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 Existence of Solutions for a Class of Elliptic Systems in RN Involving the p x , q x -Laplacia | Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2008 Article ID 612938 16 pages doi 2008 612938 Research Article Existence of Solutions for a Class of Elliptic Systems in RN Involving the p x q x -Laplacian S. Ogras R. A. Mashiyev M. Avci and Z. Yucedag Department of Mathematics Dicle University 21280 Diyarbakir Turkey Correspondence should be addressed to R. A. Mashiyev mrabil@ Received 4 April 2008 Accepted 17 July 2008 Recommended by M. Garcia Huidobro In view of variational approach we discuss a nonlinear elliptic system involving the p x -Laplacian. Establishing the suitable conditions on the nonlinearity we proved the existence of nontrivial solutions. Copyright 2008 S. Ogras et al. 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 The paper concerns the existence of nontrivial solutions for the following nonlinear elliptic system dF . - Ap x u x u v in Rn Ĩf P Q - Aq x v du x u v in Rn where p x and q x are two functions such that 1 p x q x N N 2 for every x e RN. However F e C1 RN X R2 and Ap x is the p x -Laplacian operator defined by Ap x U div Vu p x -2Vu . Using a variational approach the authors prove the existence of nontrivial solutions. Over the last decades the variable exponent Lebesgue space Lp x and Sobolev space W1 px 1-5 have been a subject of active research stimulated mainly by the development of the studies of problems in elasticity electrorheological fluids image processing flow in porous media calculus of variations and differential equations with p x -growth conditions 6-13 . Among these problems the study of p x -Laplacian problems via variational methods is an interesting topic. A lot of researchers have devoted their work to this area 14-22 . 2 Journal of Inequalities and Applications The operator Ap x u div Vu
