Existence and multiplicity of bound state solutions to a Kirchhoff type equation with a general nonlinearity

TL;DR

This paper establishes the existence and multiplicity of bound state solutions for a Kirchhoff type equation with a general nonlinearity, using a novel perturbation method and compactness techniques, without requiring classical conditions.

Contribution

It introduces a new perturbation approach and a global compactness lemma to prove solutions for Kirchhoff equations with general nonlinearities, relaxing previous assumptions.

Findings

Proves existence of solutions without Ambrosetti-Rabinowitz condition.

Shows multiplicity of solutions under broad conditions.

Extends results to nonlinearities with power p in (2,6).

Abstract

In this paper, we consider the following Kirchhoff type equation $$ -\left(a+ b\int_{\R^3}|\nabla u|^2\right)\triangle {u}+V(x)u=f(u),\,\,x\in\R^3, $$ where $a,b>0$ and $f\in C(\R,\R)$, and the potential $V\in C^1(\R^3,\R)$ is positive, bounded and satisfies suitable decay assumptions. By using a new perturbation approach together with a new version of global compactness lemma of Kirchhoff type, we prove the existence and multiplicity of bound state solutions for the above problem with a general nonlinearity. We especially point out that neither the corresponding Ambrosetti-Rabinowitz condition nor any monotonicity assumption is required for $f$. Moreover, the potential $V$ may not be radially symmetry or coercive. As a prototype, the nonlinear term involves the power-type nonlinearity $f(u) = |u|^{p-2}u$ for $p\in (2, 6)$. In particular, our results generalize and improve the results by Li and Ye (J.Differential Equations, 257(2014): 566-600), in the sense that the case $p\in(2,3]$ is left open there.