Positive solutions for nonlinear schrödinger–poisson systems with general nonlinearity

Abstract

In this paper, we study a class of Schrödinger-Poisson (SP) systems with general nonlinearity where the nonlinearity does not require Ambrosetti-Rabinowitz and Nehari monotonic conditions. We establish new estimates and explore the associated energy functional which is coercive and bounded below on Sobolev space. Together with Ekeland variational principle, we prove the existence of ground state solutions. Furthermore, when the ‘charge’ function is greater than a fixed positive number, the (SP) system possesses only zero solutions. In particular, when ‘charge’ function is radially symmetric, we establish the existence of three positive solutions and the symmetry breaking of ground state solutions.

Appendix

Let

$$\begin{aligned} \mathbf {A}_{0}:= & {} \left\{ u\in H_{r}^{1}\left( \mathbb {R}^{3}\right) :\int _{ \mathbb {R}^{3}}F\left( u\right) dx-\frac{1}{2}\left\| u\right\| _{H^{1}}^{2}>0\right\} ; \\ \overline{\mathbf {A}}_{0}:= & {} \left\{ u\in H^{1}\left( \mathbb {R}^{3}\right) :\int _{\mathbb {R}^{3}}f\left( u\right) udx-\left\| u\right\| _{H^{1}}^{2}>0\right\} \end{aligned}$$

and

$$\begin{aligned} \Lambda _{0}:= & {} \sup _{u\in \mathbf {A}_{0}}\frac{\int _{\mathbb {R} ^{3}}F\left( u\right) dx-\frac{1}{2}\left\| u\right\| _{H^{1}}^{2}}{ \int _{\mathbb {R}^{3}}\phi _{u}u^{2}dx}; \\ \overline{\Lambda }_{0}:= & {} \sup _{u\in \overline{\mathbf {A}}_{0}}\frac{\int _{ \mathbb {R}^{3}}f\left( u\right) udx-\left\| u\right\| _{H^{1}}^{2}}{ \int _{\mathbb {R}^{3}}\phi _{u}u^{2}dx}. \end{aligned}$$

Then we have the following results.

Theorem A.1

Suppose that conditions $\left( F1\right)$ and $\left( F2\right)$ hold. Then we have

$\left( i\right)$ $\mathbf {A}_{0}$ is a nonempty set.

$\left( ii\right)$ $0<\Lambda _{0}<\infty .$

Proof

$\left( i\right)$ If $q=2,$ then by $a_{q}>1$ and Fatou’s lemma, for $u\in H_{r}^{1}\left( \mathbb {R}^{3}\right)$ with $\left\| u\right\| _{H^{1}}^{2}-a_{q}\int _{\mathbb {R}^{3}}u^{2}dx<0,$ we have

$$\begin{aligned} \lim _{t\rightarrow \infty }\frac{1}{t^{2}}\left[ \frac{1}{2}\left\| tu\right\| _{H^{1}}^{2}-\int _{\mathbb {R}^{3}}F\left( tu\right) dx\right] =2\left( \left\| u\right\| _{H^{1}}^{2}-a_{q}\int _{\mathbb {R} ^{3}}u^{2}dx\right) <0, \end{aligned}$$

and so there exists $e\in H_{r}^{1}\left( \mathbb {R}^{3}\right)$ such that

$$\begin{aligned} \int _{\mathbb {R}^{3}}F\left( e\right) dx-\frac{1}{2}\left\| e\right\| _{H^{1}}^{2}>0. \end{aligned}$$

If $2<q<3,$ then by $a_{q}>0$ and Fatou’s lemma, for $u\in H_{r}^{1}\left( \mathbb {R}^{3}\right) \setminus \left\{ 0\right\} ,$ we have

$$\begin{aligned} \lim _{t\rightarrow \infty }\frac{1}{t^{q}}\left[ \frac{1}{2}\left\| tu\right\| _{H^{1}}^{2}-\int _{\mathbb {R}^{3}}F\left( tu\right) dx\right] =-a_{q}\int _{\mathbb {R}^{3}}\left| u\right| ^{q}dx<0, \end{aligned}$$

and so there exists $\widehat{e}\in H_{r}^{1}\left( \mathbb {R}^{3}\right)$ such that

$$\begin{aligned} \int _{\mathbb {R}^{3}}F\left( \widehat{e}\right) dx-\frac{1}{2}\left\| \widehat{e}\right\| _{H^{1}}^{2}>0. \end{aligned}$$

This implies that $\mathbf {A}_{0}$ is nonempty.

$\left( ii\right)$ For each $u\in \mathbf {A}_{0}$ there exists $\lambda ^{*}>0$ such that

$$\begin{aligned} \frac{1}{2}\left\| u\right\| _{H^{1}}^{2}+\frac{\lambda ^{*}}{4} \int _{\mathbb {R}^{3}}\phi _{u}u^{2}dx-\int _{\mathbb {R}^{3}}F\left( u\right) dx<0 \end{aligned}$$

or

$$\begin{aligned} \frac{\lambda ^{*}}{4}<\frac{\int _{\mathbb {R}^{3}}F\left( u\right) dx- \frac{1}{2}\left\| u\right\| _{H^{1}}^{2}}{\int _{\mathbb {R}^{3}}\phi _{u}u^{2}dx}, \end{aligned}$$

indicating that there exists $\widehat{\lambda }^{*}>0$ such that $\Lambda _{0}\ge \widehat{\lambda }^{*}.$ Next, we show that $0<\Lambda _{0}<\infty .$ By conditions $\left( F1\right)$ and $\left( F2\right) ,$ there exists $C_{1}>0$ such that

$$\begin{aligned} F\left( u\right) \le \frac{1}{2}u^{2}+C_{1}\left| u\right| ^{3}. \end{aligned}$$

(A.1)

Since

$$\begin{aligned} C_{1}\int _{\mathbb {R}^{3}}\left| u\right| ^{3}dx= & {} C_{1}\int _{ \mathbb {R}^{3}}\left( -\Delta \phi _{u}\right) \left| u\right| dx=C_{1}\int _{\mathbb {R}^{3}}\left\langle \nabla \phi _{u},\nabla \left| u\right| \right\rangle dx \\\le & {} \frac{1}{2}\int _{\mathbb {R}^{3}}\left| \nabla u\right| ^{2}dx+ \frac{C_{1}^{2}}{2}\int _{\mathbb {R}^{3}}\left| \nabla \phi _{u}\right| ^{2}dx \\= & {} \frac{1}{2}\int _{\mathbb {R}^{3}}\left| \nabla u\right| ^{2}dx+ \frac{C_{1}^{2}}{2}\int _{\mathbb {R}^{3}}\phi _{u}u^{2}dx\text { for all }u\in H^{1}(\mathbb {R}^{3}), \end{aligned}$$

by (A.1), we have

$$\begin{aligned} \frac{\int _{\mathbb {R}^{3}}F\left( u\right) dx-\frac{1}{2}\left\| u\right\| _{H^{1}}^{2}}{\int _{\mathbb {R}^{3}}\phi _{u}u^{2}dx}\le & {} \frac{C_{1}^{2}}{2}\times \frac{\frac{1}{2}\int _{\mathbb {R} ^{3}}u^{2}dx+C_{1}\int _{\mathbb {R}^{3}}\left| u\right| ^{3}dx-\frac{1 }{2}\left\| u\right\| _{H^{1}}^{2}}{C_{1}\int _{\mathbb {R} ^{3}}\left| u\right| ^{3}dx-\frac{1}{2}\int _{\mathbb {R} ^{3}}\left| \nabla u\right| ^{2}dx} \\\le & {} \frac{C_{1}^{2}}{2}\times \frac{C_{1}\int _{\mathbb {R}^{3}}\left| u\right| ^{3}dx-\frac{1}{2}\int _{\mathbb {R}^{3}}\left| \nabla u\right| ^{2}dx}{C_{1}\int _{\mathbb {R}^{3}}\left| u\right| ^{3}dx-\frac{1}{2}\int _{\mathbb {R}^{3}}\left| \nabla u\right| ^{2}dx} \\= & {} \frac{C_{1}^{2}}{2}. \end{aligned}$$

Thus,

$$\begin{aligned} 0<\Lambda _{0}:=\sup _{u\in \mathbf {A}_{0}}\frac{\int _{\mathbb {R}^{3}}F\left( u\right) dx-\frac{1}{2}\left\| u\right\| _{H^{1}}^{2}}{\int _{\mathbb {R} ^{3}}\phi _{u}u^{2}dx}\le \frac{C_{1}^{2}}{2}. \end{aligned}$$

This completes the proof. $\square$

Theorem A.2

Suppose that conditions $\left( F1\right)$ and $\left( F2\right)$ hold. Then we have

$\left( i\right)$ $\overline{\mathbf {A}}_{0}$ is a nonempty set.

$\left( ii\right)$ $0<\overline{\Lambda }_{0}<\infty .$

Proof

The proof is similar to the argument used in Theorem A.1 and is omitted here. $\square$