The existence of positive least energy solutions for a class of Schrodinger-Poisson systems involving critical nonlocal term with general nonlinearity

TL;DR

This paper proves the existence of positive least energy solutions for a Schrödinger-Poisson system with a critical nonlocal term and general nonlinearity, using new analytical methods without requiring the Ambrosetti-Rabinowitz condition.

Contribution

It introduces novel analytical techniques and a Pohožaev type manifold approach to establish solutions without the usual nonlinear growth restrictions.

Findings

Existence of positive least energy solutions under certain conditions on V(x)

No need for Ambrosetti-Rabinowitz condition or monotonicity assumptions

Applicable to Schrödinger-Poisson systems with critical nonlocal terms

Abstract

The present study is concerned with the following Schr\"{o}dinger-Poisson system involving critical nonlocal term with general nonlinearity: $$ \left\{ \begin{array}{ll} -\Delta u+V(x)u- \phi |u|^3u= f(u), & x\in\mathbb{R}^3, -\Delta \phi= |u|^5, & x\in\mathbb{R}^3,\\ \end{array} \right. $$ Under certain assumptions on non-constant $V(x)$, the existence of a positive least energy solution is obtained by using some new analytical skills and Poho\v{z}aev type manifold. In particular, the Ambrosetti-Rabinowitz type condition or monotonicity assumption on the nonlinearity is not necessary.