On nodal solutions of a nonlocal Choquard equation in a bounded domain

Abstract

In this paper, we are interested in the least energy nodal solutions to the following nonlocal Choquard equation with a local term \begin{equation*}\left\{\begin{array}{rll} -Δu&=λ|u|^{p-2}u+μϕ(x)|u|^{q-2}u\\ -Δϕ&=|u|^q\\ u&=ϕ=0 \end{array}\right. \begin{gathered}\begin{array}{rll} &\mbox{in}\ Ω,\\ &\mbox{in}\ Ω,\\ &\mbox{on}\ \partialΩ, \end{array}\end{gathered}\end{equation*} where $ λ,μ>0, p\in [2,6), q\in (1,5)$ and $Ω\subset \mathbb{R}^3$ is a bounded domain. This problem may be seen as a nonlocal perturbation of the classical Lane-Emden equation $-Δu=λ|u|^{p-2}u$ in $Ω.$ The problem has a variational functional with a nonlocal term $μ\int_Ωϕ|u|^q$. 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 $p\in (2,6)$. 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 $q\in(1,2)$ there does not exist a least energy nodal solution while for $q\in[2,5)$ 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 $λ<λ_1,$ the problem has a positive ground state and a least energy nodal solution. However, if $λ\geq λ_1,$ the problem has a ground state which changes sign. Hence it is also a least energy nodal solution.