Boundary Value Problems
Volume 2011 (2011), Article ID 214289, 19 pages
Research Article

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

Department of Mathematics, Harbin Institute of Technology, Harbin 150025, China

Received 29 June 2010; Revised 7 November 2010; Accepted 18 January 2011

Academic Editor: Sandro Salsa

Copyright © 2011 Jing Zhang and Xiaoping Xue. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


We discuss Neumann and Robin problems driven by the 𝑝 -Laplacian with jumping nonlinearities. Using sub-sup solution method, Fucík 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.