On indefinite Kirchhoff-type equations under the combined effect of linear and superlinear terms

TL;DR

This paper studies indefinite Kirchhoff-type equations with combined linear and superlinear terms, establishing existence and multiplicity of positive solutions under various conditions on parameters and the potential well.

Contribution

It provides new existence and multiplicity results for positive solutions of Kirchhoff equations with indefinite potentials and combined nonlinearities, extending previous work in the field.

Findings

Existence of at least one positive solution for certain parameter ranges.

Existence of at least two positive solutions under specific conditions.

Multiple solutions depend on the sign of an integral involving the eigenfunction.

Abstract

We investigate a class of Kirchhoff type equations involving a combination of linear and superlinear terms as follows: \begin{equation*} -\left( a\int_{\mathbb{R}^{N}}|\nabla u|^{2}dx+1\right) \Delta u+\mu V(x)u=\lambda f(x)u+g(x)|u|^{p-2}u\quad \text{ in }\mathbb{R}^{N}, \end{equation*}% where $N\geq 3,2<p<2^{\ast }:=\frac{2N}{N-2}$, $V\in C(\mathbb{R}^{N})$ is a potential well with the bottom $\Omega :=int\{x\in \mathbb{R}^{N}\ |\ V(x)=0\}$. When $N=3$ and $4<p<6$, for each $a>0$ and $\mu $ sufficiently large, we obtain that at least one positive solution exists for $% 0<\lambda\leq\lambda _{1}(f_{\Omega}) $ while at least two positive solutions exist for $\lambda _{1}(f_{\Omega })< \lambda<\lambda _{1}(f_{\Omega})+\delta_{a}$ without any assumption on the integral $% \int_{\Omega }g(x)\phi _{1}^{p}dx$, where $\lambda _{1}(f_{\Omega })>0$ is the principal eigenvalue of $-\Delta $ in $H_{0}^{1}(\Omega )$ with weight function $f_{\Omega }:=f|_{\Omega }$, and $\phi _{1}>0$ is the corresponding principal eigenfunction. When $N\geq 3$ and $2<p<\min \{4,2^{\ast }\}$, for $% \mu $ sufficiently large, we conclude that $(i)$ at least two positive solutions exist for $a>0$ small and $0<\lambda <\lambda _{1}(f_{\Omega })$; $% (ii)$ under the classical assumption $\int_{\Omega }g(x)\phi _{1}^{p}dx<0$, at least three positive solutions exist for $a>0$ small and $\lambda _{1}(f_{\Omega })\leq \lambda<\lambda _{1}(f_{\Omega})+\overline{\delta }% _{a} $; $(iii)$ under the assumption $\int_{\Omega }g(x)\phi _{1}^{p}dx>0$, at least two positive solutions exist for $a>a_{0}(p)$ and $\lambda^{+}_{a}< \lambda<\lambda _{1}(f_{\Omega})$ for some $a_{0}(p)>0$ and $\lambda^{+}_{a}\geq0$.