Bound and ground states for a class of Schrödinger–Poisson systems

Abstract

Abstract We are concerned with the following Schrödinger–Poisson system: {−Δu+u+K(x)ϕu=a(x)u3,x∈R3,−Δϕ=K(x)u2,x∈R3. $$ \textstyle\begin{cases} -\Delta u+u+K(x)\phi u=a(x)u^{3},& x\in \mathbb{R}^{3}, \\ -\Delta \phi =K(x)u^{2}, & x\in \mathbb{R}^{3}. \end{cases} $$ Assuming that K(x) $K(x)$ and a(x) $a(x)$ are nonnegative functions satisfying lim|x|→∞a(x)=a∞>0,lim|x|→∞K(x)=0, $$ \lim_{|x|\rightarrow \infty }a(x)=a_{\infty }>0, \qquad \lim _{|x|\rightarrow \infty }K(x)=0, $$ and other suitable conditions, we show the existence of bound and ground states via a global compactness lemma and the Nehari manifold. Our result extends the existence result of positive solutions for Schrödinger–Poisson system with more than three times growth by Cerami and Vaira (J. Differ. Equ. 248:521–543, 2010) to the system with three times growth.

Keywords

• Schrödinger–Poisson system
• Variational method
• Three times growth
• Bound state
• Lack of compactness