Existence of ground state solutions for Kirchhoff-type problem with variable potential

Abstract

The purpose of this paper is to study the following Kirchhoff-type equation $\{\begin{array}{rl}& -(a+b{\int }_{{\mathbb{R}}^3}|\mathrm{\nabla }u{|}^2\mathrm{d}x)\mathrm{△}u+V(x)u=f(u),x\in {\mathbb{R}}^3;\\ & u\in H^1({\mathbb{R}}^3),\end{array}$where a>0 and b>0 are constants. Suppose that the nonnegative continuous potential V is not an asymptotic constant at infinity, and f satisfies some relatively weak conditions in the absence of the usual Ambrosetti–Rabinowitz type condition or monotonicity condition on $\frac{f(t)}{t^3}$. The result of this paper can be applied to the case where $f(t)=|t{|}^{p-2}t$ with $2<p\le 4$. By using some new techniques and subtle analysis, we prove that the above problem admits at least one ground state solution. It is worth mention that our result generalize those obtained in Li and Ye [Existence of positive ground state solutions for the nonlinear Kirchhoff type equations in ${\mathbb{R}}^3$. J Differ Equ. 2014;257:566–600], Tang and Chen [Ground state solutions of Nehari-Pohozaev type for Kirchhoff-type problems with general potentials. Calc Var Partial Differ Equ. 2017;56:110–134], Guo [Ground states for Kirchhoff equations without compact condition. J Differ Equ. 2015;259:2884–2902] and some other related literatures. In particular, we give a proof for the Pohožaev type identity associated with the above equation, when V is unbounded.

Keywords:

• Kirchhoff-type problem
• ground state solution
• variational method

2010 Mathematics Subject Classifications:

• 35B30
• 35B09
• 35J50
• 35J62

Disclosure statement

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

Additional information

Funding

This work is partially supported by the National Natural Science Foundation of China [grant number 11971485] and the innovative project of graduate students of Central South University, P.R. China [grant number 202101025].