MULTIPLE POSITIVE SOLUTIONS OF SINGULAR DISCRETE p-LAPLACIAN PROBLEMS VIA VARIATIONAL METHODS RAVI P. AGARWAL, KANISHKA PERERA, AND DONAL O’REGAN Received 31 March 2005 We obtain multiple positive solutions of singular discrete p-Laplacian problems using variational methods. 1. Introduction We consider the boundary value problem −∆ ϕ p ∆u(k − 1) = f k,u(k) , k ∈ [1,n], () u(k) 0, k ∈ [1,n], u(0) = 0 = u(n + 1), where n is an integer greater than or equal to 1, [1,n] is the discrete interval {1,.,n}, ∆u(k) = u(k + 1) − u(k) is the forward difference operator, ϕ p (s) = |s| p−2 s,. | MULTIPLE POSITIVE SOLUTIONS OF SINGULAR DISCRETE p-LAPLACIAN PROBLEMS VIA VARIATIONAL METHODS RAVI P AGARWAL KANISHKA PERERA AND DONAL O REGAN Received 31 March 2005 We obtain multiple positive solutions of singular discrete p-Laplacian problems using variational methods. 1. Introduction We consider the boundary value problem -A ỹp Au k - 1 f k u k k e 1 n u k 0 k e 1 n u 0 0 u n 1 where n is an integer greater than or equal to 1 1 n is the discrete interval 1 . n Au k u k 1 - u k is the forward difference operator Ọp s s p-2s 1 p TO and we only assume that f e C 1 n X 0 to satisfies a0 k f k t a1 k t-Y k t e 1 n X 0 íq for some nontrivial functions a0 a1 0 and Y t0 0 so that it may be singular at t 0 and may change sign. Let A1 Ọ1 0 be the first eigenvalue and eigenfunction of -Aọp Au k - 1 Ảtyp u k k e 1 n u 0 0 u n 1 . Theorem . If holdsand f k t limsup7 p 1 Ả1 k e 1 n t TO tr then has a solution. Copyright 2005 Hindawi Publishing Corporation Advances in Difference Equations 2005 2 2005 93-99 DOI 94 Discrete p-Laplacian problems Theorem . If holdsand f k t1 0 k e 1 n for some t1 t0 then has a solution u1 t1. If in addition liminff _ A1 k e 1 n t M tp 1 then there is a second solution u2 u1. Example . Problem with f k t t-Y Atf has a solution for all Y 0 and A resp. A A1 A 0 if ỊỈ p - 1 resp. f p - 1 ỊỈ p - 1 by Theorem . Example . Problem with f k t t-Y et - A has two solutions for all Y 0 and sufficiently large A 0 by Theorem . Our results seem new even for p 2. Other results on discrete p-Laplacian problems can be found in 1 2 in the nonsingular case and in 3 4 5 6 in the singular case. 2. Preliminaries First we recall the weak comparison principle see . Jiang et al. 2 . Lemma . If -A Pp Au k - 1 -A Pp Av k - 1 k e 1 n u 0 v 0 u n 1 v n 1 then u v. Next we prove a local comparison result. Lemma . If -A Pp Au k - 1 -A Pp Av k - 1 u k v k u k 1 v k 1 then u k 1
