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

Zhisu Liu, Chaoliang Luo

DOI: http://dx.doi.org/10.12775/TMNA.2016.068


We study the existence of positive ground state solutions for the nonlinear Kirchhoff type equation $$ \begin{cases} \displaystyle -\bigg(a+b\int_{\mathbb R^3}|\nabla u|^2\bigg)\Delta {u}+V(x)u =f(u) & \mbox{in }\mathbb R^3, \\ \noalign{\medskip} u\in H^1(\mathbb R^3), \quad u> 0 & \mbox{in } \mathbb R^3, \end{cases} $$% 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.


Kirchhoff type problem; ground state solution; variational method

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