Solutions for a quasilinear Schrödinger equation: Subcritical and critical cases

Abstract

In this paper, we establish the existence of standing wave solutions for quasilinear Schrödinger equations involving nonlinearity with subcritical and critical growth. To apply the variational method and circumvent the “lack of compactness” of the problem, we combine the dual approach developed by Colin–Jeanjean [Nonlinear Anal. 56, 213–226 (2004)], Fang–Szulkin [J. Differ. Equations, 254, 2015–2032 (2013)], and Liu–Wang–Wang [J. Differ. Equations 187, 473–493 (2003)] with Del Pino–Felmer’s penalization technique [Calc. Var. Partial Differ. Equations 4, 121–137 (1996)], Moser’s iteration method, and an adaptation of Alves’ arguments [J. Elliptic Parabol. Equations 1, 231–241 (2015)] of the semilinear case.