On the existence of soliton solutions to quasilinear Schrödinger equations

Abstract

Variational techniques are applied to prove the existence of standing wave solutions for quasilinear Schrodinger equations containing strongly singular nonlinearities which include derivatives of the second order. Such equations have been derived as models of several physical phenomena. The nonlinearity here corresponds to the superfluid film equation in plasma physics. Direct methods of the calculus of variations and minimax methods like the Mountain Pass Theorem are used. The difficulties introduced by the nonconvex functional \(\Phi(u)=\int |\nabla u|^2 u^2\) are substantially different from the semilinear case.