Quasilinear Schrödinger equation with exponential growth on the Heisenberg group

Abstract We establish existence results for a quasilinear Schrödinger equation on the Heisenberg group, which appears naturally in several applications of mathematical physics and conformal geometry. The nonlinearity considered in the equation depends on a concave term and an exponential term that may be subcritical, critical, or supercritical in the sense of the Trudinger–Moser inequality on the Heisenberg group. In such cases, variational methods cannot be applied directly. Our approach is based on a suitable change of variables, which transforms the original problem into an equivalent semilinear one. The positive solutions to semilinear equations are then presented using an approximation scheme together with a variation of the fixed point theorem. An important feature is that there are few works in the literature for the type of problem considered here, and the Galerkin method was not used to consider quasilinear Schrödinger equations on the Heisenberg group.