Standing waves for a class of Schrödinger-Poisson equations in ${\mathbb{R}}^3$ involving critical Sobolev exponents

Abstract

We are concerned with the following Schrödinger-Poisson equation with critical nonlinearity: \[\left\{\begin{gathered} - {\varepsilon ^2}Δu + V(x)u + ψu = λ|u{|^{p - 2}}u + |u{|^4}u{\text{in}}{\mathbb{R}^3}, \hfill - {\varepsilon ^2}Δψ= {u^2}{\text{in}}{\mathbb{R}^3},{\text{}}u > 0,{\text{}}u \in {H^1}({\mathbb{R}^3}), \hfill \end{gathered} \right. \] where $\epsilon >0$ is a small positive parameter, $\lambda >0$, $3<p\le 4$. Under certain assumptions on the potential $V$, we construct a family of positive solutions $u_{\epsilon }\in H^1({\mathbb{R}}^3)$ which concentrates around a local minimum of $V$ as $\epsilon \to 0$.

40 pages