Positive Solutions for Schrödinger-Poisson Systems with Sign-Changing Potential and Critical Growth

He Peng Fei ,Hong-Min Suo

doi:10.1155/2020/3197596

Abstract

In this paper, we study the existence of positive solutions for Schrödinger-Poisson systems with sign-changing potential and critical growth. By using the analytic techniques and variational method, the existence and multiplicity of positive solutions are obtained.

Highlights

The existence, uniqueness, and multiplicity of positive solutions of systems like (2) have been extensively studied in the last few decades, such as [1,2,3,4,5,6,7,8,9,10,11,12,13,14]
Introduction and Main ResultIn this paper, we consider the existence and multiplicity of positive solutions of the Schrödinger-Poisson systems with critical nonlinearity of the following form: 8 >>< −Δu − φu = λf λ ðxÞjujq−2 u + juj4 u, in Ω, >>:−Δφ = u2, u = φ = 0, in Ω, ð1Þ on ∂Ω, where Ω is a smooth bounded domain in R3, 1 < q < 2, f λ = λf + + f −, λ > 0, and f ± = ±max f±f, 0g
The main objective of this paper is to look for new estimates and establish two positive solutions by analytic techniques

Summary

Introduction

The existence, uniqueness, and multiplicity of positive solutions of systems like (2) have been extensively studied in the last few decades, such as [1,2,3,4,5,6,7,8,9,10,11,12,13,14]. The authors studied the existence and multiplicity of positive solutions by the variational method and analytic techniques. 2. Existence of a First Positive Solution of System (1) For each u ðΩÞ, there exists a unique solution φu

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Conclusion