Multiple positive solutions of a singular gelfand type problem for second-order impulsive differential systems

K e y w o r d s S e c o n d o r d e r impulsive differential system, Singular boundary value problem, Semilinear elliptic problem, Positive solution, Radial solution, Upper solution, Lower solution, Fixed-point index. 1. I N T R O D U C T I O N In this paper , we prove the existence of mul t ip le posi t ive solut ions for a s ingular Gel fand type b o u n d a r y value p rob lem of the following second-order impuls ive differential sys tem: u"(t) + )~hl(t)f(u(t),v(t)) = 0, t e (0, 1), t ~ t l , v"(t) ÷ ~h2(t)g(u(t), v(t)) = 0, t E (0, 1), t ~ t l , ~'t~=~1 = N~(~(t~)) , av'Lt=~l = Yo(v( t , ) ) , u(0) = a > 0, v(0) = b > 0, u(1) = c > 0, v(1) d_> 0, U / where )~, ;t are posi t ive real pa rame te r s , Ault=t ~ = u(t +) u ( t l ) , and A It=tl = u'(t+) --u'(t-~). T h r o u g h o u t this paper , we assume f , g E C ( R 2, (0, c~)), I~, I . e C ( R , R ) sat isfying I~(0) = 0 = I,(O), N ~ , N , e C ( R , ( c ~ , 0 ] ) , and hi, h2 e C((0 ,1 ) , (0, cxD)). We not ice f ( 0 , 0 ) > 0, g (0 ,0 ) > 0 and h~, h2 m a y be s ingular a t t = 0 a n d / o r 1. Le t J = [0, 1], J ' = [0, 1] \ {0, 1,t l} , This work was supported by Grant R01-2003-000-11731-0 from the Basic Research Program of the Korea Science and Engineering Foundation. The authors wish to express their thanks to the referee for valuable suggestions. 0895-7177/04/$ see front matter (~) 2004 Elsevier Ltd. All rights reserved. Typeset by Jth/~S-TEX doi:10.1016/j.mcm.2003.12.007 308 E.K. LEE AND Y.-H. LEE PC[O, 1] -{u I u : [0, 1] -~ R is continuous at t # t l , left continuous at t = t l , and its right-hand limit at t = t l exists) and X ~ PC[0, 1] x PC[O, 1]. Then PC[O, 1] and X are Banach spaces with norm Ilull = supte[0,1] lu(t)l and If(u, v)l I -Ilull + Ilvll, respectively. The solution of (PT) is (u, v) e X N (C2(J ') x C2(J ')) which satisfies (Pw). In recent years, scalar second-order impulsive problems have been studied extensively in literature. (See [1-11].) Among them, for scalar problems of the form u"(t) + Aq(t)f(u(t)) = 0, t e (0, 1), t # t l ,