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

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 $N\geq 3,2 0\right) ,$ 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) $ $2 0$ 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.