On positive ground state solutions for the nonlinear Kirchhoff type equations in ℝ³

Abstract

In this article, we first give a much more concise proof on the existence of ground state solutions of Kirchhoff equation $\begin{array}{rl}-\left(a+b{\int }_{{\mathbb{R}}^3}|\mathrm{\nabla }u{|}^2\mathrm{d}x\right)\mathrm{△}u+u& =v_{\mathrm{\infty }}|u{|}^{p-1}u,x\in {\mathbb{R}}^3,\\ u& >0,x\in {\mathbb{R}}^3,\end{array}$ by rearrangement than that given in Li and Ye [Existence of positive ground state solutions for the nonlinear Kirchhoff type equations in R3. J. Differ. Equat. 2014;257(2):566-600] and Ye [Positive high energy solution for Kirchhoff equation in R3 with superlinear nonlinearities via Nehari–Pohozaev manifold. Discrete Contin. Dyn. Syst. 2015;35(8):3857-3877], where a,b>0,2<p<5. And then, we prove a nonexistence result on ground state solutions of the non-autonomous Kirchhoff equation $\begin{array}{rl}-\left(a+b{\int }_{{\mathbb{R}}^3}|\mathrm{\nabla }u{|}^2\mathrm{d}x\right)\mathrm{△}u+u& =v\left(x\right)|u{|}^{p-1}u,x\in {\mathbb{R}}^3,\\ u& >0,x\in {\mathbb{R}}^3,\end{array}$ which can be considered as a nice complement to Zhang et al. [Semiclassical ground states for a class of nonlinear Kirchhoff-type problems. Appl. Anal. 2016;96(13):2267-2284].

COMMUNICATED BY:

• Daomin Cao

Keywords:

• Kirchhoff equation
• rearrangement
• ground state solution
• nonexistence

2010 Mathematics Subject Classifications:

• 35J60
• 47J30
• 35J20

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

This work was supported by National Natural Science Foundation of China [grant nos. 11701114, 11471085, 1170113] and the program for Changjiang scholars and Innovative Research Team in university [grant no. IRT1226].