带有凹凸非线性项的Choquard方程的非平凡解

Nontrivial solutions for a nonlinear Choquard equations with concave-convex nonlinearities

研究带有凹凸非线性项的Choquard方程:-Δu+u=(I_(α)*|u|^(p))|u|^(p-2)u+μg(x,u)+λf(x,u),u∈H_(0)^(1)(Ω),其中Iα是里斯位势,Ω是R^(N)中的有界光滑区域,μ是参数,λ>0.通过变分法证明当p∈(N+α/N,N+α/(N-2))+(N≥1),α∈(0,N)及非线性扰动满足一些结构性假设时解的存在性.

In this paper,the nonlinear Choquard equation with the concave-convex nonlinearities-Δu+u=(I_(α)*|u|^(p))|u|^(p-2)u+μg(x,u)+λf(x,u),u∈H_(0)^1(1)(Ω)is considered,where I_(α)is a Riesz potential,Ωis a bounded smooth domain in R^(N),μis a parameter andλ>0.The existence of solutions by variational methods when p∈(N+α/N,N+α/(N-2))+,N≥1,α∈(0,N)and the nonlinear perturbation satisfies some structural assumptions is proved.