Multiple existence results of solutions for the Neumann problems via super- and sub-solutions

Abstract

By variational methods, we provide existence results of multiple solutions for quasilinear elliptic equations under the Neumann boundary condition. Our main result shows the existence of two constant sign solutions and a sign changing solution in the case where we do not impose the subcritical growth condition to the nonlinear term not including derivatives of the unknown function. The studied equations contain the p-Laplacian problems as a special case. Moreover, we give a result concerning a local minimizer in C 1 ( Ω ¯ ) versus W 1 , p ( Ω ) . Auxiliary results of independent interest are also obtained: a density property for the space W 1 , p ( Ω ) , a strong maximum principle of Zhangʼs type, and a Moserʼs iteration scheme depending on a parameter.