Existence results for quasilinear Schrödinger equations under a general critical growth term

We study the existence of solutions for the following quasilinear Schrödinger equation: $$ -\Delta u-\Delta (u^2)u=|u|^{2\cdot 2^*-2}u+g(u), \quad x\in \mathbb {R}^N, $$ where $N\geq 3$ and $g$ satisfies very weak growth conditions. The method is to analyze the behavior of solutions for subcritical problems from Colin and Jeanjean’s work [Nonlinear Anal. 56 (2004), 213–226] and to take the limit as the exponent approaches the critical exponent.