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

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 $\mathrm{\Omega }\subset {\mathbb{R}}^3$ is a bounded domain. This problem may be seen as a nonlocal perturbation of the classical Lane-Emden equation $-\mathrm{\Delta }u=\lambda \mathrm{\mid }u{\mathrm{\mid }}^{p-2}u$ in $\mathrm{\Omega }\mathrm{.}$ The problem has a variational functional with a nonlocal term $\mu {\int }_{\mathrm{\Omega }}\varphi \mathrm{\mid }u{\mathrm{\mid }}^q$. 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 $\lambda ≠0$. 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 $\lambda \ge {\lambda }_1,$ the problem has a ground state which changes sign. Hence it is also a least energy nodal solution.