Multiple positive solutions for a class of Kirchhoff equation on bounded domain

ABSTRACT

In this paper, we study the existence of multiple positive solutions of the following Kirchhoff equation $\{\begin{array}{ll}-(a+b{\int }_{\mathrm{\Omega }}|\mathrm{\nabla }u{|}^2\mathrm{d}x)\mathrm{\Delta }u+\lambda u=|u{|}^{p_{ϵ}-2}u,& \mathrm{in}\mathrm{\Omega },\\ u=0,& \mathrm{in}{\mathbb{R}}^3\mathrm{\setminus }\mathrm{\Omega },\\ u>0,& \mathrm{in}\mathrm{\Omega },\end{array}$where $ϵ>0,$ Ω is a regular bounded domain of ${\mathbb{R}}^3,$ $\lambda \ge 0,$ $a,b>0,$ $p_{ϵ}:=6-ϵ>4.$ We obtain the existence of $ca{t}_{\mathrm{\Omega }}(\mathrm{\Omega })+1$ positive solutions by using the variational methods and Lusternik–Schnirelmann theory.

Keywords:

• Kirchhoff equation
• Lusternik–Schnirelmann theory

2010 Mathematics Subject Classifications:

• 35J47
• 35J50

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work was supported by National Natural Science Foundation of China (NNSFC) [11671077].