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: [Formula: see text] where [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text] is a bounded domain. This problem may be seen as a nonlocal perturbation of the classical Lane–Emden equation [Formula: see text] in [Formula: see text]. The problem has a variational functional with a nonlocal term [Formula: see text]. The appearance of the nonlocal term makes the variational functional very different from the local case [Formula: see text] for which the problem has ground state solutions and least energy nodal solutions if [Formula: see text]. The problem may also be viewed as a nonlocal Choquard equation with a local perturbation term when [Formula: see text]. For [Formula: see text], we show that although ground state solutions always exist, the existence of least energy nodal solution depends on [Formula: see text]: for [Formula: see text], there does not exist a least energy nodal solution while for [Formula: see text], such a solution exists. Note that [Formula: see text] is a critical value. In the case of a linear local perturbation, i.e. [Formula: see text], if [Formula: see text], the problem has a positive ground state and a least energy nodal solution. However, if [Formula: see text], the problem has a ground state which changes sign. Hence, it is also a least energy nodal solution.