Semiclassical solutions for critical Schr\"odinger-Poisson systems involving multiple competing potentials

TL;DR

This paper investigates semiclassical solutions for a Schr"odinger-Poisson system with multiple, sign-changing potentials and critical Sobolev exponent, establishing existence, concentration behavior, and nonexistence conditions.

Contribution

It introduces a novel analysis for systems with multiple competing potentials, proving existence and concentration of ground states using advanced variational methods.

Findings

Existence of ground state solutions in the semiclassical limit.

Convergence of solutions to the limiting problem's ground state.

Conditions for nonexistence of solutions.

Abstract

In this paper, a class of Schr\"{o}dinger-Poisson system involving multiple competing potentials and critical Sobolev exponent is considered. Such a problem cannot be studied with the same argument of the nonlinear term with only a positive potential, because the weight potentials set $\{Q_i(x)|1\le i \le m\}$ contains nonpositive, sign-changing, and nonnegative elements. By introducing the ground energy function and subtle analysis, we first prove the existence of ground state solution $v_\varepsilon$ in the semiclassical limit via the Nehari manifold and concentration-compactness principle. Then we show that $v_\varepsilon$ converges to the ground state solution of the associated limiting problem and concentrates at a concrete set characterized by the potentials. At the same time, some properties for the ground state solution are also studied. Moreover, a sufficient condition for the nonexistence of the ground state solution is obtained.