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: Singular limiting solutions for elliptic problem involving exponentially dominated nonlinearity and convection term | Baraket et al. Boundary Value Problems 2011 2011 10 http content 2011 1 10 o Boundary Value Problems a SpringerOpen Journal RESEARCH Open Access Singular limiting solutions for elliptic problem involving exponentially dominated nonlinearity and convection term Sami Baraket1 Imed Abid2 Taieb Ouni2 and Nihed Trabelsi2 Correspondence smbaraket@ department of Mathematics College of Science King Saud University . Box 2455 Riyadh 11451 Saudi Arabia Full list of author information is available at the end of the article Abstract Given Q bounded open regular set of R2 and x1 x2 . xm e Q we give a sufficient condition for the problem div duV u p2f u to have a positive weak solution in Q with u 0 on dQ which is singular at each X as the parameters p l 0 tend to 0 and where f u is dominated exponential nonlinearities functions. 2000 Mathematics Subject Classification 35J60 53C21 58J05. Keywords singular limits Green s function nonlinear Cauchy-data matching method 1 Introduction and statement of the results We consider the following problem i div d u V u u p2f u in Q 1 0 on 3 where V is the gradient and o is an open smooth bounded subset of R2. The function a is assumed to be positive and smooth. In the following we take a u elu and f u elu eu egu for l 0 and g e 0 1 then problem 1 take the form J Au Ấ V u 2 u p2 eu eYu in c R2 2. 0 on 3q 2 Using the following transformation w Ấp2eu Ấ then the function w satisfies the following problem À 1 Y 1 Aw w Ấ QW Ấ in c R2 w Ấp 2 Ấ on 3 . SpringerOpen0 2011 Baraket et al licensee is an Open Access article distributed underthe 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 properly cited. Baraket et al. Boundary Value Problems 2011 2011 10 http content 2011 1 10 Page 2 of 17 with p Ip2 1-1. So when l 0 the .
