Existence and nonexistence of positive solutions for quasilinear elliptic systems

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1 ISSN , England, UK World Journal of Modelling and Simulation Vol. 4 (2008) No. 1, Existence and nonexistence of ositive solutions for uasilinear ellitic systems G. A. Afrouzi, H. Ghorbani Deartment of Mathematics, Faculty of asic Sciences, Mazandaran University, abolsar, Iran (Received February , Acceted March ) Abstract. We rove existence and nonexistence and uniueness of ositive solutions u = λf(v) + θ v = µg(u) + β on Ω where f, g : [0, ) [0, ), Ω is a bounded domain in R N with smooth boundary Ω, u = div( u 2 u), v = div( v 2 v),, > 1, and λ, µ, θ, β are ositive arameters. Keywords: ositive solutions, uasilinear ellitic systems, schauder fixed oint 1 Introduction Consider the uasilinear ellitic system u = λf(v) + θ v = µg(u) + β (I) on Ω where f, g : [0, ) [0, ), Ω is a bounded domain in R N with smooth boundary Ω, u = div( u 2 u), v = div( v 2 v),, > 1, and λ, µ, θ, β are ositive arameters. Dalmasso [2] studied existence and uniueness of ositive solutions to (I) when = = 2,θ = β = 0 and f(cg(x)) is sublinear at 0 and for every c > 0. Related results in the case when f(0) < 0 or g(0) < 0 are obtained in [5]. D. D. Hai [4] studied existence and uniueness of ositive solution to (I) when θ = β = 0. In this aer we are interested in the existence and uniueness of ositive solutions for the uasilinear system (I) when λ 1 (f(cµ 1 1 (g(x)+ β µ ) 1 1 )+ θ λ ) 1 is sublinear at 0 and for every c > 0. we also show under additional assumtions that (I) has no ositive solutions for all λ, µ, θ, β > 0. Our aroach deends on fixed oints arguments and maximum rinciles. 2 Main results (H 1 ): f, g : [0, ) [0, ) are continuous, nondecreasing, and g(x) > 0 for x > 0. (H 2 ): For each c > 0, lim su x λ (f(cµ 1 1 (g(x) + β µ ) θ λ ) 1 x = address: afrouzi@umz.ac.ir. Published by World Academic Press, World Academic Union

2 World Journal of Modelling and Simulation, Vol. 4 (2008) No. 1, (H 3 ): For each c > 0, lim inf x λ 1 (f(cµ 1 1 (g(x) + β µ ) θ λ ) 1 x = 0. (H 4 ): There exist C 1, C 2 > 0 such that f(v) C 1 v, g(u) C 2 u 1 for all u, v > 0. Then we have Theorem 1. Let (H 1 ) (H 2 ) hold. then (I) has a ositive solution (u, v) for all λ, µ > 0. Theorem 2. Let f, g satisfy (H 1 ). Suose that there exist ositive numbers r, s with rs < ( 1)( 1) such that f(x) + θ x r and g(x) + β x s are nonincreasing for x 0. then (I) has at most one ositive solution. Theorem 3. Suose that (H 4 ) hold, then there exist ositive number λ, µ > 0 such that (I) has no ositive solution for λ > λ, µ > µ. Let φ, ψ, satisfy Let D be a sub-domain of Ω with D Ω. Let { 1 x D h(x) = 0 x D and let φ, ψ be the solution of φ = 1, φ = 0 on Ω (1) φ = 1, ψ = 0 on Ω (2) { φ = h φ = 0 and { ψ = h ψ = 0 resectively. y the strong maximum rincile (see [8]), there exist ositive numbers M, m such that φ Mφ and φ, ψ, φ, ψ m in D. Without loss of generality, we assume that λ = µ = 1 in the roof of theorems 1 and 2. Proof of theorem 1. y (H 2 ), there exist ε (0, 1) such that M(f(m(g(εm) + β) 1 ε for each ω C(Ω), let u = T ω be the solution of u = f(v) + θ v = g(max(ω, εφ)) + β Then T : C( Ω) C( Ω) is comletely continuous (see e.g. [3, 4]). y (H 3 ), there exist a number R > φ such that WJMS for subscrition: info@wjms.org.uk

3 46 G. Afrouzi & H. Ghorbani: Existence and nonexistence of ositive solutions (f( ψ (g(r) + β) 1 φ R. We claim that T : (0, R) (0, R), where (0, R) denotes the closed ball centered at 0 with radius R in C( Ω). Indeed, let ω C( Ω) with ω R. Then we have v = g(max(ω, εφ)) + β g(r) + β which imlies by the maximum rinciles that Thus and therefore conseuently roving the claim. v (g(r) + β) 1 1 ψ. u = f(v) + θ f((g(r) + β) 1 1 ψ) + θ f( ψ (g(r) + β) 1 1 ) + θ u (f( ψ (g(r) + β) 1 φ u (f( ψ (g(r) + β) 1 φ R v = g(max(u, εφ)) + β { g(ɛm) + β x D y the Schauder fixed theorem, T has a fixed oint u with u R. Next we verify that u εφ. Since { g(εm) x D v = g(max(u, εφ)) + β It follows from the maximum rincile that v (g(εm) + β) 1 1 ψ. Using this in the euation for u gives { u = f(v) + θ f(m(g(εm) + β) 1 1 ) + β x D and therefore u (f(m(g(εm) + β) 1 φ M(f(m(g(εm) + β) 1 φ εφ. Since g(x) > 0 for x > 0, we have v > 0 by the strong maximum rincile. This comletes the roof of theorem 1. Proof of theorem 2. Let (u, v) and (u 1, v 1 ) be ositive solutions of (I). As in [1, 8], we define δ = su{ε > 0 : v εv 1 }. Then v δv 1. If δ < 1, then we have Since u = f(v) + θ f(δv 1 ) + θ δ r (f(v 1 ) + θ). (δ r u 1 ) = δ r (f(v 1 ) + θ), WJMS for contribution: submit@wjms.org.uk

4 World Journal of Modelling and Simulation, Vol. 4 (2008) No. 1, it follows that Using this in the euation for v gives which imlies u δ r u 1. v = g(u) + β g(δ r u 1 ) + β δ rs (g(u 1 ) + β), rs v δ ()( 1) v 1, a contradiction with the definition of δ. Thus δ = 1, i,e, v v 1, and so v = v 1, u = u 1, roving theorem 2. Proof of theorem 3. Suose that (I) has a ositive solution (u, v) for every λ, µ > 0. y (H 3 ) then we have Hence Or u = λf(v) + θ λc 1 v, v = µg(u) + β µc 2 u 1. u v λc 1, v u 1 µc 2. div( u 2 u v ) λc 1, (3) div( v 2 v u 1 ) µc 2. (4) Let = (x 1, r 2 ) and φ C 0 () with φ 0 on and φ 0. Multilying (3), (4) by φ, and integrating, we obtain u 2 u. φ v dx λc 1 φdx, v 2 v. φ u 1 dx µc 2 φdx. y Holder s ineuality, u 2 u. φ v dx v 2 v. φ u 1 dx u φ v dx 1 v 1 φ u 1 dx 1 u v dx + 1 φ dx, (5) v u dx + 1 φ dx. (6) Combining (3),(5) and (4), (6) we get 1 u v dx λc 1 φdx 1 φ dx, (7) 1 v u dxµc 2 φdx 1 φ dx. (8) Next we show that u v dx K 1, (9) v u dx K 2, (10) WJMS for subscrition: info@wjms.org.uk

5 48 G. Afrouzi & H. Ghorbani: Existence and nonexistence of ositive solutions where K 1, K 2 are constant indeendent of u, v, λ, µ. To this ends, let ψ C 0 ( 1) with ψ 0 on 1 = (x, r). Multilying (3) by ψ and (4) by ψ and integrating we obtain ψ u 2 u ψ u 1 v dx ( 1) 1 ψ v dx λc 1 ψ dx 0, 1 ψ 1 v 2 v ψ v 1 u 1 ( 1) 1 ψ u µc 2 ψ dx 0. 1 From this and holder s ineuality, we deduce u ( 1) 1 ψ v dx v ( 1) 1 ψ dx u ψ u 2 u ψ 1 v dx 1 ψ 1 v 2 v ψ 1 u 1 dx 1 1 u ψ v 1 v ψ u dx + dx ψ dx, 1 ψ dx, which imlies ( 1) 2 u 1 ψ v dx ψ dx 1 ( 1) 2 v 1 ψ u dx 1 ψ dx 1 and (9),(10) follows if we choose ψ so that ψ = 1 on. In view of (7), (8) we reach a contradiction if λ, µ is sufficiently large. This comletes the roof of theorem 3. References [1] H. rezis, S. Kamin. Sublinear ellitic euations in r n. Manuscrita Math, 1992, 74: [2] R. Dalmasso. Existence and uniueness of ositive solutions of semilinear ellitic systems. Nonlinear Anal., 2000, 39: [3] D. Gilbarg, N. S. Trudinger. Ellitic artial differential euations of second order. Sringer, [4] D. D. Hai. Existence and uniuness of solutions for uasilinear ellitic systems. in: Procedings of the american mathematical society, 2004, [5] D. D. Hai, R. Shivaji. An existence result on ositive solutions for a class of semilinear ellitic systems. Proc. Roy. Soc. Edinburgh, 2004, [6] D. D. Hai, X. Xu. On a class of uasilinear roblems with sign-changing nonlinearities. Nonlinear Anal., 2006, 64: [7] M. A. Krasnoselskii. Positive solutions of oerator euations [8] G. M. Liberman. oundary regularity for solutions of degenerate ellitic euations. Nonlinear Anal., 1988, 12: [9] J. I. Vazuez. A strong maximum for some uasilinear ellitic euations. Al. Math. otim., 1984, 12: WJMS for contribution: submit@wjms.org.uk