_k\left(u_n\right)|}-1\right)\mathrm{sgn}\left(G_k\left(u_n\right)\right)dx$$

$$\le {\int }_{{\Omega }_{n,k}}|E\left(x\right)||\nabla G_k\left(u_n\right)|\left(e^{2\lambda |G_k\left(u_n\right)|}-1\right)dx+{\int }_{\Omega }|f|\left(e^{2\lambda |G_k\left(u_n\right)|}-1\right)dx.$$

Taking into account that, for any $t\ge 0$ and any $D>1$, the following inequality holds:

$$|t^2{-1|\le D(t-1)}^2+{\displaystyle \frac{1}{D-1}},$$

and using (3), Young’s and Hölder’s inequalities, one has

$$\begin{array}{c}{\displaystyle 2\lambda \alpha {\int }_{\Omega }|\nabla G_k\left(u_n\right){|}^2{e}^{2\lambda |G_k\left(u_n\right)|}dx\le C_{\alpha \lambda D}{\int }_{{\Omega }_{n,k}}{|E\left(x\right)|}^2{\left(e^{\lambda |G_k\left(u_n\right)|}-1\right)}^{2}dx}\hfill \\ \\ {\displaystyle +\lambda \alpha {\int }_{{\Omega }_{n,k}}|\nabla G_k\left(u_n\right){|}^2{\left(e^{\lambda |G_k\left(u_n\right)|}-1\right)}^{2}dx+{\displaystyle \frac{1}{D-1}}\parallel E\left(x\right){\parallel }_{L^2(\Omega )}{\parallel \nabla u_n\parallel }_{L^2(\Omega )}}\hfill \\ \\ {\displaystyle +{D\parallel f\parallel }_{L^{N/2}\left({\Omega }_{n,k}\right)}\parallel e^{\lambda |G_k\left(u_n\right)|}{-1\parallel }_{L^{2^{\ast }}(\Omega )}^2+{\displaystyle \frac{1}{D-1}}{\parallel f\parallel }_{L^1(\Omega )}.}\hfill \end{array}$$

(94)

Noting that

$${\int }_{\Omega }{\left|\nabla \left(e^{\lambda |G_k\left(u_n\right)|}-1\right)\right|}^{2}dx={\lambda }^2{\int }_{\Omega }{|\nabla G_k\left(u_n\right)|}^2{e}^{2\lambda |G_k\left(u_n\right)|}dx$$

(95)

by a previous inequality and Theorem 1, we obtain

$$\begin{array}{c}{\displaystyle {\int }_{\Omega }{\left|\nabla \left(e^{\lambda |G_k\left(u_n\right)|}-1\right)\right|}^{2}dx}\hfill \\ \\ {\displaystyle \le {C\parallel E\left(x\right)\parallel }_{M^{s,N-s}\left({\Omega }_{n,k}\right)}^2\left(\parallel \nabla (e^{\lambda |G_k\left(u_n\right)|}-1){\parallel }_{L^2(\Omega )}^2+{\parallel e^{\lambda |G_k\left(u_n\right)|}-1\parallel }_{L^2(\Omega )}^2\right)}\hfill \\ \\ {\displaystyle +{C\parallel E\left(x\right)\parallel }_{L^2(\Omega )}{\parallel \nabla u_n\parallel }_{L^2(\Omega )}}\hfill \\ \\ {\displaystyle +{C\parallel f\parallel }_{L^{N/2}\left({\Omega }_{n,k}\right)}\parallel e^{\lambda |G_k\left(u_n\right)|}{-1\parallel }_{L^{2^{\ast }}(\Omega )}^2+C{\parallel f\parallel }_{L^1(\Omega )},}\hfill \end{array}$$

(96)

where C is a different positive constant, independent of n, on each line.

Thus, in view of (11) and Corollary 1, there exists $k_0$, independent of n, such that, using the Sobolev and Hölder’s inequality, we have

$$\begin{array}{c}{\displaystyle \parallel e^{\lambda |G_{k_0}\left(u_n\right)|}{-1\parallel }_{L^{2^{\ast }}(\Omega )}^2\le }\hfill \\ \\ {\displaystyle \le C|{\Omega }_{n,k_0}{|}^{1-\frac{2}{2^{\ast }}}{\parallel E\left(x\right)\parallel }_{M^{s,N-s}(\Omega )}^2{\parallel e^{\lambda |G_{k_0}\left(u_n\right)|}-1\parallel }_{L^{2^{\ast }}(\Omega )}^2}\hfill \\ \\ {\displaystyle +{C\parallel E\left(x\right)\parallel }_{L^2(\Omega )}{\parallel \nabla u_n\parallel }_{L^2(\Omega )}}\hfill \\ \\ {\displaystyle +{C\parallel f\parallel }_{L^{N/2}\left({\Omega }_{n,k_0}\right)}\parallel e^{\lambda |G_{k_0}\left(u_n\right)|}{-1\parallel }_{L^{2^{\ast }}(\Omega )}^2+C{\parallel f\parallel }_{L^1(\Omega )}.}\hfill \end{array}$$

(97)

Hence, by (24) and using again Corollary 1, unless the value of $k_0$ is enlarged, we have

$${\displaystyle \parallel e^{\lambda |G_{k_0}\left(u_n\right)|}{-1\parallel }_{L^{2^{\ast }}(\Omega )}^2\le C\left({\parallel E\left(x\right)\parallel }_{L^2(\Omega )}+{\parallel f\parallel }_{L^1(\Omega )}\right),}$$

(98)

with C independent of n.

• Step 2: Let us prove that, for any $\lambda >0$ and any $k>0$, there exists a positive constant C, independent of n, such that

$${\displaystyle {\int }_{\Omega }{\left(e^{\lambda |T_k\left(u_n\right)|}-1\right)}^{2^{\ast }}dx\le C.}$$

(99)

By Lemma 4, we can choose $\left(e^{2\lambda |T_k\left(u_n\right)|}-1\right)\mathrm{sgn}\left(T_k\left(u_n\right)\right)$ as a test function in the variational formulation of (18), obtaining, by (2) and (4),

$$2\lambda \alpha {\int }_{\Omega }{|\nabla T_k\left(u_n\right)|}^2{e}^{2\lambda |T_k\left(u_n\right)|}dx+\mu {\int }_{\Omega }{u}_n\left(e^{2\lambda |T_k\left(u_n\right)|}-1\right)\mathrm{sgn}\left(T_k\left(u_n\right)\right)dx$$

$$\le {\int }_{\Omega }|E\left(x\right)||\nabla u_n|\left(e^{2\lambda |T_k\left(u_n\right)|}-1\right)dx+{\int }_{\Omega }|f|\left(e^{2\lambda |T_k\left(u_n\right)|}-1\right)dx.$$

Therefore, by the analogous results of (95) and the Sobolev inequality, we obtain

$$\begin{array}{c}{\displaystyle \parallel e^{\lambda |T_k\left(u_n\right)|}{-1\parallel }_{L^{2^{\ast }}(\Omega )}^2\le C(e^{2\lambda k}-1)\left({\parallel E\left(x\right)\parallel }_{L^2(\Omega )}\parallel \nabla u_n{\parallel }_{L^2(\Omega )}+{\parallel f\parallel }_{L^1(\Omega )}\right),}\hfill \end{array}$$

(100)

which gives (99), using (37).

• Step 3. It results in

$$\begin{array}{c}{\displaystyle {\int }_{\Omega }{\left(e^{\lambda |u_n|}-1\right)}^{2^{\ast }}dx\le {\int }_{\Omega }{\left(e^{\lambda (|T_{k_0}\left(u_n\right)|+|G_{k_0}\left(u_n\right)|)}-1\right)}^{2^{\ast }}dx}\hfill \\ {\displaystyle \le C\left(e^{2^{\ast }\lambda k_0}{\int }_{\Omega }{\left(e^{\lambda |G_{k_0}\left(u_n\right)|}-1\right)}^{2^{\ast }}dx+{\int }_{\Omega }{\left(e^{\lambda |T_{k_0}\left(u_n\right)|}-1\right)}^{2^{\ast }}dx\right).}\hfill \end{array}$$

(101)

Putting together (93), (99), and (101), we have (92). □

Finally, proceeding as in [2], we obtain the following:

Theorem 6.

Assume (2)–(5). Then,

1. 

if $f\in L^1(\Omega )\cap L^m(\Omega )$, ${\left(2^{\ast }\right)}^{\prime }<m<\frac{N}{2}$, then there exists a weak solution $u\in W_0^{1,2}(\Omega )\cap L^{m^{\ast \ast }}(\Omega )$ of problem (1);

2. 

if $f\in L^{{\left(2^{\ast }\right)}^{\prime }}(\Omega )\cap L^m(\Omega )$, $m>\frac{N}{2}$, then there exists a weak solution $u\in W_0^{1,2}(\Omega )\cap L^{\infty }(\Omega )$ of problem (1);

3. 

if $f\in L^1(\Omega )\cap L^{N/2}(\Omega )$, then there exists a weak solution $u\in W_0^{1,2}(\Omega )$ of problem (1) such that $(e^{\lambda |u|}-1)\in L^{2^{\ast }}(\Omega )$ for any $\lambda >0$.

6. Uniqueness

In this section, under the following assumption:

$$|H(x,\xi )-H(x,\eta )|\le |E\left(x\right)||\xi -\eta |,\forall x\in \Omega ,\forall \xi ,\eta \in {\mathbb{R}}^N,$$

(102)

we prove the following uniqueness result.

Theorem 7.

Under hypotheses (2)–(6) and (102), let $u_1,u_2$ be solutions of (1). Then, we have $u_1=u_2$ almost everywhere in Ω.

Proof. 

We obtain the result from the same covering argument as in [2]. □

7. Concluding Remarks

In this article, we study a noncoercive nonlinear elliptic operator in an unbounded domain. We prove existence, regularity, and uniqueness theorems for a Dirichlet problem when the function that controls the first-order term is given in an appropriate space of a Morrey-type, improving previous results known in the literature. Specifically, the novelty of this work with respect to the current literature is studying the Dirichlet problem when the singular first-order term is controlled through a function in a suitable space of a Morrey type, which has been introduced to deal with problems in unbounded domains.

The essential tools for achieving our results are the boundedness and compactness of a multiplication operator (see Theorem 1) and the weak maximum principle. To obtain the existence result, we start proving the existence of a solution of the approximating problems using the surjectivity theorem. Then, by means of the weak maximum principle, we show a priori bounds. This estimate is needed to pass the limit in the variational formulation of approximating problems, obtaining the claimed existence result. Then we prove the regularity results as the summability of the datum f varies. We also establish the uniqueness. In the future, it will be possible to extend this study to systems or to the p-Laplacian. It is worth pointing out that the strengths of the used methods are the inclusion $M^N(\Omega )\subset M^{s,N-s}(\Omega )$ (which leads to an improvement of previous results), the weak maximum principle, and the related a priori estimates of solutions of an approximating problem (which provide good properties of the obtained solutions).

In addition, the results concerning existence (Theorem 5) and uniqueness (Theorem 7) can be used to analyze certain first-passage-time problems for diffusion processes with advection defined in unbounded domains. These kinds of problems deserve interest in various applications related to mathematical biology, for instance, the modeling of protein receptor motion in neurophysiology, transport problems in cells, and water diffusion through porin in outer bacterial membranes (see details in [36] and references therein). However, due to complexity and the relevance of the aspects related to problem (1), a deeper study focused on the mentioned applications will be the subject of future investigations.