Semiclassical ground state solutions for critical Schrödinger-Poisson systems with lower perturbations

Abstract

In this paper, we study the following singularly perturbed Schrödinger-Poisson system{−ε2△u+V(x)u+ϕu=f(u)+u5,x∈R3,−ε2△ϕ=u2,x∈R3, where ε is a small positive parameter, V∈C(R3,R) and f∈C(R,R) satisfies neither the usual Ambrosetti-Rabinowitz type condition nor any monotonicity condition on f(u)/u3. By using some new techniques and subtle analysis, we prove that there exists a constant ε0>0 determined by V and f such that for ε∈(0,ε0] the above system admits a semiclassical ground state solution vˆε with exponential decay at infinity. We also study the asymptotic behavior of {vˆε} as ε→0. In particular, our results can be applied to the nonlinearity f(u)∼|u|q−2u for q∈[3,4], and extend the previous work that only deals with the case in which f(u)∼|u|q−2u for q∈(4,6).