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Báo cáo hóa học: " Research Article Nonexistence of Positive Solution for Quasilinear Elliptic Problems in the Half-Space"

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 Nonexistence of Positive Solution for Quasilinear Elliptic Problems in the Half-Space | Hindawi Publishing Corporation Journal ofInequalities and Applications Volume 2007 Article ID 65126 4 pages doi 10.1155 2007 65126 Research Article Nonexistence of Positive Solution for Quasilinear Elliptic Problems in the Half-Space Sebastian Lorca Received 16 October 2006 Accepted 9 February 2007 Recommended by Robert Gilbert Liouville-type results in Rn or in the half-space Rn might be important to obtain a priori estimates for positive solutions of associated problems in bounded domains via some procedure of blow up. In this work we obtain a nonexistence result for the positive solution of up - Amu Cup in the half-space. Copyright 2007 Sebastian Lorca. 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 Consider the following problem -Amu up in Rn 1.1 where 1 m N and m - 1 p N m - 1 N - m . Mitidieri and Pohozaev proved in 1 among other results that problem 1.1 has no positive solution. On the other hand as far as we know there is not a similar result in the half-space Rn x x1 . xN e Rn xN 0 . This kind of results may be used to prove existence results for associated problems in bounded domains -Amu f x u in O u 0 on do. This is particularly useful if the problem under consideration is nonvariational see e.g. 2-4 and the references therein . Usually these a priori estimates are obtained by using a blow up technique. Suppose by contradiction that there exists a sequence un n of solutions of the associated problem with un unbounded in the L norm . Let xn be a point at which un attain their maxima. With suitable assumptions on the function f the blow up methods provide a 2 Journal of Inequalities and Applications nontrivial solution of the problem -Amu up 1.2 in RN or in the half-space. To avoid the case of the half-space it is assumed in 3 that Q is convex f does not depend on x and 1 m 2. .