Concentrating Solutions for Fractional Schrödinger–Poisson Systems with Critical Growth

Abstract

In this paper, we consider a class of fractional Schrödinger–Poisson systems (−Δ)su+λV(x)u+ϕu=f(u)+|u|2s*−2u and (−Δ)tϕ=u2 in R3, where s,t∈(0,1) with 2s+2t>3, λ>0 denotes a parameter, V:R3→R admits a potential well Ω≜intV−1(0) and 2s*≜63−2s is the fractional Sobolev critical exponent. Given some reasonable assumptions as to the potential V and the nonlinearity f, with the help of a constrained manifold argument, we conclude the existence of positive ground state solutions for some sufficiently large λ. Upon relaxing the restrictions on V and f, we utilize the minimax technique to show that the system has a positive mountain-pass type by introducing some analytic tricks. Moreover, we investigate the asymptotical behavior of the obtained solutions when λ→+∞.

Keywords

• fractional Schrödinger–Poisson systems
• steep potential well
• Sobolev critical growth
• existence
• concentration
• variational method