Standing wave solutions of a quasilinear degenerate Schrodinger equation with unbounded potential

We are concerned with the existence of entire distributional nontrivial solutions for a new class of nonlinear partial differential equations. The differential operator was introduced by A. Azzolini et al. [3, 4] and it is described by a potential with different growth near zero and at infinity. The main result generalizes a property established by P. Rabinowitz in relationship with the existence of nontrivial standing waves of the Schrodinger equation with lack of compactness. The proof combines arguments based on the mountain pass and energy estimates.