Quasilinear Schrödinger equations with linear growth: Existence and asymptotic behavior of soliton solutions

In this paper, we investigate the existence of soliton solutions to a class of generalized quasilinear Schrödinger equations involving a parameter. This class of equations appears when modeling of superfluid films in plasma physics. While the growth rate of the nonlinearity is required to be superlinear in most existing works, we propose a new variational approach based on the Strauss inequality to deal with nonlinearities of linear growth, and obtain the existence and nonexistence of nontrivial solutions depending on the parameter. Furthermore, we study the asymptotic behavior of these solutions and, as a consequence, a new existence result for the semilinear Schrödinger equation has emerged.