On the $N$-dimensional Schr\"{o}dinger--Poisson--Slater equation \`a la Brezis--Nirenberg

TL;DR

This paper establishes the existence and multiplicity of solutions for N-dimensional Schrödinger--Poisson--Slater equations with critical exponents using Pohozaev's identity and Nehari manifold methods.

Contribution

It introduces a novel approach combining Pohozaev's identity and Nehari manifold to analyze solutions with critical exponents in high-dimensional Schrödinger--Poisson--Slater equations.

Findings

Proved existence of solutions for the equations.

Established multiplicity results for solutions.

Analyzed solutions at prescribed energy levels.

Abstract

With aid of the Pohozaev's identity and Nehari manifold, we prove the existence and multiplicity of solutions to $N$-dimensional Schr\"{o}dinger--Poisson--Slater type equations involving critical exponents, by considering prescribed energy solutions.