Multiple nodal solutions for a critical nonlinear Choquard equation

Abstract

In this paper, our goal is to investigate the existence of multiple nodal solutions to the following class of problems \begin{equation*} \begin{cases} -\Delta u+ V(x) u = \big(I_{\alpha}\ast |u|^{2_{\alpha}^* }\big) |u|^{2_{\alpha}^* -2}u +\lambda g(u) & \text{in } \mathbb{R}^N,\\ u \in H^1(\mathbb{R}^N), \end{cases} \end{equation*} where $I_{\alpha}$ represents the Riesz potential, $V$ is a continuous potential, $N\geq 3$, $0< \alpha< N$, $\lambda> 0$, $2^*_{\alpha}= ({N+\alpha})/({N-2})$ is the critical exponent in the sense of the Hardy-Littlewood-Sobolev inequality and $g$ is a local or nonlocal superlinear perturbation with subcritical growth in the sense of Sobolev or Hardy-Littlewood-Sobolev, respectively. For this matter, combining variational methods with the technique introduced in \cite{primeiro}, which was adapted for Choquard equations in \cite{GuiGuo}, \cite{nodal}, we show that for any positive $k\in \mathbb{N}$ the previous problem with a nonlinearity involving critical growth has at least a radially symmetrical ground state solution changing sign exactly $k$-times.