Bifurcations for quasilinear Schrödinger equations, I Original Research Article

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.