Báo cáo hóa học: " Research Article Multiplicity of Positive and Nodal Solutions for Nonhomogeneous Elliptic Problems in Unbounded Cylinder Domains"

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 Multiplicity of Positive and Nodal Solutions for Nonhomogeneous Elliptic Problems in Unbounded Cylinder Domains | Hindawi Publishing Corporation Boundary Value Problems Volume 2009 Article ID 687385 20 pages doi 2009 687385 Research Article Multiplicity of Positive and Nodal Solutions for Nonhomogeneous Elliptic Problems in Unbounded Cylinder Domains Tsing-San Hsu Center for General Education Chang Gung University Kwei-San Tao-Yuan 333 Taiwan Correspondence should be addressed to Tsing-San Hsu tshsu@ Received 13 March 2009 Accepted 7 May 2009 Recommended by Zhitao Zhang We show that if a x and f x satisfy some suitable conditions then the Dirichlet problem -Am u afx u p-2u f x in Q has a solution that changes sign in Q. in addition to two positive solutions where Q is an unbounded cylinder domain in RN. Copyright 2009 Tsing-San Hsu. 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. 1. Introduction Throughout this paper let x y z be the generic point of RN with y e Rm z e R where 2N N m 3 m 2 1 2 p N 2 In this paper we study the multiplicity results of both positive and nodal solutions for the nonhomogeneous elliptic problems -Am u a x u p 2u f x in Q u e Hj Q where 0 e w Q Rm is a bounded smooth domain Q w X R is a smooth unbounded cylinder domain in RN . 2 Boundary Value Problems It is assumed that afix and f x satisfy the following assumptions a1 a x is continuous and a x e 0 1 on Q and lim afix 1 uniformly for y e W z o f1 f x 0 f x 0 f x e H 1 Q f2 Yf 0 in which we defined Yf infi A P-1 P-2 p - 2 l u 2 p-1 p-2 p - 1 -J fudx J a x u pdx 1Ị f3 there exist positive constants Co e0 Ro such that f x C0exp - 1 y1 e0 z J for z R0 uniformly for y e w where y1 is the first positive eigenvalue of the Dirichlet problem - A in w. For the homogeneous case that is f x 0 Zhu 1 has established the existence of a positive solution and a nodal solution of problem in H 1 RN provided a x

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