Existence of Solutions for a Modified Nonlinear Schrödinger System

Abstract EN

We are concerned with the following modified nonlinear Schrödinger system: -Δu+u-(1/2)uΔ(u2)=(2α/(α+β))|u|α-2|v|βu, x∈Ω, -Δv+v-(1/2)vΔ(v2)=(2β/(α+β))|u|α|v|β-2v, x∈Ω, u=0, v=0, x∈∂Ω, where α>2, β>2, α+β<2·2*, 2*=2N/(N-2) is the critical Sobolev exponent, and Ω⊂ℝN (N≥3) is a bounded smooth domain.

By using the perturbation method, we establish the existence of both positive and negative solutions for this system.