Existence Results for a System of Kirchhoff–Schrödinger–Maxwell Equations

Abstract

In this paper, we study existence, nonexistence, and properties of solutions for some Kirchhoff–Schrödinger–Maxwell systems as (1.3). The solutions can be seen as saddle points of functionals which are unbounded both from above and from below.

Appendix: Basic Results and Existence for Bounded Data

In this Appendix, we will prove some results concerning the first equation of system (1.3), and the whole system in the case of bounded data.

Proposition 3.3

Let v be a function in $W_0^{1,2}(\Omega )$, let $\varphi \ge 0$ be a function in $L^{1}(\Omega )$, let f be a function in $L^{\infty }(\Omega )$, and let $r > 1$. Then, there exists a unique weak solution u of :

$$\begin{aligned} u \in W_0^{1,2}(\Omega ): -{\text {div}}\left( \left[ a(x) + \int _{\Omega }|\nabla v|^{2}\right] \nabla u\right) + \varphi \,|u|^{r-2}\,u = f\,; \end{aligned}$$

(3.11)

that is, $\varphi \,|u|^{r}$ belongs to $L^{1}(\Omega )$, and

$$\begin{aligned} \int _{\Omega }\left[ a(x) + \int _{\Omega }|\nabla v|^{2}\right] \nabla u \nabla \eta + \int _{\Omega }\varphi \,|u|^{r-2}\,u\,\eta = \int _{\Omega }f\,\eta \,, \quad \forall \eta \in W_0^{1,2}(\Omega )\cap L^{\infty }(\Omega )\,. \end{aligned}$$

Furthermore:

(3.12)

for some positive constant C independent on v.

Proof.  The existence and uniqueness of a solution u in $W_0^{1,2}(\Omega )$ of (3.11) follows from well-known results on semilinear elliptic equations (see, for example, [5]), since (3.11) is of the kind:

$$\begin{aligned} -{\text {div}}(Q(x)\nabla w) + b(x)|w|^{r-2}w = g\,, \end{aligned}$$

with Q a uniformly elliptic and bounded matrix, $r > 1$, b a function in $L^{1}(\Omega )$, and g a function belonging to some Lebesgue space.

As for (3.12), we begin by choosing $T_{k}(u)$ as test function. Dropping two positive terms, and using (1.4), we obtain:

$$\begin{aligned} \alpha \int _{\Omega }|\nabla T_{k}(u)|^{2} \le \int _{\Omega }f\,T_{k}(u)\,. \end{aligned}$$

Letting k tend to infinity, and using Fatou lemma in the left-hand side, and Lebesgue theorem in the right-hand one, we obtain:

$$\begin{aligned} \alpha \int _{\Omega }|\nabla u|^{2} \le \int _{\Omega }f\,u\,, \end{aligned}$$

and from this inequality, it is easy to prove (using Sobolev embedding and Hölder inequality) that:

which is one half of (3.12). The second half can be obtained by choosing $G_{k}(T_{h}(u))$, letting h tend to infinity, and then following the proof of Théorème 4.1 of the paper [11] by G. Stampacchia. $\square$

We can now prove an existence result for solutions of (1.3), in the case of bounded data f.

Theorem 3.4

Let f be a function in $L^{\infty }(\Omega )$, and let $r > 1$. Then, there exist u and $\psi$, weak solutions of the system :

$$\begin{aligned} \left\{ \begin{array}{l} \displaystyle u \in W_0^{1,2}(\Omega ): -{\text {div}}\left( \left[ a(x) + \int _{\Omega }|\nabla u|^{2}\right] \nabla u\right) + \psi \,|u|^{r-2}\,u = f\,, \\ \displaystyle \psi \in W_0^{1,2}(\Omega ): -{\text {div}}(M(x)\nabla \psi ) = |u|^{r}\,. \end{array} \right. \end{aligned}$$

(3.13)

Furthermore, u and $\psi$ belong to $L^{\infty }(\Omega )$, and $\psi \ge 0$.

Proof.  Let $\sigma$ in ${\mathbb {N}}$ and v in $W_0^{1,2}(\Omega )$; by Lax–Milgram theorem, there exists a unique solution $\varphi$ of :

$$\begin{aligned} \varphi \in W_0^{1,2}(\Omega ): -{\text {div}}(M(x)\nabla \varphi ) = |T_{\sigma }(v)|^{r}\,. \end{aligned}$$

(3.14)

By standard elliptic estimates, one has that:

(3.15)

for some positive constant C. By the maximum principle, $\varphi \ge 0$, so that, by Proposition 3.3, there exists a unique solution u of:

$$\begin{aligned} u \in W_0^{1,2}(\Omega ): -{\text {div}}\left( \left[ a(x) + \int _{\Omega }|\nabla v|^{2}\right] \nabla u\right) + \varphi \,|u|^{r-2}\,u = f\,. \end{aligned}$$

Since, by (3.12), one has:

the ball $B_{R}(0)$ of $W_0^{1,2}(\Omega )$ is invariant for the map $S : v \mapsto u$. We are going to prove that the map S is completely continuous.

We begin by proving that if $\{v_{n}\}$ is bounded in $W_0^{1,2}(\Omega )$, then there exists a subsequence of $\{u_{n}= S(v_{n})\}$ which is strongly convergent in $W_0^{1,2}(\Omega )$. Indeed, since the ball $B_{R}(0)$ is invariant for S, the sequence $\{u_{n}\}$ is bounded in $W_0^{1,2}(\Omega )$; hence, up to subsequences, it will converge, weakly in $W_0^{1,2}(\Omega )$, strongly in (for example) $L^{2}(\Omega )$, and almost everywhere in $\Omega$, to a function u. Since, by (3.12), the sequence $\{u_{n}\}$ is bounded in $L^{\infty }(\Omega )$, we have that $u_{n}$ strongly converges to u in $L^{p}(\Omega )$ for every $p \ge 1$. Furthermore, if $\varphi _{n}$ is the solution of (3.14) with datum $|T_{\sigma }(v_{n})|^{r}$, then by (3.15) the sequence $\{\varphi _{n}\}$ is bounded in $W_0^{1,2}(\Omega )$, so that, up to subsequences, it strongly converges to some function $\varphi$ in (for example) $L^{2}(\Omega )$.

We now choose $u_{n}- u$ as test function in the equation solved by $u_{n}$ to have that:

$$\begin{aligned} \int _{\Omega }\left[ a(x) + \int _{\Omega }|\nabla v_{n}|^{2}\right] \nabla u_{n}\nabla (u_{n}- u) + \int _{\Omega }\varphi _{n}\,|u_{n}|^{r-2}u_{n}\,(u_{n}- u) = \int _{\Omega }f\,(u_{n}- u)\,. \end{aligned}$$

Adding and subtracting the term:

$$\begin{aligned} \int _{\Omega }\left[ a(x) + \int _{\Omega }|\nabla v_{n}|^{2}\right] \nabla u \nabla (u_{n}- u)\,, \end{aligned}$$

we obtain, using (1.4) and dropping a positive term:

$$\begin{aligned} \begin{array}{l} \displaystyle \alpha \int _{\Omega }|\nabla (u_{n}- u)|^{2} + \int _{\Omega }\varphi _{n}\,|u_{n}|^{r-2}u_{n}\,(u_{n}- u) \\ \displaystyle \qquad \le \int _{\Omega }f\,(u_{n}- u) + \int _{\Omega }\left[ a(x) + \int _{\Omega }|\nabla v_{n}|^{2}\right] \nabla u \nabla (u_{n}- u)\,. \end{array} \end{aligned}$$

Since f belongs to $L^{\infty }(\Omega )$, and $u_{n}- u$ tends to zero in (for example) $L^{1}(\Omega )$, we have that:

$$\begin{aligned} \lim _{n \rightarrow +\infty }\,\int _{\Omega }f\,(u_{n}- u) = 0\,, \end{aligned}$$

while, since $a(x)\nabla u$ belongs to $(L^{2}(\Omega ))^{N}$, $\{v_{n}\}$ is bounded in $W_0^{1,2}(\Omega )$, and $\nabla (u_{n}- u)$ tends to zero weakly in $(L^{2}(\Omega ))^{N}$, we have that:

$$\begin{aligned} \begin{array}{l} \displaystyle \lim _{n \rightarrow +\infty }\,\int _{\Omega }\left[ a(x) + \int _{\Omega }|\nabla v_{n}|^{2}\right] \nabla u \nabla (u_{n}- u) = \lim _{n \rightarrow +\infty }\,\int _{\Omega }a(x)\nabla u \nabla (u_{n}- u) \\ \displaystyle \qquad + \lim _{n \rightarrow +\infty }\,\left[ \int _{\Omega }|\nabla v_{n}|^{2}\right] \int _{\Omega }\nabla u \nabla (u_{n}- u) = 0\,. \end{array} \end{aligned}$$

On the other hand, since $\varphi _{n}$ is strongly convergent to $\varphi$ in $L^{2}(\Omega )$, and $|u_{n}|^{r-2}u_{n}\,(u_{n}- u)$ is strongly convergent to zero in $L^{2}(\Omega )$, as well (recall that $\{u_{n}\}$ strongly converges to u in every $L^{p}(\Omega )$), we have that:

$$\begin{aligned} \lim _{n \rightarrow +\infty }\,\int _{\Omega }\varphi _{n}\,|u_{n}|^{r-2}u_{n}\,(u_{n}- u) = 0\,. \end{aligned}$$

These three convergences imply that:

$$\begin{aligned} \lim _{n \rightarrow +\infty }\,\alpha \int _{\Omega }|\nabla (u_{n}- u)|^{2} = 0\,, \end{aligned}$$

so that the sequence $\{u_{n}= S(v_{n})\}$ is strongly convergent in $W_0^{1,2}(\Omega )$, up to subsequences. This proves that the map S is compact. To prove its continuity, let $\{v_{n}\}$ be a sequence strongly convergent to v in $W_0^{1,2}(\Omega )$, and let $u_{n}=S(v_{n})$. If $\varphi _{n}$ is the solution of:

$$\begin{aligned} \varphi _{n} \in W_0^{1,2}(\Omega ): -{\text {div}}(M(x)\nabla \varphi _{n}) = |T_{\sigma }(v_{n})|^{r}\,, \end{aligned}$$

since $\{|T_{\sigma }(v_{n})|^{r}\}$ is strongly convergent in (for example) $L^{2}(\Omega )$, then $\varphi _{n}$ is strongly convergent in $W_0^{1,2}(\Omega )$ to $\varphi$, the unique solution of:

$$\begin{aligned} \varphi \in W_0^{1,2}(\Omega ): -{\text {div}}(M(x)\nabla \varphi ) = |T_{\sigma }(v)|^{r}\,. \end{aligned}$$

On the other hand, since the sequence $\{v_{n}\}$ is bounded in $W_0^{1,2}(\Omega )$, the result proved before yields that, up to subsequences, the sequence $\{u_{n}\}$ is strongly convergent in $W_0^{1,2}(\Omega )$ to some function u, and the sequence $\{\varphi _{n}\,|u_{n}|^{r-2}\,u_{n}\}$ is strongly convergent in (at least) $L^{1}(\Omega )$ to $\varphi \,|u|^{r-2}\,u$. Therefore, if $\eta$ belongs to $W_0^{1,2}(\Omega )\cap L^{\infty }(\Omega )$, one can pass to the limit in n in the identities:

$$\begin{aligned} \int _{\Omega }\left[ a(x) + \int _{\Omega }|\nabla v_{n}|^{2}\right] \nabla u_{n}\nabla \eta + \int _{\Omega }\varphi _{n}\,|u_{n}|^{r-2}\,u_{n}\,\eta = \int _{\Omega }f\,\eta \,, \end{aligned}$$

to have that u is such that:

$$\begin{aligned} \int _{\Omega }\left[ a(x) + \int _{\Omega }|\nabla v|^{2}\right] \nabla u \nabla \eta + \int _{\Omega }\varphi \,|u|^{r-2}\,u\,\eta = \int _{\Omega }f\,\eta \,, \quad \forall \eta \in W_0^{1,2}(\Omega )\cap L^{\infty }(\Omega ). \end{aligned}$$

This implies, by uniqueness of the solution, that $u = S(v)$, and that the whole sequence $\{u_{n}\}$ is strongly convergent in $W_0^{1,2}(\Omega )$ to u; therefore, S is continuous, and so, S is completely continuous.

Therefore, by Schauder fixed point theorem, there exists u in $W_0^{1,2}(\Omega )$, such that $S(u) = u$; if we define $\psi$ as the unique solution of:

$$\begin{aligned} \psi \in W_0^{1,2}(\Omega ): -{\text {div}}(M(x)\nabla \psi ) = |T_{\sigma }(u)|^{r}\,, \end{aligned}$$

we have proved that for every $\sigma$ in ${\mathbb {N}}$, there exist weak solutions $u_{\sigma }$ and $\psi _{\sigma }$ of the system:

$$\begin{aligned} \left\{ \begin{array}{l} \displaystyle u_{\sigma } \in W_0^{1,2}(\Omega ): -{\text {div}}\Big (\left[ a(x) + \int _{\Omega }|\nabla u_{\sigma }|^{2}\right] \nabla u_{m}\Big ) + \psi _{\sigma }\,|u_{\sigma }|^{r-2}\,u_{\sigma } = f\,, \\ \displaystyle \psi _{\sigma } \in W_0^{1,2}(\Omega ): -{\text {div}}(M(x)\nabla \psi _{\sigma }) = |T_{\sigma }(u_{\sigma })|^{r}\,. \end{array} \right. \end{aligned}$$

We observe now that, by (3.12), we have that:

Therefore, if $\sigma \ge R$, we have $T_{\sigma }(u_{\sigma }) = u_{\sigma }$, so that the functions $u {\mathop {=}\limits ^{\mathrm{def}}}u_{\sigma }$ and $\psi {\mathop {=}\limits ^{\mathrm{def}}}\psi _{\sigma }$ are weak solutions of the system:

$$\begin{aligned} \left\{ \begin{array}{l} \displaystyle u \in W_0^{1,2}(\Omega ): -{\text {div}}\Big (\left[ a(x) + \int _{\Omega }|\nabla u|^{2}\right] \nabla u\Big ) + \psi \,|u|^{r-2}\,u = f\,, \\ \displaystyle \psi \in W_0^{1,2}(\Omega ): -{\text {div}}(M(x)\nabla \psi ) = |u|^{r}\,. \end{array} \right. \end{aligned}$$

Furthermore, both u and $\psi$ (by Théorème 4.1 of [11]) belong to $L^{\infty }(\Omega )$, and $\psi \ge 0$, as desired. $\square$