Uniqueness of Positive Solutions for a Class of p-laplacian Systems with Multiple Parameters

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1 Int. Journal of Math. Analysis, Vol. 2, 2008, no. 2, Uniqueness of Positive Solutions for a Class of p-laplacian Systems with Multiple Parameters G. A. Afrouzi and E. Graily Department of Mathematics, Faculty of Basic Sciences Mazandaran University, Babolsar, Iran afrouzi@umz.ac.ir Abstract We prove uniqueness of positive solution for the system Δ p u = λf(v) in Ω, Δ q v = μg(u) in Ω u = v =0 on Ω where Ω is a bounded domain in R N, f,g : R + R + and f(x) x p, g(x) x q at for some positive numbers p, q with p, q > Mathematics Subject Classification: 35J60, 35B30, 35B40 Keywords: uniqueness of positive solutions; p-laplacian systems; multiple parameters Introduction Consider the boundary value problem Δ p u = λf(v); Δ q v = μg(u); u(x) =v(x) =0 x Ω x Ω x Ω () where the p-laplacian operator Δ p z = div( z p 2 z),λ,μ are positive parameters and Ω is bounded domain in R N with smooth boundary Ω and f,g : R + R +. In [2,6] the authors consider the existence of positive solutions for the p- laplacian system with large λ:

2 006 G. A. Afrouzi and E. Graily Δ p u = λf(v) Δ p v = λg(u) u = v =0 in Ω in Ω on Ω (2) Δu = λf(v) Δv = λg(u) u = v =0 in Ω in Ω on Ω (3) Dalmasso in [3], discussed the system (3) when p = q = 2 and f,g are increasing and f,g 0. In [2]Dalmasso proved existence and uniqueness of positive solutions to () when the composition fo(cg) is sublinear at and superlinear at 0 for each c>0. The arguments in [5] rely on the fact the positive solutions to (3) in a ball are radially symmetric and decreasing (see [4,0]). D.D.Hai in [] discussed about uniqueness of positive solutions for semilinear elliptic of the system (3). In this paper we shall extend the results in [],[5] to the case of a bounded domain in R N and for a class of p-laplacian systems. Results for the single equation case were obtained in [7,9]. Our approach is based on sub- and supersolution, and maximum principle, and weak comparison principle. 2. Existence and uniqueness results We make the following assumptions (H.) f,g : R + R + are nondecreasing, continuous, C on (0, ) lim sup 0 + x f(x) <, lim sup 0 + x g(x) < (H.2) There exist positive β, δ, p, q with p, q > such that βx p f(x) δx p, for all x 0,and for p >p,q >q, βx q g(x) δx q f(x) x p and g(x) x q are nonincreasing for x large. Our main result is Theorem. Let (H.) (H.2) hold. Then system () has a unique positive solution for min (λ q μ p,λ q μ p ) large. The next lemma provides estimates for solutions to (). When Ω is a ball, it was established in [5].

3 Uniqueness of positive solutions 007 We shall denote the norm in C k (Ω) by. k. Lemma. Let (u, v) be a positive solution of (). Then there exist positive M and M i, i 4, such that M (λ q μ p ) (p+q) d(x, Ω) u(x) M 2 (λ q μ p ) (p+q) d(x, Ω) M 3 (λ q μ p ) (p+q) d(x, Ω) v(x) M 4 (λ q μ p ) (p+q) d(x, Ω) for min (λ q μ p,λ q μ p ) >M. Here d(x, Ω) denotes the distance for x to Ω. Proof. Let (u, v) be a positive solution for (). We first establish the upper estimate for v. In what follows, we shall denote by C i positive constant independent of λ, μ, u, v. Using the equations for u, v, we obtain u(x) =λ K(x, y)f(v(y))dy, v(x) Ω =μ K(x, y)g(u(y))dy Ω where K(x, y) denotes the Green, s function of Δ p ( Δ q ) with Dirichlet boundary conditions. Thus, by (H.2), Δ p u = λf(v) λδv p (4) and with follow from weak comparison principle we have and and From (5), (7), it follow that u 0 C (λδ v p 0) p (5) Δ q v = μg(u) λδu q (6) v 0 C (μδ u q 0 ) q (7) u 0 C 2 (λ q μ p ) (p+q) (8) Which, we follow from [] and with (H.2) and weak comparison principle and regularity estimates, implies v C 3 (μ g(u) 0 ) q C3 (μδ u q 0 ) q C3 (μδ[c 2 (λ q μ p ) (p+q) ] q ) M 4 (λ q μ p ) (p+q) q

4 008 G. A. Afrouzi and E. Graily and v(x) M 4 (λ q μ p ) (p+q) d(x, Ω) (9) Follows from the mean value theorem. The upper estimate for u follows in the same manner. Next, let x 0 Ω and R>0besuch that B B(x 0,R) Ω.Here B(x 0,R) denotes the open ball centered at x 0 with radius R. Then (u, v) is a supersolution for Δ p u = λf(v) in B Δ p v = λg(u) in B (0) u = v =0 on B We shall construct a positive subsolution (u 0,v 0 ) for (0) with u 0 u and v o v. To this end, let ɛ>0 and let ũ, ṽ be the solution of Δũ = ṽ p in B Δṽ = ũ q in B () ũ = ṽ =0 on B whose existence follows from [3, 5]. Define u 0 = ɛ pq ũ, v0 = μβɛ p ṽ, suppose that u 0 p 2, v 0 q 2,β is given by (H.2). A direct calculation gives Δ p u 0 = ( u 0 p 2 u 0 )= u 0 p 2 Δu 0 = u 0 p 2 (ɛ pq )ṽ p If λ q μ p > and ɛ sufficiently small, and λβ(μβɛ p ṽ) p λf(βɛ p ṽ) = λf(v 0 ) Δ q v 0 = v 0 q 2 Δv 0 = μβɛ p ũq = μβ(ɛ pq ũ) q μg(ɛ pq ũ) = μg(u 0 ), i.e., (u 0,v 0 ) is a subsolution for (0). Clearly u 0 u andv 0 in B for small ɛ. Hence there exists a solution (u, v) to (0) with u u, v v. Since u is radially symmetric, it follows from [5,Lemma 4] that

5 Uniqueness of positive solutions 009 u(x) M (λ q μ p ) (p+q) for x x 0 R 2 (2) for min (λ q μ p,λ q μ p ) large, where M is a positive constant independent of u, v, λ, μ. Let Ω =Ω\ B(x 0, R ) and let ψ be the solution of 2 Δ p ψ =0 in Ω, ψ =0 on Ω, (3) ψ = on B(x 0, R) 2 Since Δ p u 0 in Ω,the maximum principle (see, e.g., [8, 0]) implies u(x) ( M (λ q μ p ) (p+q) ) p ψ(x) M (λ q μ p ) (p+q) d(x, Ω) in Ω, where M is a positive constant satisfying M ψ(x) M d(x, Ω) for x Ω. Combine this and (2), we obtain the lower estimate for u. This completes the proof of lemma. Lemma 2. Let (u, v) be a solution to () and satisfy { Δp w 0 = g(u) in Ω, (4) w 0 =0 on Ω, Then for min (λ q μ p,λ q μ p ) large, there exist a positive number C independent of u, v, λ, μ, such that w 0 (x) Cd(x, Ω) for x Ω. Proof. Let ɛ 0 > 0. It follows from (H.2) and lemma that for (λ q μ p,λ q μ p ) large, g(u(x)) β(u(x)) q β[m (λ q μ p ) (p+q) ] q > if d(x, Ω) >ɛ 0.Thus Δ p w 0 { if d(x, Ω) >ɛ0, w 0 =0 if d(x, Ω) ɛ 0, (5)

6 00 G. A. Afrouzi and E. Graily and the lemma follows from the maximum principle and weak comparison principle. Proof of theorem. The existence part of follows from [3]. Let (u, v) and (u,v 2 ) be solutions () and suppose that min (λ q μ p,λ q μ p ) is large enough so lemma, 2 apply. By Lemma, M M 2 u u M 2 M u in Ω. Let α = sup{c >o: u cu inω}. Then α 0 α α 0, where α 0 = M M 2. We claim thatα. Suppose to the contrary that α<. Let q,q 2,p,p 2 >o be such that q 2 >q >q,p 2 >p >pand p, q >. Let A>0 be such that g(x) x q is nonincreasing for x>a and define Ω = {x Ω: u (x) > A α 0 }. Then g(αu (x)) α q g(u (x)) x Ω, (6) While if x Ω \ Ω, g(u (x)) g(αu (x)) K( α), where K = α 0 sup{ xǵ(x) : o<x A α 0 }, which implies g(u (x)) g(αu (x)) K( α) for x Ω \ Ω (7) Define the operator T : C(Ω) C(Ω) by T (z) =w if Δ p w = z in Ω, w =0 on Ω. Let w = Tg(αu ). Then it follows from (6),(7) and maximum principle that w w, where w satisfies α q g(u ) in Ω, Δ p w = g(u (x)) + K( α) in Ω \ Ω, w =0 on Ω (8) Let w 0 = Tg(u ). Then Δ p w 0 = g(u ) and therefore Δ p (w α q w 0 )= { 0 in Ω, (α q )g(u )+K( α) in Ω \ Ω, (9)

7 Uniqueness of positive solutions 0 Note that there exists a positive constant K depending 0nly on A, α 0,K,q, such that (α q )g(u )+K( α) K ( α) in Ω \ Ω we follow from [] and weak comparison principle, we obtain, Since w α q w 0 [K ( α)] p C. (20) it follows that M (λ q μ p ) (p+q) d(x, Ω) u (x) A α 0 on Ω \ Ω d(x, Ω) A α 0 M (λ q μ p ) (p+q) for x Ω \ Ω and therefore the right-hand side of (20) goes to 0 as λ q μ p. Let ɛ> 0, then it follows from (20) and the mean value theorem and weak comparison principle that w(x) α q w 0 [ɛ( α)] p d(x, Ω), x Ω, for min(λ q μ p,λ q μ p ) large, which implies by Lemma 2 that w(x) α q 2 w 0 (x) (α q α q 2 )w 0 (x) [ɛ( α)] p d(x, Ω) cα q ( α q 2 q )d(x, Ω) [ɛ( α)] p d(x, Ω) 0 [min(,q 2 q )cα q 0 ɛ p ( α) 2 p p ]( α)d(x, Ω) > if ɛ sufficiently small. Consequently, w(x) w(x) α q 2 w 0 (x), or Since Tg(αu ) α q Tg(u ). (2) Δ q v = μg(u) μg(αu ), it follows from (2) that v (μt g(αu ) q (μα q 2 Tg(u )) q = α q 2 v.

8 02 G. A. Afrouzi and E. Graily This implies Now similarly Δ p u = λf(v) λf(α q 2 v ). (22) (22),(23) and the maximum principle imply Tf(α q 2 v ) α p 2q 2 Tf(v ). (23) u (λt f(α q 2 v )) p (λα p 2 q 2 Tf(v )) p = α p 2 q 2 u, Which is a contradiction since α p 2q 2 >α. Thus α, i.e., u u and therefore u = u. Similarly, v = v, completing the proof of Theorem. References [] D. D. Hai, Uniqueness positive solutions for a class of semilinear elliptic systems, J.Math.Anal.Apple. 33 (2006) [2] M. Chhetri, D. D. Hai, R. Shivaji, On positive solutions for class of p-laplacian semipositone systems, Discrete Continious Dyn.Syst. 9(4) (2003) [3] Dalmasso, Existence and uniqueness of positive solutions of semilinear elliptic system, Nonlinear. Anal. 39 (2000), [4] D. G. Defigueredo, Monotonicity and symmetry of positive solution 0f elliptic systems in jeneral domain, NoDEA.Nonlinear Differential Equations Appl. (994) [5] D. D. Hai, Uniqueness positive solutions for a class of semilinear elliptic systems, Nonlinear. Anal. 52 (2003) [6] D. D. Hai, R. Shivaji, An existence result on positive solutions for a class of p-laplacian systems for, Nonlinear Anal. 56 (2004) [7] S. S. Lin, On the number of positive solutions for a class of elliptic equations when a parameter is large, Nonlinear Anal. 6 (99)

9 Uniqueness of positive solutions 03 [8] H. Amann, Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces, SIAM Rev. 8 (976) [9] E. N. Dancer, on the number of positive solutions of a weakly nonlinear elliptic equation when a parameter is large, Proc.London Math. Soc. (986) [0] W. C. Troy, Symmetric properties in systems of semilinear elliptic equations, J.Differentional Equations 42 (98) Received: March 2, 2008