Semiclassical Solutions for a Kind of Coupled Schrödinger Equations

Abstract

In this paper, we are concerned with the following coupled Schrödinger equations −λ2Δu+a1xu=cxv+a2xup−2u+a3xu2∗−2u,x∈ℝN,−λ2Δv+b1xv=cxu+b2xvp−2v+b3xv2∗−2v,x∈ℝN, where 2<p<2∗,2<q<2∗,2∗=2N/N−2, andN≥3; λ>0 is a parameter; and a1,a2,a3,b1,b2,b3,c∈CℝN,ℝ and u,v∈H1ℝN. Under some suitable conditions that a10=infa1=0 or b10=infb1=0 and cx2≤ϑa1xb1x with ϑ∈0,1, the above coupled Schrödinger system possesses nontrivial solutions if λ∈0,λ0, where λ0 is related to a1,a2,a3,b1,b2,b3, and N.