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

TL;DR

This paper addresses the existence of infinitely many radial sign-changing solutions for a Kirchhoff-type nonlocal problem in $\mathbb{R}^3$ with nonlinearity $|u|^{p-2}u$ for $2<p<4$. The authors develop a variational approach based on a novel Nehari-type manifold $N_k^-$ for an auxiliary system, enabling a gluing construction across annuli to produce solutions with exactly $k$ nodal domains and showing energy monotonicity in $k$. They prove that for each $k$ there exists a small $b^*>0$ such that, for $b\in(0,b^*)$, a radial solution $u_k^b$ with $k$ sign changes exists and satisfies $I_b(u_{k+1})>I_b(u_k)$; moreover, as $b\to0^+$, $u_k^b$ converges in $\mathcal{H}$ to the least-energy radial solution $u_k^0$ of the limiting local equation $-\\\Delta u+V(x)u=|u|^{p-2}u$. This work extends previous nodal-solution results to the challenging $2<p<4$ regime and provides a robust framework for nonlocal Kirchhoff problems via constrained Nehari manifolds and annular gluing.

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.

Theorems & Definitions (18)

• Theorem 1.1
• Remark 1.2
• Theorem 1.3
• Theorem 1.4
• Remark 1.5
• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Lemma 2.3