Positive solutions for a class of singular quasilinear Schrödinger equations with critical Sobolev exponent

Abstract

In this paper we prove the existence of positive solutions of the following singular quasilinear Schrödinger equations at critical growth \begin{eqnarray*} -Δu-λc(x)u-κα(Δ(|u|^{2α}))|u|^{2α-2}u = |u|^{q-2}u+|u|^{2^*-2}u,\quad u\in{D^{1,2}(\mathbb{R}}^N), \end{eqnarray*} via variational methods, where $\lambda \ge 0$, $0<\alpha <1/2$, $2<q<2^{\ast }$. It is interesting that we do not need to add a weight function to control $|u{|}^{q-2}u$.

42 pages