The existence of least energy nodal solutions for some class of Kirchhoff equations and Choquard equations

TL;DR

This paper establishes the existence of least energy nodal solutions for certain nonlocal Kirchhoff and Choquard equations using variational methods on sign-changing manifolds, revealing new insights into their solution structures.

Contribution

It introduces a novel variational approach on the sign-changing Nehari manifold to find least energy nodal solutions for Kirchhoff and Choquard equations, which are nonlocal in nature.

Findings

Existence of least energy nodal solutions for Kirchhoff equations.

Existence of sign-changing solutions for Choquard equations.

Sign-changing solutions have energy between the least energy and twice the least energy.

Abstract

In this paper, we study the existence of least energy nodal solutions for some class of Kirchhoff type problems. Since Kirchhoff equation is a nonlocal one, the variational setting to look for sign-changing solutions is different from the local cases. By using constrained minimization on the sign-changing Nehari manifold, we prove the Kirchhoff problem has a least energy nodal solution with its energy exceeding twice the least energy. As a co-product of our approaches, we obtain the existence of least energy sign-changing solution for Choquard equations and show that the sign-changing solution has an energy strictly larger than the least energy and less than twice the least energy.