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Tuyển tập các báo cáo nghiên cứu về sinh học được đăng trên tạp chí sinh học Journal of Biology đề tài: Research Article Sign-Changing and Extremal Constant-Sign Solutions of Nonlinear Elliptic Neumann Boundary Value Problems | Hindawi Publishing Corporation Boundary Value Problems Volume 2010 Article ID 139126 22 pages doi 10.1155 2010 139126 Research Article Sign-Changing and Extremal Constant-Sign Solutions of Nonlinear Elliptic Neumann Boundary Value Problems Patrick Winkert Institut fur Mathematik Technische Universitat Berlin Strafe des 17. Juni 136 10623 Berlin Germany Correspondence should be addressed to Patrick Winkert winkert@math.tu-berlin.de Received 23 November 2009 Accepted 15 June 2010 Academic Editor Pavel Drabek Copyright 2010 Patrick Winkert. 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. Our aim is the study of a class of nonlinear elliptic problems under Neumann conditions involving the p-Laplacian. We prove the existence of at least three nontrivial solutions which means that we get two extremal constant-sign solutions and one sign-changing solution by using truncation techniques and comparison principles for nonlinear elliptic differential inequalities. We also apply the properties of the Fucik spectrum of the p-Laplacian and in particular we make use of variational and topological tools for example critical point theory Mountain-Pass Theorem and the Second Deformation Lemma. 1. Introduction Let Q c RN be a bounded domain with Lipschitz boundary dQ. We consider the following nonlinear elliptic boundary value problem. Find u e W1 p Q 0 and constants a e R b e R such that -àpu f x u - u p 2u in Q Vu p 2 a u p 1 - b u p 1 g x u on dQ 1.1 where -àpu - div Vu p-2Vu 1 p TO is the negative p-Laplacian du dv denotes the outer normal derivative of u and u max u 0 as well as u max -u 0 are the positive and negative parts of u respectively. The nonlinearities f Q X R R and g dQ X R R are some Carathéodory functions which are bounded on bounded sets. For 2 Boundary Value Problems reasons of simplification we drop the
