Multiple solutions for Schr{\"o}dinger-Poisson-Slater equations with critical growth

TL;DR

This paper proves the existence of multiple solutions for a critical growth Schrödinger-Poisson-Slater equation in three dimensions, using a simplified approach to verify the Palais-Smale condition, with results extending to related equations with external potentials.

Contribution

It introduces a new, simpler method to verify the Palais-Smale condition for critical growth equations, leading to multiple solutions for Schrödinger-Poisson-Slater equations.

Findings

Multiple solutions established for the zero mass Schrödinger-Poisson-Slater equation.

Verification of the (PS)$_c$ condition using a simplified method.

Extension of results to related equations with external potentials.

Abstract

We obtain multiple solutions for the zero mass Schr{\"o}dinger-Poisson-Slater equation \[ - \Delta u + \left( \frac{1}{4 \pi | x |} \ast u^2 \right) u = \lambda g (x) | u |^{p - 2} u + | u |^{6 - 2} u \text{, \ \ \ \ } u \in \mathcal{D}^{1, 2} (\mathbb{R}^3) \text{} \] for $\lambda \gg 1$, where $p \in (4, 6)$ and $g \in L^{6 / (6 - p)} (\mathbb{R}^3)$. The crucial (PS)$_c$ condition is verified using a simpler method. Similar multiplicity result is also obtained for related equation with an external potential.