Bulletin of the
Korean Mathematical Society
BKMS

ISSN(Print) 1015-8634 ISSN(Online) 2234-3016

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Bull. Korean Math. Soc. 2022; 59(4): 961-977

Published online July 31, 2022 https://doi.org/10.4134/BKMS.b210567

Copyright © The Korean Mathematical Society.

Positive solution and ground state solution for a Kirchhoff type equation with critical growth

Caixia Chen, Aixia Qian

Mathematics Department of Jining University; Qufu Normal University

Abstract

In this paper, we consider the following Kirchhoff type equation on the whole space $$\left\{\aligned &-(a+b\int_{\mathbb{R}^{3}}|\nabla u|^{2}dx)\triangle u = u^{5} + \lambda k(x)g(u), \ x\in \mathbb{R}^{3},\\ & u\in\mathcal{D}^{1,2}(\mathbb{R}^{3}),\endaligned\right.$$ where $\lambda>0$ is a real number and $k, g$ satisfy some conditions. We mainly investigate the existence of ground state solution via variational method and concentration-compactness principle.

Keywords: Kirchhoff type equation, critical nonlinearity, concentration-compactness principle, ground state solution

MSC numbers: Primary 35J20, 35J60, 58E05

Supported by: This work was financially supported by SSF ZR2021MA096, ZR2020MA005.