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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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.

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

First available in Project Euclid: 11 April 2017

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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.

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