부산대학교 도서관

This article investigates the existence and properties of ground state solutions to the following nonlocal Hamiltonian elliptic system: (-Δ)12u+V0u=g(v),x∈R(-Δ)12v+V0v=f(u),x∈R,(-Δ)12V0>0RS(u,v)∈SL∞(R)SH12(R)×H12(R)(u,v)∈Sϵ(-Δ)12φ+V(x)φ=g(ψ),x∈Rϵ(-Δ)12ψ+V(x)ψ=f(φ),x∈R,ϵ>0V∈C(R)ϵ→0where (-Δ)12u+V0u=g(v),x∈R(-Δ)12v+V0v=f(u),x∈R,(-Δ)12V0>0RS(u,v)∈SL∞(R)SH12(R)×H12(R)(u,v)∈Sϵ(-Δ)12φ+V(x)φ=g(ψ),x∈Rϵ(-Δ)12ψ+V(x)ψ=f(φ),x∈R,ϵ>0V∈C(R)ϵ→0 is the square root Laplacian operator, (-Δ)12u+V0u=g(v),x∈R(-Δ)12v+V0v=f(u),x∈R,(-Δ)12V0>0RS(u,v)∈SL∞(R)SH12(R)×H12(R)(u,v)∈Sϵ(-Δ)12φ+V(x)φ=g(ψ),x∈Rϵ(-Δ)12ψ+V(x)ψ=f(φ),x∈R,ϵ>0V∈C(R)ϵ→0 and f,  g have critical exponential growth in (-Δ)12u+V0u=g(v),x∈R(-Δ)12v+V0v=f(u),x∈R,(-Δ)12V0>0RS(u,v)∈SL∞(R)SH12(R)×H12(R)(u,v)∈Sϵ(-Δ)12φ+V(x)φ=g(ψ),x∈Rϵ(-Δ)12ψ+V(x)ψ=f(φ),x∈R,ϵ>0V∈C(R)ϵ→0. Using minimization technique over some generalized Nehari manifold, we show that the set (-Δ)12u+V0u=g(v),x∈R(-Δ)12v+V0v=f(u),x∈R,(-Δ)12V0>0RS(u,v)∈SL∞(R)SH12(R)×H12(R)(u,v)∈Sϵ(-Δ)12φ+V(x)φ=g(ψ),x∈Rϵ(-Δ)12ψ+V(x)ψ=f(φ),x∈R,ϵ>0V∈C(R)ϵ→0 of ground state solutions is non empty. Moreover for (-Δ)12u+V0u=g(v),x∈R(-Δ)12v+V0v=f(u),x∈R,(-Δ)12V0>0RS(u,v)∈SL∞(R)SH12(R)×H12(R)(u,v)∈Sϵ(-Δ)12φ+V(x)φ=g(ψ),x∈Rϵ(-Δ)12ψ+V(x)ψ=f(φ),x∈R,ϵ>0V∈C(R)ϵ→0, u,  v are uniformly bounded in (-Δ)12u+V0u=g(v),x∈R(-Δ)12v+V0v=f(u),x∈R,(-Δ)12V0>0RS(u,v)∈SL∞(R)SH12(R)×H12(R)(u,v)∈Sϵ(-Δ)12φ+V(x)φ=g(ψ),x∈Rϵ(-Δ)12ψ+V(x)ψ=f(φ),x∈R,ϵ>0V∈C(R)ϵ→0 and uniformly decaying at infinity. We also show that the set (-Δ)12u+V0u=g(v),x∈R(-Δ)12v+V0v=f(u),x∈R,(-Δ)12V0>0RS(u,v)∈SL∞(R)SH12(R)×H12(R)(u,v)∈Sϵ(-Δ)12φ+V(x)φ=g(ψ),x∈Rϵ(-Δ)12ψ+V(x)ψ=f(φ),x∈R,ϵ>0V∈C(R)ϵ→0 is compact in (-Δ)12u+V0u=g(v),x∈R(-Δ)12v+V0v=f(u),x∈R,(-Δ)12V0>0RS(u,v)∈SL∞(R)SH12(R)×H12(R)(u,v)∈Sϵ(-Δ)12φ+V(x)φ=g(ψ),x∈Rϵ(-Δ)12ψ+V(x)ψ=f(φ),x∈R,ϵ>0V∈C(R)ϵ→0 up to translations. Furthermore under locally lipschitz continuity of f and g we obtain a suitable Pohožaev type identity for any (-Δ)12u+V0u=g(v),x∈R(-Δ)12v+V0v=f(u),x∈R,(-Δ)12V0>0RS(u,v)∈SL∞(R)SH12(R)×H12(R)(u,v)∈Sϵ(-Δ)12φ+V(x)φ=g(ψ),x∈Rϵ(-Δ)12ψ+V(x)ψ=f(φ),x∈R,ϵ>0V∈C(R)ϵ→0. We deduce the existence of semi-classical ground state solutions to the singularly perturbed system (-Δ)12u+V0u=g(v),x∈R(-Δ)12v+V0v=f(u),x∈R,(-Δ)12V0>0RS(u,v)∈SL∞(R)SH12(R)×H12(R)(u,v)∈Sϵ(-Δ)12φ+V(x)φ=g(ψ),x∈Rϵ(-Δ)12ψ+V(x)ψ=f(φ),x∈R,ϵ>0V∈C(R)ϵ→0where (-Δ)12u+V0u=g(v),x∈R(-Δ)12v+V0v=f(u),x∈R,(-Δ)12V0>0RS(u,v)∈SL∞(R)SH12(R)×H12(R)(u,v)∈Sϵ(-Δ)12φ+V(x)φ=g(ψ),x∈Rϵ(-Δ)12ψ+V(x)ψ=f(φ),x∈R,ϵ>0V∈C(R)ϵ→0 and (-Δ)12u+V0u=g(v),x∈R(-Δ)12v+V0v=f(u),x∈R,(-Δ)12V0>0RS(u,v)∈SL∞(R)SH12(R)×H12(R)(u,v)∈Sϵ(-Δ)12φ+V(x)φ=g(ψ),x∈Rϵ(-Δ)12ψ+V(x)ψ=f(φ),x∈R,ϵ>0V∈C(R)ϵ→0 satisfy the assumption (V) given below (see Sect. 1). Finally as (-Δ)12u+V0u=g(v),x∈R(-Δ)12v+V0v=f(u),x∈R,(-Δ)12V0>0RS(u,v)∈SL∞(R)SH12(R)×H12(R)(u,v)∈Sϵ(-Δ)12φ+V(x)φ=g(ψ),x∈Rϵ(-Δ)12ψ+V(x)ψ=f(φ),x∈R,ϵ>0V∈C(R)ϵ→0, we prove the existence of minimal energy solutions which concentrate around the closest minima of the potential V.