The concentration behavior of ground state solutions for a critical fractional Schrödinger–Poisson system

Abstract

In this paper, we study the following critical fractional Schrödinger–Poisson system trueright115.20007pt{ε2s(−Δ)su+Vfalse(xfalse)u+ϕu=Pfalse(xfalse)ffalse(ufalse)+Qfalse(xfalse)|u|2s∗−2u,indouble-struckR3,ε2t(−Δ)tϕ=u2,indouble-struckR3,where ε>0 is a small parameter, s∈(34,1),t∈false(0,1false) and 2s+2t>3, 2s∗:=63−2s is the fractional critical exponent for 3‐dimension, Vfalse(xfalse)∈C(double-struckR3) has a positive global minimum, and Pfalse(xfalse),Qfalse(xfalse)∈C(double-struckR3) are positive and have global maximums. We obtain the existence of a positive ground state solution by using variational methods, and we determine a concrete set related to the potentials V,P and Q as the concentration position of these ground state solutions as ε→0+. Moreover, we consider some properties of these ground state solutions, such as convergence and decay estimate.

Concentration of positive solutions for critical quasilinear Schrödinger equation with competing potentials

“…In [7], the authors found new concentration phenomena for Dirac equations with competing potentials and subcritical or critical nonlinearities, respectively. See also [8,31,32] for other related results.…”

Existence and multiplicity of normalized solutions for a class of fractional Schrödinger–Poisson equations

“…By the monotone trick, concentration-compactness principe and a global compactness lemma he establishes the existence of ground state solutions. For other existence results we refer to [13,19,25,26,27,29,35,36,37,39] and the references therein.…”

Existence and concentration of solution for Schrödinger-Poisson system with local potential

“…In this case, V and Q all want to attract solutions to their local extreme point, the we call it local competing potential functions. In fact, if the condition is global, there have been some results for Schrödinger-Poisson system [34,35,38] and fractional Schrödinger-Poisson system [40,43].…”