tailieunhanh - Báo cáo hóa học: " Research Article A Class of p-q-Laplacian Type Equation with Potentials Eigenvalue Problem in RN"

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 A Class of p-q-Laplacian Type Equation with Potentials Eigenvalue Problem in RN | Hindawi Publishing Corporation Boundary Value Problems Volume 2009 Article ID 185319 19 pages doi 2009 185319 Research Article A Class of p-q-Laplacian Type Equation with Potentials Eigenvalue Problem in RN Mingzhu Wu1 and Zuodong Yang1 2 1 School of Mathematics Science Institute of Mathematics Nanjing Normal University Jiangsu Nanjing 210097 China 2 College of Zhongbei Nanjing Normal University Jiangsu Nanjing 210046 China Correspondence should be addressed to Zuodong Yang zdyang_jin@ Received 20 October 2009 Accepted 6 December 2009 Recommended by Wenming Zou The nonlinear elliptic eigenvalue problem -div IVuIp-2Vu - div IVuIq-2Vu Xa x u p-2u Xb x u q-2u f x u u e W1 p n W1 q RN where 2 q p N and a x e LN p RN b x e LN q RN a x b x 0 are studied. The key ingredient is a special constrained minimization method. Copyright 2009 M. Wu and Z. Yang. 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 In this paper we are interested in finding nontrivial weak solutions for the nonlinear eigenvalue problem div IVuIp-2V ij div IVu q 2V ij a v u p-2u b v u q-2u f x uì v I V w I V w v I V Í I u a x I Í I Í x I Í I u J x lu u e W 1 p n W 1 q RN u 0 where 2 q p N and a x e LN p RN b x e LN q RN a x b x 0 inf ajx inf b x 0 f Jx Ù satisfy the following conditions A f e C Rn X R R lim 0 f x t t p- 1 0 and limI f x f IfIp-1 p2 N 0 uniformly in x e RN B limIx f x f f f uniformly for t in bounded subsets of R. 2 Boundary Value Problems Remark . We can see if a x e LN p RN b x e LN q RN then a x u pdx RN RN 1-p p TO p p b x u qdx f b x N q i f uq TO RN RN RN where p Np N - p and q Nq N - q . Problem comes for example from a general reaction-diffusion system ut div D u Vu c x u where D u Vu p-2 Vu q-2 . This system has a wide range of applications in physics and related sciences