On the existence of infinitely many solutions of a nonlinear Neumann problem involving the m-Laplace operator

Abstract

This paper surveys the existence of infinitely many solutions of a nonlinear Neumann problem involving the m-Laplace operator, where the constant m satisfies certain alternative inequalities, and some functions f(x,u) and g(x,u) continuous on \overline{\Omega}\times\RR and on \partial\Omega\times\RR, respectively, and odd with respect to u. We work on a domain \Omega bounded in \RR^{N} with smooth boundary. More specifically, we demonstrate the existence of a sequence of solutions which diverge to infinity provided that the nonlinear term is locally superlinear and the existence of a sequence of solutions which converge to zero provided that the nonlinear term is locally sublinear.