Multiplicity and concentration of positive solutions for quasilinear Schrödinger-Poisson system with critical nonlinearity

Abstract

In this paper, we are interested in the following Schrödinger-Poisson system { − ε p ∆ p u + V ( x ) | u | p − 2 u + ϕ | u | p − 2 u = f ( u )+ | u | p ∗ − 2 u in R 3 , − ε 2 ∆ ϕ = | u | p in R 3 , where ε>0 is a parameter, 3 2 < p < 3 , ∆ p u = div ( | ∇ u | p − 2 ∇ u ) , p ∗ = 3 p 3 − p , V : R 3 → R is a positive function with a local minimum and f is subcritical growth. Based on the penalization method, Nehari manifold techniques and Ljusternik-Schnirelmann category theory, we obtain the multiplicity and concentration of positive solutions.