tailieunhanh - báo cáo hóa học:" Research Article On a Perturbed Dirichlet Problem for a Nonlocal Differential Equation of Kirchhoff Type"

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 On a Perturbed Dirichlet Problem for a Nonlocal Differential Equation of Kirchhoff Type | Hindawi Publishing Corporation Boundary Value Problems Volume 2011 Article ID 891430 10 pages doi 2011 891430 Research Article On a Perturbed Dirichlet Problem for a Nonlocal Differential Equation of Kirchhoff Type Giovanni Anello Department of Mathematics University of Messina S. Agata 98166 Messina Italy Correspondence should be addressed to Giovanni Anello ganello@ Received 24 May 2010 Accepted 26 July 2010 Academic Editor Feliz Manuel Minhos Copyright 2011 Giovanni Anello. 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. We study the existence of positive solutions to the following nonlocal boundary value problem -K u 2 Au Xus-1 f x u in Q u 0 on dQ where s e 1 2 f Q X R R is a Caratheodory function K R R is a positive continuous function and 1 is a real parameter. Direct variational methods are used. In particular the proof of the main result is based on a property of the infimum on certain spheres of the energy functional associated to problem -K u 2 Au Xus-1 in Q U 3Q 0. 1. Introduction This paper aims to establish the existence of positive solutions in w0 2 Q to the following problem involving a nonlocal equation of Kirchhoff type -k u 2 Au 1us 1 f x u in Q u 0 on dQ. W Here Q is an open bounded set in RN with smooth boundary dQ s e 1 2 f Q X 0 x 0 x is a Carathéodory function K R R is a positive continuous function 1 is a real parameter and uH JQ Vu 2dx 1 2 is the standard norm in W01 2 Q . In what follows for every real number t we put t t t 2. By a positive solution of Pa we mean a positive function u e W01 2 Q nC0 Q which is a solution of Px in the weak sense that is such that k u 2 J fVu x Nv xy dx -J xu x s 1 f x u x Ỷỳ v x dx 0 2 Boundary Value Problems for all v e W01 2 Q . Thus the weak solutions of Pf are exactly the positive critical points of the associated energy .