On the indefinite Kirchhoff type problems with local sublinearity and linearity

TL;DR

This paper investigates indefinite Kirchhoff problems with local sublinear and linear behaviors, establishing conditions for existence and multiplicity of solutions using variational methods.

Contribution

It introduces new existence and multiplicity results for Kirchhoff problems with locally sublinear and linear nonlinearities, including criteria for three solutions.

Findings

Existence of nontrivial solutions under certain conditions.

Multiple solutions including at least three solutions.

Application of mountain pass theorem and Ekeland variational principle.

Abstract

The purpose of this paper is to study the indefinite Kirchhoff type problem: \begin{equation*} \left\{ \begin{array}{ll} M\left( \int_{\mathbb{R}^{N}}(|\nabla u|^{2}+u^{2})dx\right) \left[ -\Delta u+u\right] =f(x,u) & \text{in }\mathbb{R}^{N}, \\ 0\leq u\in H^{1}\left( \mathbb{R}^{N}\right), & \end{array} \right. \end{equation*} where $N\geq 1$, $M(t)=am\left( t\right) +b$, $m\in C(\mathbb{R}^{+})$ and $ f(x,u)=g(x,u)+h(x)u^{q-1}$. We require that $f$ is \textquotedblleft local\textquotedblright\ sublinear at the origin and \textquotedblleft local\textquotedblright\ linear at infinite. Using the mountain pass theorem and Ekeland variational principle, the existence and multiplicity of nontrivial solutions are obtained. In particular, the criterion of existence of three nontrivial solutions is established.