Multiple nodal solutions of planar Stein-Weiss equations

TL;DR

This paper proves the existence of multiple sign-changing solutions for planar Stein-Weiss equations with nonlinearities of subcritical or critical growth, using a combination of gluing and Nehari manifold methods.

Contribution

It introduces a novel approach combining gluing and Nehari manifold techniques to establish multiple nodal solutions for Stein-Weiss problems.

Findings

Existence of at least one radially symmetric ground state solution for each positive integer k.

Solutions change sign exactly k times, demonstrating multiple nodal solutions.

Applicable to nonlinearities with subcritical or critical Trudinger-Moser growth.

Abstract

In this paper, our goal is to investigate the existence of multiple nodal solutions to a class of planar Stein-Weiss problems involving a nonlinearity $f$ with subcritical or critical growth in the sense of Trudinger-Moser. To achieve this, we combine a gluing approach with the Nehari manifold argument. We demonstrate that for any positive integer $k\in \mathbb{N}$, the problem studied has at least one radially symmetrical ground state solution that changes sign exactly $k$-times.