Infinitely Many Positive Nonradial Solutions for the Kirchhoff Equation

Abstract

We are concerned with the existence of positive nonradial solutions to the following Kirchhoff equation:

$\begin{array}{c}-\left(a+b{\int }_{{ℝ}^3}|\nabla u{|}^{2}dx\right)\Delta u+V\left(\right|x\left|\right)u=Q\left(\right|x\left|\right)|u{|}^{p-1}u,x\in {ℝ}^3,\end{array}$

where $a,b>0,1<p<5$ and $V,Q$ are radial functions having the following expansions:

$V\left(\right|x\left|\right)=V_0+\frac{d_1}{|x{|}^m}+O\left(\frac{1}{|x{|}^{m+\theta }}\right),Q\left(\right|x\left|\right)=Q_0+\frac{d_2}{|x{|}^n}+O\left(\frac{1}{|x{|}^{n+\kappa }}\right)\text{as}\left|x\right|\to \infty$

with $V_0,Q_0,\theta ,\kappa ,d_1>0$ and $d_2\in ℝ$. By introducing the Miranda theorem and developing some delicate analysis, we construct infinitely many positive nonradial multibump solutions of this equation under suitable numbers $m,n$ via the Lyapunov–Schmidt reduction method, whose maximum points lie on the top and bottom circles of a cylinder close to infinity. These nonradial multibump solutions are different from the ones obtained in a previous study. This result complements and extends the previous results in the literature.