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A variational argument to finding global solutions of a quasilinear Schrödinger equation
Soliton solutions for quasilinear Schrödinger equations involving supercritical exponent in $\mathbb R^N$
1.  Department of Mathematics, University of British Columbia, Vancouver, BC, Canada 
$\epsilon \Delta u+V(x)u\epsilon k(\Delta(u^2))u=g(u), \quad u>0,x \in \mathbb R^N,$
where g has superlinear growth at infinity without any restriction from above on its growth. Mountain pass in a suitable Orlicz space is employed to establish this result. These equations contain strongly singular nonlinearities which include derivatives of the second order which make the situation more complicated. Such equations arise when one seeks for standing wave solutions for the corresponding quasilinear Schrödinger equations. Schrödinger equations of this type have been studied as models of several physical phenomena. The nonlinearity here corresponds to the superfluid film equation in plasma physics.
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