Positive Ground State Solutions for Generalized Quasilinear Schrodinger Equations with Critical Growth

This paper concerns the existence of positive ground state solutions for generalized quasilinear Schrodinger equations in 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(R-N). We use the method of Nehari manifold for the modified equation, establish the minimax characterization, and then prove that 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 Brezis and Nirenberg (Commun PureAppl Math 36:437-477, 1983), we obtain that the bounded Palais-Smale sequence has a nonvanishing behavior. Finally, we establish the existence of a positive ground state solution under some appropriate assumptions.