Infinitely many solutions for a new Kirchhoff-type equation with subcritical exponent

ABSTRACT

In this article, we consider the following new Kirchhoff-type problem: $\begin{array}{rl}-\left(a-b{\int }_{\mathrm{\Omega }}|\mathrm{\nabla }u{|}^2\mathrm{d}x\right)\mathrm{\Delta }u& =|u{|}^{p-2},\mathrm{i}\mathrm{n}\mathrm{\Omega },\\ u& =0,\mathrm{o}\mathrm{n}\mathrm{\partial }\mathrm{\Omega },\end{array}$ where a and b are positive constants, $\mathrm{\Omega }\subset {\mathbb{R}}^N$ is a bounded domain with $C^1$ boundary $\mathrm{\partial }\mathrm{\Omega }$, $p\in [2,2^{\ast })$ with $2^{\ast }=2N/\left(N-2\right)$ if $N\ge 3$, and $2^{\ast }=+\mathrm{\infty }$ if N = 1, 2. We show that the problem possesses infinitely many sign-changing solutions by using combination of invariant sets of descent flow and the Ljusternik–Schnirelman type minimax method. And an example for p = 2 is illustrated our results.

Keywords:

• New Kirchhoff-type problem
• sign-changing solution
• minimax method

2010 Mathematics Subject Classifications:

• 35A09
• 35A15
• 40C10

Acknowledgments

The authors would like to thank Prof. Wei Wei from Guizhou Education University, anonymous reviewers and the editors for their valuable comments and suggestions, which led to an improvement of the original manuscript.

Disclosure statement

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

Additional information

Funding

This research is supported by the National Natural Science Foundation of China (grant numbers 11761021, 11661021, 11861021), the Innovation Group Major Research Program Funded by Guizhou Provincial Education Department of China(grant number KY[2016]029), and the Natural Science Foundation of Science and Technology of Guizhou Province (grant number KJ [2019]1163).