tailieunhanh - Báo cáo hóa học: " Research Article Multiple Solutions of p-Laplacian with Neumann and Robin Boundary Conditions for Both Resonance and Oscillation Problem"

Tuyển tập báo cáo các nghiên cứu khoa học quốc tế ngành hóa học dành cho các bạn yêu hóa học tham khảo đề tài: Research Article Multiple Solutions of p-Laplacian with Neumann and Robin Boundary Conditions for Both Resonance and Oscillation Problem | Hindawi Publishing Corporation Boundary Value Problems Volume 2011 Article ID 214289 19 pages doi 2011 214289 Research Article Multiple Solutions of p-Laplacian with Neumann and Robin Boundary Conditions for Both Resonance and Oscillation Problem Jing Zhang and Xiaoping Xue Department of Mathematics Harbin Institute of Technology Harbin 150025 China Correspondence should be addressed to Jing Zhang zhangjing127math@ Received 29 June 2010 Revised 7 November 2010 Accepted 18 January 2011 Academic Editor Sandro Salsa Copyright 2011 J. Zhang and X. 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 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. 1. Introduction Let Q be a bounded domain of Rn with smooth boundary ÔQ we consider the following problems i Neumann problem -àpu a u p 2u f x u in Q du 0 on dQ ii Robin problem -àpu a u p 2u f x u in Q Vu p 2ũd b x u p 2u 0 on dQ dv p1 p2 2 Boundary Value Problems where Apu div Vu p-2Vu is the p-Laplacian operator of u with 1 p TO a 0 b x e LTO ÔQ b x 0 and b x 0 on ÔQ f x 0 0 for . x e Q and du dv denotes the outer normal derivative of u with respect to ÔQ. Our purpose is to show the multiplicity of solutions to p1 and p2 . It is known that p1 and p2 are the Euler-Lagrange equations of the functionals T z 1 p 1 . a p J1 u - Vulpdx - ulpdx - F x u dx T X 1 p a p 1 p J2 u - Vulpdx - ulpdx - b x lulpds - F x u dx respectively .