Soliton solutions for quasilinear Schrödinger equations: The critical exponential case

Quasilinear elliptic equations in R²${\mathbb{R}}^2$ of second order with critical exponential growth are considered. By using a change of variable, the quasilinear equations are reduced to semilinear equations, whose respective associated functionals are well defined in H¹(R²)$H^1\left({\mathbb{R}}^2\right)$ and satisfy the geometric hypotheses of the mountain pass theorem. Using this fact, we obtain a Cerami sequence converging weakly to a solution v$v$. In the proof that v$v$ is nontrivial, the main tool is the concentration–compactness principle [P.L. Lions, The concentration compactness principle in the calculus of variations. The locally compact case. Part I and II, Ann. Inst. H. Poincaré Anal. Non. Linéaire 1 (1984) 109–145, 223–283] combined with test functions connected with optimal Trudinger–Moser inequality.