Nehari-type ground state solutions for Kirchhoff type problems in ℝ^N

ABSTRACT

In the present paper, by employing Non-Nehari manifold method, we make use of a weak condition on the existence of Nehari-type ground state solutions for the following Kirchhoff-type problem (0.1) $$\begin{array}{c}\hfill \left\{\begin{array}{cc}-\left(a+b{\int }_{{\mathbb{R}}^N}{|\mathrm{\nabla }u|}^2\mathrm{d}x\right)▵u+V(x)u=f(x,u),\hfill & x\in {\mathbb{R}}^N\text{;}\hfill \\ u\in H^1({\mathbb{R}}^N),\hfill \end{array}\right.\end{array}$$(0.1)

which weakens both the variant Ambrosetti–Rabinowitz condition and the variant Nehari-type condition related to Problem (0.1). Furthermore, we allow $f(x,u)u-4F(x,u)$ to be sign-changing, which is different from the previous literature.

KEYWORDS:

• Kirchhoff-type problems
• Nehari-type ground state solutions
• non-Nehari manifold method

AMS SUBJECT CLASSIFICATIONS:

• 35J20
• 35J65

Notes

No potential conflict of interest was reported by the authors.

Additional information

Funding

This work is partially supported by the NNFC [grant number 11571370] and Yunnan Local Colleges Applied Basic Research Projects (2017).