Sign-changing radial solutions for the Schr\"odinger-Poisson-Slater problem

TL;DR

This paper proves the existence of sign-changing radial solutions with a specified number of nodes for the Schr"odinger-Poisson-Slater system in 3 and related nonlocal equations, including solutions with prescribed nodal properties.

Contribution

It establishes the existence of infinitely many radially symmetric solutions with prescribed nodal counts for both the SPS system in 3 and a nonlocal parabolic problem.

Findings

Existence of solutions with exactly k sign changes in 3

Construction of solutions with k+1 nodal regions in the parabolic setting

Extension of results to nonlocal equations in bounded domains

Abstract

We consider the Schr\"odinger-Poisson-Slater (SPS) system in $\R^3$ and a nonlocal SPS type equation in balls of $\mathbb R^3$ with Dirichlet boundary conditions. We show that for every $k\in\mathbb N$ each problem considered admits a nodal radially symmetric solution which changes sign exacly $k$ times in the radial variable. Moreover when the domain is the ball of $\mathbb R^3$ we obtain the existence of radial global solutions for the associated nonlocal parabolic problem having $k+1$ nodal regions at every time.