Bifurcations for quasilinear Schrödinger equations, I

This paper is concerned with bifurcation problems for quasilinear Schrödinger type equations such as the modified nonlinear Schrödinger equations which have appeared as several models from mathematical physics. We give a new approach for dealing with bifurcation problems of nonlinear potential operators. The novelty of our approach lies in establishing multiple bifurcations without making a reduction to finite dimensional problems. The finite dimensional reduction procedures that have been used in the past work for semilinear problems but are not suitable for quasilinear problems. We give a unified approach requiring much less regularity of the problems. As applications we provide multiple bifurcation results for the modified nonlinear Schrödinger equations.