Multiple nodal and semi-nodal solutions to a nonlinear Choquard-type system

Abstract

We study the nonlinearly coupled Choquard-type system(0.1){−Δu1+μ1u1=a1(Iα⁎|u1|p)|u1|p−2u1+β(Iα⁎|u2|p)|u1|p−2u1,x∈Ω,−Δu2+μ2u2=a2(Iα⁎|u2|p)|u2|p−2u2+β(Iα⁎|u1|p)|u2|p−2u2,x∈Ω,u1=u2=0on∂Ω, where Ω is a bounded smooth domain in RN with N≥3, p∈(N+αN,N+αN−2), α∈(0,N), Iα is the Riesz potential, and μ1,μ2,a1,a2,β are positive constants. For every k∈N, we prove that there exists βk>0 such that system (0.1) possesses k nodal solutions and k semi-nodal solutions for β∈(0,βk) and p>2. Additionally, the existence of least energy nodal solutions is also obtained.