Abstract:

We are concerned with the following non-autonomous Kirchhoff type equation in ${{\Bbb R}} ^{3}$ $\begin{eqnarray*} M\left(\int_{{{\Bbb R}} ^{N}}|\nabla u(x)|^{2}+\int_{{{\Bbb R}} ^{N}}V(x)|u(x)|^{2}\right) \left(-\Delta u+V(x)u \right)=\lambda K(x)f(u)+u^{5}, \end{eqnarray*}$ with vanishing potentials $V(x), K(x)$ at infinity and a Sobolev critical term $u^{5}$, where $\lambda\geq 0$ is a parameter. We prove that there exists $\lambda^*>0$ such that the equation has a nontrivial solution $u_{\lambda}$ for all $\lambda\geq\lambda^*$.

Key words: Kirchhoff type equation, Critical growth, Variational methods