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

TL;DR

This paper investigates the existence and properties of least energy nodal solutions for a nonlocal Choquard equation with local perturbations in a bounded domain, revealing how solution existence depends on parameters like q and λ.

Contribution

It provides new existence results for least energy nodal solutions in a nonlocal Choquard equation, highlighting the critical role of the parameter q and the perturbation term.

Findings

Existence of least energy nodal solutions depends on q: no solutions for q in (1,2), solutions exist for q in [2,5).

Ground state solutions always exist regardless of parameters.

Critical value q=2 influences the existence of nodal solutions.

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} -\Delta u&=\lambda|u|^{p-2}u+\mu \phi(x)|u|^{q-2}u\\ -\Delta \phi&=|u|^q\\ u&=\phi=0 \end{array}\right. \begin{gathered}\begin{array}{rll} &\mbox{in}\ \Omega,\\ &\mbox{in}\ \Omega,\\ &\mbox{on}\ \partial\Omega, \end{array}\end{gathered}\end{equation*} where $ \lambda,\mu>0, p\in [2,6), q\in (1,5)$ and $\Omega\subset \mathbb{R}^3$ is a bounded domain. This problem may be seen as a nonlocal perturbation of the classical Lane-Emden equation $-\Delta u=\lambda|u|^{p-2}u$ in $\Omega.$ The problem has a variational functional with a nonlocal term $\mu\int_{\Omega}\phi|u|^q$. The appearance of the nonlocal term makes the variational functional very different from the local case $\mu=0$, 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 $\lambda \not =0$. For $\mu>0$, 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 $\lambda<\lambda_1,$ the problem has a positive ground state and a least energy nodal solution. However, if $\lambda\geq \lambda_1,$ the problem has a ground state which changes sign. Hence it is also a least energy nodal solution.