An upper bound for the least energy of a sign-changing solution to a zero mass problem

TL;DR

This paper establishes upper bounds for the least energy of sign-changing solutions to a nonlinear scalar field equation in high dimensions, revealing new energy thresholds depending on the dimension.

Contribution

It provides the first explicit upper bounds for the least energy of sign-changing solutions in a zero mass problem with specific nonlinearities.

Findings

Existence of nonradial sign-changing solutions with energy below certain thresholds.

Energy bounds are smaller than 12c_0 for N=5,6 and smaller than 10c_0 for N≥7.

Results depend on the dimension and the nature of the nonlinearity.

Abstract

We give an upper bound for the least energy of a sign-changing solution to the the nonlinear scalar field equation $$-\Delta u = f(u), \qquad u\in D^{1,2}(\mathbb{R}^{N}),$$ where $N\geq5$ and the nonlinearity $f$ is subcritical at infinity and supercritical near the origin. More precisely, we establish the existence of a nonradial sign-changing solution whose energy is smaller that $12c_0$ if $N=5,6$ and smaller than $10c_0$ if $N\geq 7$, where $c_0$ is the ground state energy.