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

作者：Miyajima Shizuo; Motreanu Dumitru; Tanaka Mieko^*

来源：Journal of Functional Analysis, 2012, 262(4): 1921-1953.

DOI：10.1016/j.jfa.2011.11.028

• Quasilinear elliptic equations
• Neumann boundary value condition
• Super-solution and sub-solution for the Neumann problems
• Mountain pass theorem
• Second deformation lemma

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)((ohm) over bar) versus W(1,p)(ohm). Auxiliary results of independent interest are also obtained: a density property for the space W(1,p)(ohm), a strong maximum principle of Zhang%26apos;s type, and a Moser%26apos;s iteration scheme depending on a parameter.

• 出版日期2012-2-15