Positive solutions for a critical quasilinear Schrödinger equation

Abstract

In our current work we investigate the following critical quasilinear Schrödinger equation $ -\Delta \Theta+\mathcal V(x)\Theta-\Delta (\Theta^2)\Theta = |\Theta|^{22^*-2}\Theta+\lambda \mathcal K(x)g(\Theta), \ x \ \in \mathbb R^N, $ where $ N\geq 3 $, $ \lambda > 0 $, $ \mathcal V, \ \mathcal K\in C(\mathbb R^N, \mathbb R^+) $ and $ g\in C(\mathbb R, \mathbb R) $ has a quasicritical growth condition. We use the dual approach and the mountain pass theorem to show that the considered problem has a positive solution when $ \lambda $ is a large parameter.

Keywords

• quasilinear schrödinger equations
• quasicritical growths
• dual approaches