Multiple solutions for a class of Kirchhoff equation with singular nonlinearity

TL;DR

This paper studies the existence and multiplicity of solutions for a Kirchhoff equation with singular nonlinearity and sign-changing potential, using variational methods like the Nehari manifold and concentration-compactness.

Contribution

It introduces new results on multiple solutions for a Kirchhoff equation with singular nonlinearity and sign-changing potential, employing advanced variational techniques.

Findings

Existence of solutions established using Nehari manifold method.

Multiple solutions demonstrated via concentration-compactness principle.

Potential $k(x)$ allowed to change sign, broadening applicability.

Abstract

In this article, we investigate the existence and multiplicity of solutions of Kirchhoff equation \begin{equation*} \left\{ \begin{aligned} -(1+b \int_{\mathbb{R}^3}|\nabla u|^2)\Delta u= k(x)\frac{|u|^2 u}{|x|} +\lambda h(x)u,~~x\in\mathbb{R}^3\\ u(x)\rightarrow 0 ~~~~~~~~~~~~~~~~~~~~~~~as ~~~~|x|\rightarrow\infty \end{aligned} \right. \end{equation*} where the potential $k(x)$ allows sign changing. Making use of Nehari manifold method and Concentration-compactness principle, we obtain the existence and multiplicity of solutions for this equation.