Multiple Solutions of p-Laplacian with Neumann and Robin Boundary Conditions for Both Resonance and Oscillation Problem

Xiaoping Xue,Jing Zhang

doi:10.1155/2011/214289

Abstract

We discuss Neumann and Robin problems driven by the -Laplacian with jumping nonlinearities. Using sub-sup solution method, Fucik spectrum, mountain pass theorem, degree theorem together with suitable truncation techniques, we show that the Neumann problem has infinitely many nonconstant solutions and the Robin problem has at least four nontrivial solutions. Furthermore, we study oscillating equations with Robin boundary and obtain infinitely many nontrivial solutions.

Highlights

Let Ω be a bounded domain of Rn with smooth boundary ∂Ω, we consider the following problems:i Neumann problem:−Δpu α|u|p−2u f x, u, in Ω, ∂u 0, on ∂Ω, p1∂ν ii Robin problem:|∇u|p−2 ∂u b x |u|p−2u 0, on ∂Ω, p2 ∂νBoundary Value Problems where Δpu div |∇u|p−2∇u is the p-Laplacian operator of u with 1 < p < ∞, α > 0, b x ∈ L∞ ∂Ω, b x ≥ 0, and b x / 0 on ∂Ω, f x, 0 0 for a.e. x ∈ Ω, and ∂u/∂ν denotes the outer normal derivative of u with respect to ∂Ω
Assume that f is satisfied as in Theorem 1.3 and (F ), one can have infinitely many sign-changing solutions for p2 which are of mountain pass type or not mountain pass type but with positive local degree
Lemma 2.3 Mountain pass theorem in half-order intervals, sup-solutions case see 9

Summary

Jing Zhang and Xiaoping Xue

We discuss Neumann and Robin problems driven by the p-Laplacian with jumping nonlinearities. Using sub-sup solution method, Fucık spectrum, mountain pass theorem, degree theorem together with suitable truncation techniques, we show that the Neumann problem has infinitely many nonconstant solutions and the Robin problem has at least four nontrivial solutions. We study oscillating equations with Robin boundary and obtain infinitely many nontrivial solutions

Introduction

Consider the problem

Is u

Consider the truncated functions

Ji u