Existence and Concentration Behavior of Solutions of the Critical Schrödinger–Poisson Equation in R3

Jichao Wang,Ting Yu

doi:10.3390/math9050464

Abstract

In this paper, we study the singularly perturbed problem for the Schrödinger–Poisson equation with critical growth. When the perturbed coefficient is small, we establish the relationship between the number of solutions and the profiles of the coefficients. Furthermore, without any restriction on the perturbed coefficient, we obtain a different concentration phenomenon. Besides, we obtain an existence result.

Highlights

In Theorem 1, we obtain the existence of spikes on the strict global maximum of h
We pointed out that, when we seek multiplicity of solutions, it is crucial to prove the compactness of the Palais–Smale sequence
Many authors solved the problem by imposing the Rabinowitz type assumption, which is restrictive

Summary

Introduction

Under suitable assumptions on λ, V, b and f , they proved the existence and concentration behavior of positive ground state solutions. In [16,17], the authors considered the existence, multiplicity and concentration behavior of the critical Schrödinger–Poisson equation (8). Motivated by the above results, in this paper, we study the multiplicity and concentration behavior of positive solutions of (1). There exists ε∗ > 0 such that problem (1) has at least k different positive solutions wiε , i = 1, 2, . In Theorem 1, we obtain the existence of spikes (multiple solutions concentrating at a single point) on the strict global maximum of h. If V is non-radial, K ≡ 1, f = |u| p−2 u, the authors in [20,28] obtained the existence of ground state solutions of (10).

Preliminary Lemmas

Proof of Theorem 1

Proof of Theorems 2 and 3

Conclusions