Topological Methods in Nonlinear Analysis

Existence of positive ground state solutions for Kirchhoff type equation with general critical growth

Zhisu Liu and Chaoliang Luo

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Abstract

We study the existence of positive ground state solutions for the nonlinear Kirchhoff type equation $$ \begin{cases} \displaystyle -\bigg(a+b\int_{\mathbb R^3}|\nabla u|^2\bigg)\Delta {u}+V(x)u =f(u) & \mbox{in }\mathbb R^3, \\ u\in H^1(\mathbb R^3), \quad u> 0 & \mbox{in } \mathbb R^3, \end{cases} $$ where $a,b> 0$ are constants, $f\in C(\mathbb R,\mathbb R)$ has general critical growth. We generalize a Berestycki-Lions theorem about the critical case of Schrödinger equation to Kirchhoff type equation via variational methods. Moreover, some subcritical works on Kirchhoff type equation are extended to the current critical case.

Article information

Source
Topol. Methods Nonlinear Anal., Volume 49, Number 1 (2017), 165-182.

Dates
First available in Project Euclid: 11 April 2017

Permanent link to this document
https://projecteuclid.org/euclid.tmna/1491876025

Digital Object Identifier
doi:10.12775/TMNA.2016.068

Mathematical Reviews number (MathSciNet)
MR3635641

Zentralblatt MATH identifier
1375.35188

Citation

Liu, Zhisu; Luo, Chaoliang. Existence of positive ground state solutions for Kirchhoff type equation with general critical growth. Topol. Methods Nonlinear Anal. 49 (2017), no. 1, 165--182. doi:10.12775/TMNA.2016.068. https://projecteuclid.org/euclid.tmna/1491876025


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