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EXISTENCE OF A POSITIVE SOLUTION FOR A p-LAPLACIAN SEMIPOSITONE PROBLEM MAYA CHHETRI AND R. SHIVAJI

EXISTENCE OF A POSITIVE SOLUTION FOR A p-LAPLACIAN SEMIPOSITONE PROBLEM MAYA CHHETRI AND R. SHIVAJI Received 30 September 2004 and in revised form 13 January 2005 We consider the boundary value problem −∆ p u = λ f (u) in Ω satisfying u = 0 on ∂Ω, where u = 0 on ∂Ω, λ 0 is a parameter, Ω is a bounded domain in Rn with C 2 boundary ∂Ω, and ∆ p u := div(|∇u| p−2 ∇u) for p 1. Here, f : [0,r] → R is a C 1 nondecreasing function for some r 0 satisfying f. | EXISTENCE OF A POSITIVE SOLUTION FOR A p-LAPLACIAN SEMIPOSITONE PROBLEM MAYA CHHETRI AND R. SHIVAJI Received 30 September 2004 and in revised form 13 January 2005 We consider the boundary value problem -ApU Af u in D satisfying u 0 on dD where u 0 on dD A 0 is a parameter D is a bounded domain in R with C2 boundary dD and ApU div Vu p-2 Vu for p 1. Here f 0 r R is a C1 nondecreasing function for some r 0 satisfying f 0 0 semipositone . We establish a range of A for which the above problem has a positive solution when f satisfies certain additional conditions. We employ the method of subsuper solutions to obtain the result. 1. Introduction Consider the boundary value problem -Apu Af u in D u 0 in D 1.1 u 0 on dD where A 0 is a parameter D is a bounded domain in R with C2 boundary dD and Apu div Vu p-2Vu for p 1. We assume that f e C1 0 r is a nondecreasing function for some r 0 such that f 0 0 and there exist f e 0 r such that f s s - f 0 for s e 0 r . To precisely state our theorem we first consider the eigenvalue problem -Apv Alvlp-2v in D 1.2 v 0 on dD. Let Ộ1 e C1 D be the eigenfunction corresponding to the first eigenvalue A1 of 1.2 such that Ộ1 0 in D and II01II 1. It can be shown that dộ1 dp 0 on dD and hence depending on D there exist positive constants m 8 Ơ such that V01 p - A1ộp m on Dg Ộ1 Ơ on D Dg 1.3 where Dg x e D l d x dD 8 . Copyright 2006 Hindawi Publishing Corporation Boundary Value Problems 2005 3 2005 323-327 DOI 10.1155 BVP.2005.323 324 Positive solution for p-Laplacian semipositone problems We will also consider the unique solution e e C1 Q of the boundary value problem -Ape 1 in Q e 0 on dQ 1.4 to discuss our result. It is known that e 0 in Q and de dp 0 on dQ. Now we state our theorem. Theorem 1.1. Assume that there exist positive constants l1 l2 e p r satisfying a l2 kl1 b f 0 A1 mf l1 1 and c lp-1 f l2 p lp-1 f l1 where k k Q A1 p-1 p p - 1 ơ p-1 p e ro and p p Q p e TO p - 1 p-1 A1 ơp . Then there exist A A such that 1.1 has a positive .