Positive solutions for a nonlinear nonlocal elliptic transmission problem

K e y w o r d s E l l i p t i c system, Transmission problem, Positive solution. 1. I N T R O D U C T I O N Let 12 be a smooth bounded domain of R n, n > 2, and let ~1 c gl be a subdomain with smooth boundary ~ satisfying ~1 C i2. Writing F = c9~ and gl2 = ~\~l l we have ~ = ~1 tJ 122 and 0i'~2 = ~, tJ F. We are concerned with the existence of positive solutions to the following system of nonlinear elliptic equations: -a(/~ IVul2dx) Au-f(x,u), i n ~ l , 1 -b (~ IVv,2dx) Av=g(x,v), in 122, 2 v = 0, on F, (1.1) (1.2) (1.3) The authors wish to thank Professor C. O. Alves for helpful comments during the preparation of this paper. Partially supported by CNPq, Brazil. 0893-9659/03/$ see front matter (~) 2003 Elsevier Science Ltd. All rights reserved. Typeset by .Ah,~S-TEX PII: S0893-9659(02)00186-6 244 T.F. MA AND J. E. MUfiOZ RIVERA with the transmission conditions u = v, on ~, a 1 IVu[ 2 dx -~ = b 2 IVy[ 2 dx O--~'Ov on E. (1.4) (1.5) Here a, b are positive continuous functions and f : ~1 x R ~ R, g : ~2 x R ~ R are locally Lipschitz functions with subcritical growth. I ~ E F ~21 m ~ = ~1 Ugh20I2=FUE , y = outward normal to f~2