On the solvability of an indefinite nonlinear Kirchhoff equation via associated eigenvalue problems

TL;DR

This paper investigates the existence, multiplicity, and non-existence of positive solutions for a nonlinear Kirchhoff equation with indefinite potential and sign-changing weights, using eigenvalue problem techniques.

Contribution

It provides new conditions for positive solutions of indefinite nonlinear Kirchhoff equations, especially regarding parameter ranges and eigenvalue thresholds.

Findings

At least two positive solutions exist under certain parameter conditions.

Non-existence results are established for specific parameter regimes.

Existence of solutions near the first eigenvalue of the associated eigenvalue problem.

Abstract

We study the non-existence, existence and multiplicity of positive solutions to the following nonlinear Kirchhoff equation:% \begin{equation*} \left\{ \begin{array}{l} -M\left( \int_{\mathbb{R}^{3}}\left\vert \nabla u\right\vert ^{2}dx\right) \Delta u+\mu V\left( x\right) u=Q(x)\left\vert u\right\vert ^{p-2}u+\lambda f\left( x\right) u\text{ in }\mathbb{R}^{N}, \\ u\in H^{1}\left( \mathbb{R}^{N}\right) ,% \end{array}% \right. \end{equation*}% where $N\geq 3,2<p<2^{\ast }:=\frac{2N}{N-2},M\left( t\right) =at+b$ $\left( a,b>0\right) ,$ the potential $V$ is a nonnegative function in $\mathbb{R}% ^{N}$ and the weight function $Q\in L^{\infty }\left( \mathbb{R}^{N}\right) $ with changes sign in $\overline{\Omega }:=\left\{ V=0\right\} .$ We mainly prove the existence of at least two positive solutions in the cases that $% \left( i\right) $ $2<p<\min \left\{ 4,2^{\ast }\right\} $ and $0<\lambda <% \left[ 1-2\left[ \left( 4-p\right) /4\right] ^{2/p}\right] \lambda _{1}\left( f_{\Omega }\right) ;$ $\left( ii\right) $ $p\geq 4,\lambda \geq \lambda _{1}\left( f_{\Omega }\right) $ and near $\lambda _{1}\left( f_{\Omega }\right) $ for $\mu >0$ sufficiently large, where $\lambda _{1}\left( f_{\Omega }\right) $ is the first eigenvalue of $-\Delta $ in $% H_{0}^{1}\left( \Omega \right) $ with weight function $f_{\Omega }:=f|_{% \overline{\Omega }},$ whose corresponding positive principal eigenfunction is denoted by $\phi _{1}.$ Furthermore, we also investigated the non-existence and existence of positive solutions if $a,\lambda $ belongs to different intervals.