A local and nonlocal coupling model involving the $p$-Laplacian

TL;DR

This paper extends previous work to a model coupling local and nonlocal p-Laplacian operators, establishing existence and uniqueness of solutions through energy minimization.

Contribution

It introduces a new coupled local and nonlocal p-Laplacian model and proves the existence and uniqueness of solutions using energy functional methods.

Findings

Existence of solutions established

Uniqueness of solutions proven

Solution obtained via energy minimization

Abstract

In this paper we extend some results presented in \cite{julio} to the case of the $p$-Laplacian operator. More precisely, we consider a model that couples a local $p$-Laplacian operator with a nonlocal $p$-Laplacian operator through source terms in the equation. The resulting problem is associated with an energy functional. We establish the existence and uniqueness of a solution, which is obtained via the direct minimization of the corresponding energy functional.

Equations

$$\left\{\begin{array}[c]{ll}f\left(x,u\right)=-\Delta_{p}u+{\textstyle\int\limits_{{\small\Omega}_{n\ell}}}J(x,y)\left|u(x)-u(y)\right|^{p-2}\left(u(x)-u(y)\right)dy&\text{in }\Omega_{\ell},\\ \partial_{\nu}u=0&\text{in }\partial\Omega_{\ell}\cap\Omega,\\ u=0&\text{in }\partial\Omega\cap\partial\Omega_{\ell},\end{array}\right.$$

$$\left\{\begin{array}[c]{ll}f\left(x,u\right)=\int_{\mathbb{R}^{N}\setminus\Omega_{n\ell}}J(x,y)\left|u(x)-u(y)\right|^{p-2}\left(u(x)-u(y)\right)dy\vskip 3.0pt plus 1.0pt minus 1.0pt&\text{in }\Omega_{n\ell},\\ \qquad\qquad\,\,+2\int_{\Omega_{n\ell}}J(x,y)\left|u(x)-u(y)\right|^{p-2}\left(u(x)-u(y)\right)dy&\\ u=0&\text{in }\mathbb{R}^{N}\setminus\Omega.\end{array}\right.$$

$$\left|f\left(x,\xi \right)\right|\le a\left(x\right)+b\left(x\right){\left|\xi \right|}^q\text{for }a.e.x\in \Omega \text{, }\xi \in \mathbb{R},$$

$$T_{J,q}(u):={\int }_{\Omega }J(x,y)u(y)dy,$$

$$W:=\left\{u\in L^p\left(\Omega \right):u_{|{\Omega }_{\ell }}\in W^{1,p}\left({\Omega }_{\ell }\right),u=0\text{ in }{\mathbb{R}}^N\setminus \Omega \right\},$$

$$\Vert u{\Vert }_W:=\Vert u{\Vert }_{L^p(\Omega )}+\Vert |\nabla u|{\Vert }_{L^p({\Omega }_{\ell })}.$$

$$E(u):={\int }_{{\Omega }_{\ell }}\frac{{\left|\nabla u\right|}^p}{p}+\frac{1}{p}{\int }_{{\Omega }_{n\ell }}{\int }_{{\mathbb{R}}^N}J(x,y){\left|u(x)-u(y)\right|}^{p}dydx-{\int }_{\Omega }\tilde{F}\left(x,u\right)dx.$$

$${\int }_U{\int }_{U}J(x,y){\left|u(x)-u(y)\right|}^{p}dydx=0\text{,}$$

$$\underset{n\to \infty }{\lim }{\int }_{{\Omega }_{n\ell }}{\int }_{\Omega }J(x,y){\left|u_n(x)-u_n(y)\right|}^{p}dydx=0,$$

$$\underset{n\to \infty }{\lim }{\int }_{{\Omega }_{n\ell }}{\left|u_n(x)\right|}^{p}dx=0,$$

$$\underset{n\to \infty }{\lim }{\int }_{{\Omega }_{n\ell }}{\int }_{\Omega }J(x,y){\left|u_n(x)\right|}^{p}dydx=0,$$

$$\begin{array}[c]{l}\int_{\Omega_{n\ell}}\int_{\Omega}J(x,y)\left|u_{n}(x)-u_{n}(y)\right|^{p}dydx\vskip 6.0pt plus 2.0pt minus 2.0pt\\ =\int_{\Omega}J(x,y)\left|u_{n}(x)-u_{n}(y)\right|^{p-M}\left|u_{n}(x)-u_{n}(y)\right|^{M}dydx\vskip 6.0pt plus 2.0pt minus 2.0pt\\ \geq C_{p}\int_{\Omega_{n\ell}}\int_{\Omega}J(x,y)\left(\left|u_{n}(x)\right|^{p-M-2}u_{n}(x)-\left|u_{n}(y)\right|^{p-M-2}u_{n}(y)\right)\vskip 6.0pt plus 2.0pt minus 2.0pt\\ \quad\times\left(u_{n}(x)-u_{n}(y)\right)\left|u_{n}(x)-u_{n}(y)\right|^{M}dydx\vskip 6.0pt plus 2.0pt minus 2.0pt\\ \geq C_{p}\int_{\Omega_{n\ell}}\int_{\Omega}J(x,y)\left(\left|u_{n}(x)\right|^{p-M-2}u_{n}(x)-\left|u_{n}(y)\right|^{p-M-2}u_{n}(y)\right)\vskip 6.0pt plus 2.0pt minus 2.0pt\\ \quad\times\left(u_{n}(x)-u_{n}(y)\right)\left(\left|u_{n}(x)\right|-\left|u_{n}(y)\right|\right)^{M}dydx\vskip 6.0pt plus 2.0pt minus 2.0pt\\ =C_{p}\int_{\Omega_{n\ell}}\int_{\Omega}J(x,y)\vskip 6.0pt plus 2.0pt minus 2.0pt\\ \quad\times\left[\left|u_{n}(x)\right|^{p-M}+\left|u_{n}(y)\right|^{p-M}-u_{n}(x)u_{n}(y)\left(\left|u_{n}(x)\right|^{p-M-2}+\left|u_{n}(y)\right|^{p-M-2}\right)\right]\vskip 6.0pt plus 2.0pt minus 2.0pt\\ \quad\times\sum_{j=0}^{M}\binom{M}{j}(-1)^{j}\left|u_{n}(y)\right|^{j}\left|u_{n}(x)\right|^{M-j}dydx\vskip 6.0pt plus 2.0pt minus 2.0pt\\ =C_{p}\left(\sum_{j=0}^{M}\binom{M}{j}(-1)^{j}\int_{\Omega_{n\ell}}\int_{\Omega}J(x,y)\left|u_{n}(x)\right|^{p-j}\left|u_{n}(y)\right|^{j}dydx\right.\vskip 6.0pt plus 2.0pt minus 2.0pt\\ \quad+\sum_{j=0}^{M}\binom{M}{j}(-1)^{j}\int_{\Omega_{n\ell}}\int_{\Omega}J(x,y)\left|u_{n}(y)\right|^{p-M+j}\left|u_{n}(x)\right|^{M-j}dydx\vskip 6.0pt plus 2.0pt minus 2.0pt\\ \quad+\sum_{j=0}^{M}\binom{M}{j}(-1)^{j+1}\int_{\Omega_{n\ell}}\int_{\Omega}J(x,y)\left|u_{n}(x)\right|^{p-j-2}u_{n}(x)\left|u_{n}(y)\right|^{j}u_{n}(y)dydx\vskip 6.0pt plus 2.0pt minus 2.0pt\\ \quad+\left.\sum_{j=0}^{M}\binom{M}{j}(-1)^{j+1}\int_{\Omega_{n\ell}}\int_{\Omega}J(x,y)\left|u_{n}(y)\right|^{p-M+j-2}u_{n}(y)\left|u_{n}(x)\right|^{M-j}u_{n}(x)dydx\right)\vskip 6.0pt plus 2.0pt minus 2.0pt\\ =C_{p}\left(\int_{\Omega_{n\ell}}\int_{\Omega}J(x,y)\left|u_{n}(x)\right|^{p}dydx+\sum_{j=1}^{M}\binom{M}{j}(-1)^{j}\left\langle T_{J,p/j}(|u_{n}|^{j}),|u_{n}|^{p-j}\right\rangle\right.\vskip 6.0pt plus 2.0pt minus 2.0pt\\ \quad+\int_{\Omega_{n\ell}}\int_{\Omega}J(x,y)\left|u_{n}(y)\right|^{p}dydx+\sum_{j=0}^{M-1}\binom{M}{j}(-1)^{j}\left\langle T_{J,p/(p-M+j)}(|u_{n}|^{p-M+j}),|u_{n}|^{M-j}\right\rangle\vskip 6.0pt plus 2.0pt minus 2.0pt\\ \quad+\sum_{j=0}^{M}\binom{M}{j}(-1)^{j+1}\left\langle T_{J,p/(j+1)}(\left|u_{n}\right|^{j}u_{n}),\left|u_{n}\right|^{p-j-2}u_{n}\right\rangle\vskip 6.0pt plus 2.0pt minus 2.0pt\\ \quad+\left.\sum_{j=0}^{M}\binom{M}{j}(-1)^{j+1}\left\langle T_{J,p/(p-M+j-1)}(\left|u_{n}\right|^{p-M+j-2}u_{n}),\left|u_{n}\right|^{M-j}u_{n}\right\rangle\right).\end{array}$$

$$T_{J,p/r}(|u_n{|}^r)\to 0\text{and}{T}_{J,p/r}(|u_n{|}^{r-1}{u}_n)\to 0,$$

$$(p/r{)}^{\prime }=\frac{p/r}{p/r-1}=\frac{p/r}{(p-r)/r}=p/(p-r),$$

$$⟨T_{J,p/r}(|u_n{|}^r),|u_n{|}^{p-r}⟩\to 0,⟨T_{J,p/r}(|u_n{|}^{r-1}{u}_n),|u_n{|}^{p-r-1}{u}_n⟩\to 0.$$

$$\text{if }1$$

$$\text{if }0$$

$$\begin{array}[c]{l}0=\lim_{n\rightarrow\infty}\int_{\Omega_{n\ell}}\int_{\Omega}J(x,y)\left|u_{n}(x)-u_{n}(y)\right|^{p}dydx\vskip 6.0pt plus 2.0pt minus 2.0pt\\ \geq C_{p}\lim_{n\rightarrow\infty}\left(\int_{\Omega_{n\ell}}\int_{\Omega}J(x,y)\left|u_{n}(x)\right|^{p}dydx+\int_{\Omega_{n\ell}}\int_{\Omega}J(x,y)\left|u_{n}(y)\right|^{p}dydx\right)\\ \geq 0,\end{array}$$

$$\underset{n\to \infty }{\lim }{\int }_{{\Omega }_{n\ell }}{\int }_{\Omega }J(x,y){\left|u_n(x)\right|}^{p}dydx=0.$$

$$A_{\delta }^0:=\left\{x\in {\Omega }_{n\ell }:dist(x,{\Omega }_{\ell })<\delta \right\}.$$

$${\int }_{{\Omega }_{n\ell }}{\int }_{{\Omega }_{\ell }}J(x-y)|u_n(x){|}^{p}dydx$$

$$\ge mC{\int }_{A_{\delta }^0}|u_n(x){|}^{p}dx.$$

$$\underset{n\to \infty }{\lim }{\int }_{{\Omega }_{n\ell }\setminus \overline{A_{\delta }^0}}{\int }_{A_{\delta }^0}J(x-y)|u_n(x){|}^{p}dydx=0.$$

$$A_{\delta }^1=\{x\in {\Omega }_{n\ell }\setminus \overline{A_{\delta }^0}:dist(x,A_{\delta }^0)<\delta \},$$

$$A_{\delta }^j=\left\{x\in {\Omega }_{n\ell }\setminus \overline{\bigcup _{0\le i<j}{A}_{\delta }^i}:dist(x,\bigcup _{0\le i<j}{A}_{\delta }^i)<\delta \right\}.$$

$${\Omega }_{n\ell }=\bigcup _{0\le i<J}{A}_{\delta }^i$$

$${\int }_{{\Omega }_{\ell }}\frac{{\left|\nabla u\right|}^p}{p}+\frac{1}{p}{\int }_{{\Omega }_{n\ell }}{\int }_{{\mathbb{R}}^N}J(x,y){\left|u(x)-u(y)\right|}^{p}dydx\ge c{\Vert u\Vert }_{L^p\left(\Omega \right)}^p$$

$${\int }_{{\Omega }_{\ell }}\frac{{\left|\nabla u_n\right|}^p}{p}+\frac{1}{p}{\int }_{{\Omega }_{n\ell }}{\int }_{{\mathbb{R}}^N}J(x,y){\left|u_n(x)-u_n(y)\right|}^{p}dydx\to 0.$$

$${\int }_{{\Omega }_{n\ell }}{\int }_{{\mathbb{R}}^N}J(x,y){\left|u_n(x)-u_n(y)\right|}^{p}dydx\to 0,$$

$${\int }_{{\Omega }_{n\ell }}{\int }_{{\Omega }_{n\ell }}J(x,y){\left|u(x)-u(y)\right|}^{p}dydx$$

Full text

1 Introduction and main results

Nonlocal models can describe phenomena that are not well represented by classical PDE’s, for example, problems which have long-range interactions and/or discontinuities. For instance, in the context of diffusion, long-range interactions effectively describe anomalous diffusion, while in the context of mechanics, cracks formation results in material discontinuities.

Nonlocal operators are defined through integration against an appropriate kernel, which implies that their values at a given point depend on the entire domain rather than just a neighbourhood around that point, as is typical for differential operators. One of the most important examples is the fractional Laplacian.

For general references on nonlocal models we refer e.g. to [4, 6, 7, 9, 10, 11, 24, 27, 29, 30] and its references, while the articles [8, 14, 17, 19, 20, 23, 28] focus on the study of nonlocal $p$-Laplacian operators.

In recent years there has been growing interest in models that combine local and nonlocal effects, as they are capable of capturing more complex and realistic phenomena. In such cases, nonlocal effects may arise in certain regions of the domain, while in other regions the behavior is governed by classical differential operators. See, for instance, [2, 3, 12, 13, 15, 16, 21, 25, 26] and the references therein.

The study of nonlinear partial differential equations with $p$-Laplacian operators has gained significant attention due to its broad range of applications in fields such as physics, engineering, and image processing. In this work, we analyze an elliptic equation that couples the local $p$-Laplacian operator with a nonlocal $p$-Laplacian operator through source terms. This coupling results in a variational structure, and we establish the existence and uniqueness of solutions by minimizing the corresponding energy functional. Our results extend some of those presented in [1], where the classical Laplacian is considered both in its local and nonlocal forms.

For coupling local and nonlocal models the previous strategies treat the coupling condition as an optimization objective (the goal is to minimize the mismatch of the local and nonlocal solutions on the overlap of their sub-domains). Another approach is based on a partitioned procedure as a general coupling strategy for heterogeneous systems, the system is divided into sub-problems in their respective sub-domains, which communicate with each other via the transmission conditions. As far as we are aware, the literature lacks studies addressing this approach for models that involve $p$-Laplacian operators.

1.1 Statement of the main result

We assume that $\Omega\subset\mathbb{R}^{N}$ is an open bounded domain, such that $\Omega$ is divided into two disjoint subdomains: a local region that we will denote by $\Omega_{\ell}$ and a nonlocal region, $\Omega_{n\ell}$. Thus we have $\Omega_{\ell},\Omega_{n\ell}\subset\Omega\subset\mathbb{R}^{N}$ with $\Omega=(\overline{\Omega}_{\ell}\cup\overline{\Omega}_{n\ell})^{\circ}$. Further, we assume that:

(1) $\Omega_{\ell}$ is connected and has a Lipschitz boundary.

(2) $\Omega_{n\ell}$ is $\delta$-connected. As in [1], for $\delta>0$, we say that an open set $U\subset\mathbb{R}^{N}$ is $\delta$-connected if it cannot be written as a disjoint union of two (relatively) open nontrivial sets that are at distance greater or equal than $\delta$.

(3) $dist(\Omega_{\ell},\Omega_{n\ell})<\delta.$

Our aim is to consider the following local-nonlocal problem, under suitable hypothesis on the nonlinearity $f$ and the kernel $J$:

[TABLE]

and the following nonlocal equation in $\Omega_{n\ell}$,

[TABLE]

Here $f:\Omega\times\mathbb{R}\rightarrow\mathbb{R}$ is a Carathéodory function (that is, $f\left(x,\cdot\right)$ is continuous for $a.e.$ $x\in\Omega$ and $f\left(\cdot,\xi\right)$ is measurable for all $\xi\in\mathbb{R}$) that satisfies the following growth condition:

[TABLE]

where $0\leq q<p-1$, and $a$ and $b$ are nonnegative functions such that $a\in L^{s}(\Omega)$, with $s>p^{\prime}$ (where, as usual, $1/p+1/p^{\prime}=1$) and $b\in L^{\infty}(\Omega)$. Regarding the hypothesis on $J$, we shall assume that:

(J1) $J$ is symmetric, and there exists $C>0$ such that $J(z)>C$ for all $z$ such that $\left|z\right|\leq 2\delta$.

(J2) Let $1<q<\infty$. For $u\in L^{q}$ we have that

[TABLE]

defines a compact operator in $L^{q}(\Omega_{n\ell})$. For sufficient conditions on $J$ for $T_{J,q}$ to be a compact operator we refer e.g. to [18, Theorem 1] or [5, Chapter VI].

Let $p>1$. We next consider the space

[TABLE]

which is a Banach space equipped with the norm

[TABLE]

Let $\widetilde{F}\left(x,u\right):=\int_{0}^{u}f\left(x,\xi\right)d\xi$. Defined in $W$ we have the energy functional $E:W\rightarrow\mathbb{R}$ given by

[TABLE]

It is easy to check that this functional is Fréchet differentiable.

We can now state our main result:

Theorem 1.1**.**

Let $p>1$, and assume (1), (2), (3), (f), (J1) and (J2). Then there exists a minimizer of $E$ in $W$. Moreover, the minimizer is a weak solution of (1.1) and (1.2). Furthermore, if $\xi\rightarrow f\left(x,\xi\right)$ is strictly concave for $a.e.$ $x\in\Omega$, then the minimizer of the functional $E$ is unique.

2 Proof of the main result

In order to prove Theorem 1.1 we first need to prove some auxiliary results. We start with the following lemma which is a direct adaptation of [1, Lemma 3.1]. This result will be used to prove Lemma 2.3.

Lemma 2.1**.**

Let $U\subset\mathbb{R}^{N}$ be an open $\delta$-connected set and $u\in L^{p}\left(U\right)$. If

[TABLE]

then there exists a constant $k\in\mathbb{R}$ such that $u\left(x\right)=k$ $a.e.$ $x\in U.$

The next lemma will also be necessary in order to prove Lemma 2.3. Lemma 2.2 is crucial and presents the greatest challenge in adapting the ideas developed in [1].

Lemma 2.2**.**

Let $1<p<\infty$ and $u_{n}:\Omega\rightarrow\mathbb{R}$ be a sequence such that $u_{n}\rightarrow 0$ strongly in $L^{p}(\Omega_{\ell})$ and weakly in $L^{p}(\Omega_{n\ell})$. If in addition

[TABLE]

then

[TABLE]

that is, $u_{n}\rightarrow 0$ in $L^{p}(\Omega_{n\ell})$ and hence in $L^{p}(\Omega)$.

Proof. First we prove that

[TABLE]

Let $M\in\mathbb{N}_{0}$ such that $M+1<p\leq M+2$. Then, using inequality (III) in [22, Page 71] we get

[TABLE]

Since $u_{n}\rightarrow 0$ weakly in $L^{p}(\Omega_{n\ell})$, if $0<r<p$, we get that $|u_{n}|^{r}$ is bounded in $L^{p/r}(\Omega_{n\ell})$, and then $|u_{n}|^{r}\rightarrow 0$ weakly in $L^{p/r}(\Omega_{n\ell})$. By compactness of $T_{J,p/r}$ in $L^{p/r}(\Omega_{n\ell})$, we have that

[TABLE]

both convergences in $L^{p/r}(\Omega_{n\ell})$. On the other hand

[TABLE]

and since $|u_{n}|^{p-r}$ converges weakly to zero in $L^{p/(p-r)}(\Omega_{n\ell})$, by [5, Proposition 3.5], we get

[TABLE]

Now, we observe that $M+1<p\leq M+2$. Then: if $1\leq j\leq M$, $1<p/j$; if $0\leq j<M$, $1<p/(p-M+j)$; if $0\leq j\leq M$, $1<p/(j+1)$ and $1<p/(p-M+j-1)$. Therefore

[TABLE]

Then

[TABLE]

and thus

[TABLE]

Let us next define

[TABLE]

Notice that thanks to property (3) and to the fact that $\Omega_{n\ell}$ is open we see that $A_{\delta}^{0}$ is open and nonempty. In particular it has positive $N$-dimensional measure. For any $x\in\overline{A_{\delta}^{0}}$ we consider the continuous and strictly positive function $g(x):=|B_{2\delta}(x)\cap\Omega_{\ell}|$. Since $\overline{A_{\delta}^{0}}$ is a compact set, there exists a constant $m>0$ such that $g(x)\geq m$ for any $x\in\overline{A_{\delta}^{0}}$. As a consequence

[TABLE]

Therefore, thanks to (2.2), $u_{n}\rightarrow 0$ in $L^{p}(A_{\delta}^{0})$. In order to iterate this argument we notice that at this point we know that $u_{n}\rightarrow 0$ strongly in $A_{\delta}^{0}$ and weakly in $\Omega_{n\ell}\setminus\overline{A_{\delta}^{0}}$, hence again from (2.1) we get

[TABLE]

Since $\Omega_{n\ell}$ is $\delta$ connected, $dist(\Omega_{n\ell}\setminus\overline{A_{\delta}^{0}},A_{\delta}^{0})<\delta$. Considering now

[TABLE]

and proceeding as before, we obtain, from (2.3), that $u_{n}\rightarrow 0$ strongly in $A_{\delta}^{1}$. This argument can be repeated and gives strong converge in $L^{p}(A_{\delta}^{j})$ for

[TABLE]

Since $\Omega_{n\ell}$ is bounded, we have, for a finite number $J\in\mathbb{N}$,

[TABLE]

and therefore the proof is complete. ∎

Lemma 2.3**.**

There is a constant $c>0$ such that

[TABLE]

for all $u\in W$.

Proof. We proceed by contradiction. Assume there exists $u_{n}\in H$ such that $\left\|u_{n}\right\|_{L^{p}\left(\Omega\right)}=1$ and

[TABLE]

Then, $\int_{\Omega_{\ell}}\left|\nabla u_{n}\right|^{p}\rightarrow 0$ and $\int_{\Omega_{n\ell}}\int_{\mathbb{R}^{N}}J(x,y)\left|u_{n}(x)-u_{n}(y)\right|^{p}dydx\rightarrow 0$. Since $u_{n}$ is bounded in $L^{p}\left(\Omega_{\ell}\right)$ and $\int_{\Omega_{\ell}}\left|\nabla u_{n}\right|^{p}\rightarrow 0$, by the Sobolev imbedding theorem, passing to a subsequence we get that $u_{n}\rightarrow k_{1}$ in $W^{1,p}\left(\Omega_{\ell}\right)$ for some $k_{1}\in\mathbb{R}$. We argue next in the nonlocal part $\Omega_{n\ell}$. Since $u_{n}$ is bounded in $L^{p}\left(\Omega_{n\ell}\right)$, passing to another subsequence we have that $u_{n}\rightharpoonup u$ in $L^{p}\left(\Omega_{n\ell}\right)$. Furthermore, since

[TABLE]

we get that the limit $u$ verifies that

[TABLE]

and

[TABLE]

From (2.4), using Lemma 2.1 and the fact that $\Omega_{n\ell}$ is an open $\delta$-connected set, we deduce that $u=k_{2}$ in $\Omega_{n\ell}$ for some $k_{2}\in\mathbb{R}$. On the other side, from (2.5) we obtain

[TABLE]

and so, recalling conditions (3) and (J1) we must have $k_{1}=k_{2}$. We next see that $k_{1}=0$. We have that $u_{n}=0$ in $\mathbb{R}^{N}\diagdown\Omega$. If $\partial\Omega\cap\partial\Omega_{\ell}\not=\emptyset$, then $u_{n\mid\partial\Omega\cap\partial\Omega_{\ell}}=0$; and from the convergence $u_{n}\rightarrow k_{1}$ in $W^{1,p}\left(\Omega_{\ell}\right)$, we conclude that $k_{1}=0$. If $\partial\Omega\cap\partial\Omega_{\ell}=\emptyset$, then in this case we have that $dist\left(\Omega_{n\ell},\mathbb{R}^{N}\diagdown\Omega\right)=0$. Now, using that $u_{n}=0$ in $\mathbb{R}^{N}\diagdown\Omega$,

[TABLE]

and $u_{n}\rightharpoonup k_{2}$ in $L^{p}\left(\Omega_{n\ell}\right)$, we derive that $k_{2}=0$. Summing up, we have proved that $u_{n}\rightarrow 0$ in $W^{1,p}\left(\Omega_{\ell}\right)$ and $u_{n}\rightharpoonup 0$ in $L^{p}\left(\Omega_{n\ell}\right)$. Then, Lemma 2.2 says that $u_{n}\rightarrow 0$ in $L^{p}\left(\Omega\right)$. Since $1=\left\|u_{n}\right\|_{L^{p}\left(\Omega\right)}$ for all $n$ we get a contradiction. ∎

We are now in position to prove the Theorem 1.1

Proof of Theorem 1.1. By hypothesis $(f)$ we have

[TABLE]

where $a\in L^{s}(\Omega)$, for some $s>p^{\prime}$, $b\in L^{\infty}(\Omega)$ and $q<p-1$.

By Lemma 2.3 we have that

[TABLE]

for some $C,C^{\prime},C^{\prime\prime}>0$, and so, since $q<p-1$, $E$ is bounded from below and coercive.

Let $1<r=\min\{p/s^{\prime},p/(q+1)\}$. Then $\widetilde{F}(\cdot,u)\in L^{r}(\Omega)$ for all $u\in L^{p}(\Omega)$. Suppose now that $\{u_{n}\}$ is a sequence that converges weakly to a function $u$ in $W$. On one hand the functional $F$ given by

[TABLE]

is convex and therefore $F$ is weakly lower semicontinuous. On the other side, we can take a subsequence $\{u_{n_{j}}\}$ such that

[TABLE]

Also, since $\{u_{n}\}$ is bounded in $L^{p}\left(\Omega\right)$, $\left\{\widetilde{F}(\cdot,u_{n})\right\}$ is bounded in $L^{r}(\Omega)$ and so there exists some $\left\{u_{n_{j_{k}}}\right\}$ such that $\left\{\widetilde{F}(\cdot,u_{n_{j_{k}}})\right\}$ converges weakly to $\widetilde{F}(\cdot,u)$ in $L^{r}\left(\Omega\right)$. Hence,

[TABLE]

and $E$ is a weakly lower semicontinuous functional. Therefore, it is easy to check that there exists a minimizer $u\in W$ by the direct method of the calculus of variations. Next, we prove that $u$ is a weak solution of (1.1) and (1.2). Let $\phi$ be a smooth function with $\phi=0$ in $\mathbb{R}^{N}\diagdown\Omega$. Then, for all $t\in\mathbb{R}$ we have that $\frac{\partial}{\partial t}E\left(u+t\phi\right)\left|{}_{t=0}\right.=0$. In other words,

[TABLE]

Now, we observe that

[TABLE]

and so, using that $J$ is symmetric and Fubini’s theorem we get

[TABLE]

On the other side,

[TABLE]

Therefore, recalling that $u=\phi=0$ in $\mathbb{R}^{N}\diagdown\Omega$ we have that

[TABLE]

and then $u$ is a weak solution of (1.1) and (1.2). Finally if $\xi\rightarrow f\left(x,\xi\right)$ is strictly concave for $a.e.$ $x\in\Omega$, then $E$ is a strictly convex functional in $W$ and the minimizer $u$ is unique. ∎

Acknowledgement. We would like to thank to Julio Rossi for suggesting us this problem.