Nodal Solutions for Neumann Problems with a Nonhomogeneous Differential Operator

Abstract

Abstract. We consider a nonlinear elliptic Neumann problem driven by a nonhomogeneous di¤erential operator, which is strictly monotone and incorporates as special cases the p-Laplacian, the ð p; qÞ-di¤erential operator and the generalized pmean curvature di¤erential operator. Using variational methods coupled with suitable truncation and comparison techniques and Morse theory (critical groups), we show that the problem has at least three nontrivial smooth solutions, one positive, the second negative and the third nodal. Also we show that the problem has extremal nontrivial constant sign solutions.

Topological degree methods for a Neumann problem governed by nonlinear elliptic equation

“…By combining variational methods based on critical point theory, with suitable truncation techniques and flow invariance arguments. Some important and interesting results can be found in, for example [1,3,14,16,20,27]. Moreover, S. Liu [25] by using Morse theory, have established the existence of weak solutions in the case a(x, u, ∇u) = |∇u| p−2 u and H(x, u, ∇u) = f (x, ∇u) with Dirichlet boundary conditions.…”

Positive solutions for resonant (<i>p</i>, <i>q</i>)-equations with concave terms

“…Concave-convex problems for nonlinear Neumann and Robin problems were investigated by Papageorgiou-Radulescu [24]. Recent works of Filippakis-Papageorgiou [9] and Marano-Mosconi-Papageorgiou [22] deal with resonant (p, q)-equations. A nice survey of the recent existence and multiplicity results for (p, q)-equations can be found in Marano-Mosconi [19].…”