2017/10/13 by Gui, Changfeng, Guo, Hui
#Analysis of PDEs (math.AP) #FOS: Mathematics
paper · doi:10.48550/arxiv.1710.05040
In this paper, we are interested in the least energy nodal solutions to the following nonlocal Choquard equation with a local term \-Δu · amp;=λ|u|p-2u+μϕ(x)|u|q-2u
-Δϕ · amp;=|u|q
u · amp;=ϕ=0. \begingathered· amp;in Ω,
· amp;in Ω,
· amp;on ∂Ω,\endgathered where λ,μ>0, p∈ [2,6), q∈ (1,5) and Ω⊂ ℝ3 is a bounded domain. This problem may be seen as a nonlocal perturbation of the classical Lane-Emden equation -Δu=λ|u|p-2u in Ω. The problem has a variational functional with a nonlocal term μ∫Ωϕ|u|q. The appearance of the nonlocal term makes the variational functional very different from the local case μ=0, for which the problem has ground state solutions and least energy nodal solutions if p∈ (2,6). The problem may also be viewed as a nonlocal Choquard equation with a local pertubation term when λ\not =0. For μ>0, we show that although ground state solutions always exist, the existence of least energy nodal solution depends on q: for q∈(1,2) there does not exist a least energy nodal solution while for q∈[2,5) such a solution exists. Note that q=2 is a critical value. In the case of a linear local perturbation, i.e., p=2, if λ