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

TL;DR

This paper constructs positive standing wave solutions for a Schrödinger-Poisson equation with critical Sobolev exponent in three-dimensional space, showing concentration around potential minima as a small parameter tends to zero.

Contribution

It introduces a method to find positive solutions concentrating at potential minima for Schrödinger-Poisson equations with critical nonlinearity in D.

Findings

Constructed a family of positive solutions concentrating at potential minima.

Established existence results under certain conditions on the potential.

Analyzed the asymptotic behavior as the small parameter approaches zero.

Abstract

We are concerned with the following Schr\"odinger-Poisson equation with critical nonlinearity: \[\left\{\begin{gathered} - {\varepsilon ^2}\Delta u + V(x)u + \psi u = \lambda |u{|^{p - 2}}u + |u{|^4}u{\text{in}}{\mathbb{R}^3}, \hfill - {\varepsilon ^2}\Delta \psi = {u^2}{\text{in}}{\mathbb{R}^3},{\text{}}u > 0,{\text{}}u \in {H^1}({\mathbb{R}^3}), \hfill \end{gathered} \right. \] where $\varepsilon > 0$ 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 ${u_\varepsilon} \in {H^1}({\mathbb{R}^3})$ which concentrates around a local minimum of $V$ as $\varepsilon \to 0$.