Positive ground state solutions for generalized quasilinear Schr\"odinger equations with critical growth

Abstract

This paper concerns the existence of positive ground state solutions for generalized quasilinear Schr\"odinger equations in $\mathbb{R}^N$ with critical growth which arise from plasma physics, as well as high-power ultrashort laser in matter. By applying a variable replacement, the quasilinear problem reduces to a semilinear problem which the associated functional is well defined in the Sobolev space $H^1(\mathbb{R}^N)$. We use the method of Nehari manifold for the modified equation, establish the minimax characterization, then obtain each Palais-Smale sequence of the associated energy functional is bounded. By combining Lions's concentration-compactness lemma together with some classical arguments developed by Br\'ezis and Nirenberg \cite{bn}, we establish that the bounded Palais-Smale sequence has a nonvanishing behaviour. Finally, we obtain the existence of a positive ground state solution under some appropriate assumptions. Our results extend and generalize some known results.