Existence of Solutions for Quasilinear Schrödinger Equations Under Non-Coercive Conditions

In this paper, we study the existence of solutions for a class of quasilinear Schrödinger equations under general potential conditions. The quasilinear term causing the functional to be ill-defined in H1(RN) is addressed via a dual approach involving a change of variables. To overcome the lack of coercivity in the potential V, a class of weighted function spaces is introduced, and their compact embedding properties are established. Moreover, the boundedness of the Cerami sequence is obtained with new conditions on the nonlinearity, which can be unbounded in the variable x. Recent results from the literature are improved and extended.