On nodal solutions with a prescribed number of nodes for a Kirchhoff-type problem

TL;DR

This paper investigates the existence, properties, and asymptotic behavior of multiple sign-changing solutions with a prescribed number of nodes for a Kirchhoff-type nonlinear PDE in three dimensions, extending previous work to the case where 2<p<4.

Contribution

It introduces a new analysis technique and a novel Nehari manifold definition to establish solutions with exactly k nodes for the Kirchhoff problem in the challenging case 2<p<4.

Findings

Existence of solutions with exactly k nodes for all positive integers k.

Energy of solutions increases with the number of nodes.

Asymptotic behaviors of solutions are characterized.

Abstract

We are concerned with the existence and asymptotic behavior of multiple radial sign-changing solutions with the nodal characterization for a Kirchhoff-type problem involving the nonlinearity $|u|^{p-2}u(2<p<4)$ in $\mathbb{R}^3$. By developing some useful analysis techniques and introducing a novel definition of the Nehari manifold for the auxiliary system of the equations, we show that, for any positive integer $k$, the problem has a sign-changing solution $u_k^b$ changing signs exactly $k$ times. Furthermore, the energy of $u_k^b$ is strictly increasing in $k$, as well as some asymptotic behaviors of $u_k^b$ are obtained. Our result is a complement of [Deng Y, Peng S, Shuai W, {\it J. Funct. Anal.}, {\bf269}(2015), 3500-3527], where the case $2<p<4$ is left open.