On Nodal Solutions with A Prescribed Number of Nodes for a Schrödinger-Poisson system

Abstract

We study a class of Schrödinger-Poisson problems involving the nonlinearity $f(x)|u|^{p-2}u$ $(2 < p < 4)$ in $\mathbb{R}^3.$ Such problems cannot be studied directly by applying the general Nehari manifold, because $(PS)$ sequence may not be bounded. In this paper, by developing some useful analytical techniques and introducing a novel definition of the Nehari manifold for the auxiliary system of equations, we resolve an open question posed in [S. Kim and J. Seok, Commun. Contemp. Math., 14 (2012)] of whether there exists a sign-changing solution $u_k^\lambda$ changing signs exactly $k$ times to the problem above for the remaining range $2 < p < 4$. Furthermore, the energy of $u_k^\lambda$ is strictly increasing in $k,$ as well as some asymptotic behaviors of $u_k^\lambda$ are obtained.