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: Existence and multiplicity of solutions for nonlocal p(x)-Laplacian equations with nonlinear Neumann boundary conditions | Guo and Zhao Boundary Value Problems 2012 2012 1 http content 2012 1 1 o Boundary Value Problems a SpringerOpen Journal RESEARCH Open Access Existence and multiplicity of solutions for nonlocal p x -Laplacian equations with nonlinear Neumann boundary conditions Erlin Guo and Peihao Zhao Correspondence guoerlin@lzu. School of Mathematics and Statistics Lanzhou University Lanzhou 730000 P. R. China QT1r V u p x u pxP dx p x Abstract In this article we study the nonlocal p x -Laplacian problem of the following form flCC px VufW u p x dx div Vu p x 2Vu u p x 2u b fn F x u dx f x u in .d u . . V ufW 2 g x u on 3 dv where Q is a smooth bounded domain and V is the outward normal vector on the boundary dQ and F x u f x t dt. By using the variational method and the theory of the variable exponent Sobolev space under appropriate assumptions on f g a and b we obtain some results on existence and multiplicity of solutions of the problem. Mathematics Subject Classification 2000 35B38 35D05 35J20. Keywords critical points p x -Laplacian nonlocal problem variable exponent Sobolev spaces nonlinear Neumann boundary conditions 1 Introduction In this article we consider the following problem P fa P1J Vu p x u p x dx div Vup Vu u p x 2u b fa F x u dx f x u in a fa -7-- Vupx u p x dx Vu -2 g x u on 90 where o is a smooth bounded domain in RN p e C with 1 p- info p x p x p supo p x N a t is a continuous real-valued function f o X R R g u do X R R satisfy the Caratheodory condition and F x u f x t dt. Since the 0 equation contains an integral related to the unknown u over o it is no longer an identity pointwise and therefore is often called nonlocal problem. Springer 2012 Guo and Zhao licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License http licenses by which permits unrestricted use distribution and reproduction in any medium provided the original work is .
