On the nonlinear Schrödinger-Poisson systems with sign-changing potential. (English) Zbl 1329.35292
Authors’ abstract: In this paper we study a nonlinear Schrödinger-Poisson system
\[
\begin{cases}-\Delta u+V_{\lambda}(x)u+\mu K(x)\phi u=f(x,u)\;\text{ in } \mathbb{R}^3\\ -\Delta\phi=K(x)u^2\;\text{ in } \mathbb{R}^3\end{cases}
\]
where \(\mu >0\) is a parameter, \(V_{\lambda}\) is allowed to be sign-changing and \(f\) is an indefinite function. We require that \(V_\lambda:=\lambda V^+-V^-\) with \(V^+ \)having a bounded potential well \(\Omega\) whose depth is controlled by \(\lambda\) and \(V^-\geq 0\) for all \(x\in \mathbb{R}^3\). Under some suitable assumption on \(K\) and \(f\), the existence and the nonexistence of nontrivial solutions are obtained by using variational methods Furthermore, the phenomenon of concentration of solutions is explored as well.
Reviewer: Qin Meng Zhao (Beijing)
MSC:
| 35Q55 | NLS equations (nonlinear Schrödinger equations) |
| 35J50 | Variational methods for elliptic systems |
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