Nodal solutions for an asymptotically linear Kirchhoff-type problem in ℝ^N

ABSTRACT

In this paper, we are concerned with nodal solutions for a class of Kirchhoff-type problems $-(a+b{\int }_{{\mathbb{R}}^N}|\mathrm{\nabla }u{|}^2\mathrm{d}x)\mathrm{△}u+u=f(u),\mathrm{in}{\mathbb{R}}^N,$where $N\ge 3$, a, b>0, f satisfies some asymptotically linear growth conditions. First of all, when N = 3, b>0 and $N\ge 4$, b>0 sufficiently small, we attain infinitely many nodal solutions by an equivalent transformation. Based on this, via a new and direct approach, the least energy sign-changing radial solutions can be obtained for the above problem. What's more, we also establish non-existence results of nodal solutions for $N\ge 4$ and b large enough.

KEYWORDS:

• Kirchhoff-type problem
• asymptotically linear
• nodal solutions
• ground state

AMS CLASSIFICATIONS:

• 35D30
• 35J50

Acknowledgements

The author is grateful to Prof. David G. Costa for his helpful discussion and so many supports for her researches. The author also wishes to thank the anonymous referees for their valuable suggestions and comments.

Disclosure statement

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

Additional information

Funding

The paper is supported by the National Natural Science Foundation of China [grant number 11801465], Chongqing Research Program of Basic Research and Frontier Technology [grant number cstc2017jcyjAX0331] and China Scholarship Council [grant number 201906995023].