Positive solutions to Schrödinger-Kirchhoff equations with inverse potential

Abstract

This paper is concerned with the existence of positive solutions to Schrödinger-Kirchhoff-type equations $\left(\mathcal{P}\right)\left\{\begin{array}{ll}-\left(a+b{\int }_{{\mathbb{R}}^3}|\mathrm{\nabla }u{|}^2\right)\mathrm{\Delta }u+V\left(x\right)u=|u{|}^{p-1}u& \mathrm{i}\mathrm{n}{\mathbb{R}}^3,\\ u\in H^1\left({\mathbb{R}}^3\right),\end{array}\right.$ where a and b are two positive constants, $p\in (1,5)$ and $V:{\mathbb{R}}^3\to \mathbb{R}$ is a potential function. Under certain assumptions on V, we prove that $\left(\mathcal{P}\right)$ has no ground state solution. However, we can show $\left(\mathcal{P}\right)$ has a positive bound state solution by applying a new version of the global compactness lemma and the linking theorem.

COMMUNICATED BY:

• T. Bartsch

Keywords:

• Bound state solutions
• ground state solutions
• Schrödinger-Kirchhoff-type equations
• linking theorem
• pohozaev identity

AMS SUBJECT CLASSIFICATIONS:

• 35J20
• 35J65

Acknowledgments

We would like to express our sincere gratitude to the referees for their insightful comments and valuable corrections.

Disclosure statement

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

Additional information

Funding

This research is partially supported by NSFC [grant numbers 11601363, 11671295] and Science Foundation of Shanxi Province [grant number 201601D021011].