Multiple Sign-Changing Solutions for Quasilinear Elliptic Equations via Perturbation Method

We consider existence and multiplicity of sign-changing solutions for a class of quasilinear problems which has received considerable attention in the past, including the so called Modified Nonlinear Schrödinger Equations. We develop a new variational approach to treat this class of quasilinear equations by proposing a p-Laplacian regularization process. By establishing necessary estimates we show the solutions to the perturbation problems converge in a sense to solutions of the original problems. We show that the new approach is especially effective for dealing with issues of multiple solutions and sign-changing solutions.