MAXIMUM PRINCIPLE AND EXISTENCE OF POSITIVE SOLUTIONS FOR NONLINEAR SYSTEMS ON R N

Electronic Journal of Differential Equations, Vol. 20052005, No. 85, pp. 1 12. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu login: ftp MAXIMUM PRINCIPLE AND EXISTENCE OF POSITIVE SOLUTIONS FOONLINEAR SYSTEMS ON HASSAN M. SERAG, EADA A. EL-ZAHRANI Abstract. In this paper, we study the following non-linear system on pu = ax u p 2 u + bx u α v β v + f qv = cx u α v β u + dx v q 2 v + g x x lim ux = lim vx = 0, u, v > 0 in RN x x where pu = div u p 2 u with p > 1 and p 2 is the p-laplacian, α, β > 0, p, q > 1, and f, g are given functions. We obtain necessary and sufficient conditions for having a maximum principle; then we use an approximation method to prove the existence of positive solution for this system. 1. Introduction The operator p occurs in problems arising in pure mathematics, such as the theory of quasiregular and quasiconformal mappings and in a variety of applications, such as non-newtonian fluids, reaction-diffusion problems, flow through porous media, nonlinear elasticity, glaciology, petroleum extraction, astronomy, etc see [14, 16]. We are concerned with existence of positive solutions and with the following form of the maximum principle: If f, g 0 then u, v 0 for any solution u, v of 1.3. The maximum principle for linear elliptic systems with constant coefficients and the same differential operator in all the equations, have been studied in [7, 9]. Systems defined on unbounded domains and involving Schrödinger operators have been considered in [1, 2, 22]. In [18, 19], the authors presented necessary and sufficient conditions for having the maximum principle and for existence of positive solutions for linear systems involving Laplace operator with variable coefficients. These results have been extended in [15], to the nonlinear system n p u i = a ij u j p 2 u j + f i u i in Ω 1.1 j=1 u i = 0 on Ω 2000 Mathematics Subject Classification. 35B45, 35J55, 35P65. Key words and phrases. Maximum principle; nonlinear elliptic systems; p-laplacian; sub and super solutions. c 2005 Texas State University - San Marcos. Submitted May 19, 2005. Published July 27, 2005. 1

2 H. M. SERAG, E. EL-ZAHRANI EJDE-2005/85 In [6], it has been proved the validity of the maximum principle and the existence of positive solutions for the following system defined on bounded domain Ω of, and with cooperative constant coefficients a, b, c, d: Here, we study system p u = a u p 2 u + b u α v β v + f, q v = c u α v β u + d v p 2 v + g, u = v = 0 on Ω p u = ax u p 2 u + bx u α v β v + f q v = cx u α v β u + dx v q 2 v + g in Ω in Ω x x lim ux = lim vx = 0, u, v > 0 in RN x x 1.2 1.3 where p u = div u p 2 u with p > 1 and p 2 is the p-laplacian, α, β > 0, p, q > 1, and f, g are given functions. System 1.3 is a generalization for 1.2 to the whole space and the coefficients ax, bx, cx, dx are variables. We obtain necessary and sufficient conditions on the coefficients for having a maximum principle for system 1.3. Then using the method of sup and super solutions, we prove the existence of positive solutions under some conditions on the functions f and g. This article is organized as follows: In section 2, we give some assumptions on the coefficients ax, bx, cx, dx and on the functions f, g to insure the existence of a solution of 1.3 in a suitable Sobolev space. We also introduce some technical results and some notation, which are established in [3, 4, 16]. Section 3 is devoted to the maximum principle of system 1.3, while section 4 is devoted to the existence of positive solutions. 2. Technical results In this section, we introduce some technical results concerning the eigenvalue problem see [16] where Ψ p u = u p 2 u and gx satisfies p u = λgxψ p u in 2.1 ux 0 as x, u > 0 in gx L N/p L, gx 0 almost everywhere in 2.2 For 1 < p < N, let p = pn N p be the critical Sobolev exponent of p. Let us introduce the Sobolev space D 1,p defined as the completion of C0 with respect to the norm 1/p u D 1,p = u p It can be shown that D 1,p = { u L Np N p : u L p N } and that there exists > 0 such that for all u D 1,p, u L Np/N p u D 1,p 2.3

EJDE-2005/85 MAXIMUM PRINCIPLE AND EXISTENCE OF POSITIVE SOLUTIONS 3 Clearly D 1,p is a reflexive Banach space, which is embedded continuously in L Np/N p see [11]. Lemma 2.1. i If {u n } is a sequence in D 1,p, with u n u wealy, then there is a subsequence, denoted again by {u n }, such that Bu n Bu ii If B u = 0, then Bu = 0, where Bu = gu u p dx. Theorem 2.2. Let g satisfy 2.2. Then 2.1 admits a positive first eigenvalue λ g p. Moreover, it is characterized by λ g p gu u p u p D 2.4 1,p Theorem 2.3. Let g satisfy 2.2. Then i the eigenfunction associated to λ g p is of constant sign; i.e., λ g p is a principal eigenvalue. ii λ g p is the only eigenvalue of 2.1 which admits positive eigenfunction. 3. Maximum Principle We assume that 1 < p, q < N and that the coefficients ax, bx, cx, and dx are smooth positive functions such that and where α + 1 p + β + 1 q ax, dx L p/n L 3.1 bx < ax α+1/p dx β+1/q cx < ax α+1/p dx β+1/q, = 1, α + β + 2 < N, 1 p + 1 p = 1, 3.2 1 q + 1 q = 1 3.3 Theorem 3.1. Assume that 3.1 and 3.2 hold. For f L Np 1+p, g Nq L Nq 1+q, system 1.3 satisfies the maximum principle if the following conditions are satisfied: λ a p > 1, λ d q > 1 3.4 Np λa p 1 α+1/p λd q 1 β+1/q 1 > 0 3.5 Conversely, if the maximum principle holds, then 3.4 holds and bx α+1/p cx β+1/q [λ a p 1] α+1/p [λ d q 1] β+1/q > Θ, 3.6 ax dx where Θ = inf x φ p /ψ q α+1 β+1 sup x φp /ψ q α+1 p and φ respectively ψ is the positive eigenfunction associated to λ a p respectively λ d q. β+1 q

4 H. M. SERAG, E. EL-ZAHRANI EJDE-2005/85 Proof. The necessary condition: Assume that λ a p 1, then the function f := ax1 λ a pφ p 1 and g := 0 are nonnegative; nevertheless φ, 0 satisfies 1.3, which contradicts the maximum principle. Similarly, if λ d q 1, then the functions g := dx1 λ d qψ q 1 and f := 0 are nonnegative; nevertheless 0, ψ satisfies 1.3, which contradicts the maximum principle. Now suppose that λ a p > 1, λ d q > 1 and 3.6 does not hold; i.e., bx α+1/p cx β+1/q [λ a p 1] α+1/p [λ d q 1] β+1/q Θ, ax dx Then, there exists ξ > 0 such that ax λ ap 1 bx Let ξ = D q α+1 β+1 C p So α+1/p φ p α+1 β+1 ψ q ξ with C, D > 0. Then ax λ ap 1 α+1/p Cφ p α+1 p bx Dψ q cx dx β+1/q Cφ p 1 λ d q 1 β+1/q cx β+1/q dx φ p α+1 β+1 λ d q 1 β+1/q ψ q β+1 q α+1 β+1 Dψ q axλ a p 1Cφ p β+1/q bxdψ β+1 3.7 dxλ d q 1Dψ q α+1/p cxcφ α+1 3.8 From the two expression above, we have Hence axλ a p 1Cφ p 1 bxdψ β+1 Cφ α, dxλ d q 1Dψ q 1 cxcφ α+1 Dψ β. f = axλ a p 1Cφ p 1 + bxdψ β+1 + Cφ α 0, g = dxλ d q 1Dψ q 1 + cxdψ β + Cφ α+1 0 Since f and g are nonnegative functions, and Cφ, Dψ is a solution of 1.3 and the maximum principle does not hold. Now, we show that the condition is sufficient. Assume that 3.4 and 3.5 hold. If u, v is a solution of 1.3, then for f, g 0, we obtain by multiplying the first equation of 1.3 by u := max0, u and integrating over : u p = ax u p + bx u α+1 v + β v + R N bx u α+1 v β v + fu. Then u p ax u p + bx u α+1 v β+1.

EJDE-2005/85 MAXIMUM PRINCIPLE AND EXISTENCE OF POSITIVE SOLUTIONS 5 From 2.4, we get λ a p 1 ax u p bx u α+1 v β+1 Applying Holder inequality and using 3.2, we find λ a p 1 ax u p ax u p α+1/p dx v q Hence [ β+1/q λ a p 1 ax u p R N β+1/q ] α+1/p dx v q ax u p 0 If ax u p α+1/p R = 0, then u = 0. If not, we have N λ a p 1 α+1/p ax u p α+1 β+1 dx v q Similarly λ d q 1 β+1/q dx v q α+1 β+1 ax u p β+1 α+1 β+1 α+1 From 3.9 and 3.10, we obtain λ a p 1 α+1/p λ d q 1 β+1/q 1 dx v q ax u p β+1/q β+1 α+1 3.9 3.10 From 3.5, we have u = v = 0 and hence u 0, v 0 i.e. the maximum principle holds. Corollary 3.2. If p = q, then the maximum principle holds for system 1.3 if and only if conditions 3.4 and 3.5 are satisfied. 4. Existence of positive solutions By an approximation method used in [5], we prove now that the system 1.3 has a positive solution in the space D 1,p D 1,q. For ɛ 0, 1, we introduce the system u ɛ p 2 u ɛ p u ɛ = ax 1 + ɛ 1/p u ɛ p 1 + bx v ɛ β v ɛ u ɛ α + f in 1 + ɛ 1/q v ɛ β+1 1 + ɛ 1/p u ɛ α RN v ɛ q 2 v ɛ q v ɛ = dx 1 + ɛ 1/q v ɛ q 1 + cx v ɛ β u ɛ α u ɛ + g in 1 + ɛ 1/q v ɛ β 1 + ɛ 1/p u ɛ α+1 RN 0 lim u ɛ = lim v ɛ = 0, u ɛ, v ɛ > 0 in x x 4.1 Letting ξ, η = u ɛ, v ɛ then system above can be written as p ξ = hξ, η + f q η = ξ, η + g in in ξ, η 0 as x ξ, η > 0 in

6 H. M. SERAG, E. EL-ZAHRANI EJDE-2005/85 where ξ p 2 ξ hξ, η = ax 1 + ɛ 1/p ξ p 1 + bx η β η ξ α 1 + ɛ 1/q η β+1 1 + ɛ 1/p ξ α, η β ξ α ξ ξ, η = cx 1 + ɛ 1/q η β 1 + ɛ 1/p ξ α+1 + dx η q 2 η 1 + ɛ 1/q η q 1. From 3.1 and 3.2 hξ, η, ξ, η are bounded functions; i.e., there exists M > 0 such that hξ, η M, and ξ, η M for all ξ, η. Then, as in [6], we can prove the following lemma. Lemma 4.1. If ξ, η ξ, η, wealy in L Np/N p L Nq/N q, then ξ x p 2 ξ x ax 1 + ɛ 1/p ξ x p 1 ξx p 2 ξx 1 + ɛ 1/p ξx p 1 LNp/Np 1+pRN 0 4.2 η β η bx 1 + ɛ 1/q η β+1 ξ α 1 + ɛ 1/p ξ α η β η ξ α 4.3 1 + ɛ 1/p η β+1 1 + ɛ 1/p ξ α approaches 0 under the norm of L Np/Np 1+p, a.e. in as approaches infinity in L Nq/Nq 1+q. η x q 2 η x dx 1 + ɛ 1/q η x q 1 ηx q 2 ηx 0 4.4 1 + ɛ 1/q ηx q 1 ξ α ξ η β cx 1 + ɛ 1/p ξ α+1 1 + ɛ 1/q η β ξ α ξ η β 0 1 + ɛ 1/p ξ α+1 1 + ɛ 1/q η β 4.5 a.e. in as in L Nq/Nq 1+q Lemma 4.2. System 4.1 has a solution U ɛ =: u ɛ, v ɛ in D 1,p D 1,q. Proof. We complete the proof in four steps: Step 1. Construction of sub-super solutions of 4.1: Let ξ 0 D 1,p respectively η 0 D 1,q be a solution of p ξ 0 = M + f resp. q η 0 = M + g 4.6 and let ξ 0 D 1,p respectively η 0 D 1,q be a solution of p ξ 0 = M + f resp. q η 0 = M + g 4.7 Then η 0, ξ 0 is a super solution of 4.1 and η 0, ξ 0 is a sub solution since p ξ 0 hξ 0, η f p ξ 0 M f = 0 η [η 0, η 0 ] p ξ 0 hξ 0, η f p ξ 0 M f = 0 η [η 0, η 0 ] p η 0 hξ, η 0 g q η 0 M + g = 0 ξ [ξ 0, ξ 0 ] p η 0 hξ, η 0 g q η 0 M + g = 0 ξ [ξ 0, ξ 0 ] Let us assume that K = [ξ 0, ξ 0 ] [η 0 η 0 ].

EJDE-2005/85 MAXIMUM PRINCIPLE AND EXISTENCE OF POSITIVE SOLUTIONS 7 Step 2. Definition of the operator T : We define the operator T : ξ, η w, z, where w, z is the solution of the system p w = hξ, η + f q z = ξ, η + g w = z 0 in in as x 4.8 Sept 3. Construction of an invariant set under T. We have to prove that T K: From 4.6 and 4.8, we get p w p ξ 0 hξ, η M 4.9 Multiplying this equation by w ξ 0 + and integrating over, we obtain [Ψ p w Ψ p ξ 0 ][ w ξ 0 + ] hξ, η Mw ξ 0 + 0. By monotonicity of p-laplacian, we have w ξ 0 + = 0 and hence w ξ 0. Similarly w ξ 0, so that step is complete. Step 4. T is completely continuous: We prove that T maps wealy convergent sequence to strongly convergence ones. From 4.8, we get p w p w [ ξ p 2 ξ = ax 1 + ɛ 1/p ξ p 1 ξ p 2 ξ ] 1 + ξ 1/p ξ p 1 [ η β η ξ α + bx 1 + ɛ 1/q η β+1 1 + ɛ 1/p ξ α η β η ξ i α ] 1 + ɛ 1/q η β+1 1 + ɛ 1/p ξ i α Multiplying by w w and integrating over, we obtain [Ψ p w Ψ p w][ w w] [ ξ p 2 = ax 1 + ɛ 1/p ξ p 1 ξ p 2 ξ 1 + ξ 1/p ξ p 1 [ η β η ξ α + bx 1 + ɛ 1/q η β+1 1 + ɛ 1/p ξ α η β η ξ i α ] w 1 + ɛ 1/q η β+1 1 + ɛ 1/p ξ i α w Using Hölder s inequality, we obtain [Ψ p w Ψ p w][ w w] ] w w [ ξ p 2 ax 1 + ɛ 1/p ξ p 1 ξ p 2 ξ L 1 + ξ 1/p ξ ] p 1 Np/Np 1+p w w L Np/N p [ + bx η β η 1 + ɛ 1/q η β+1 ξ α 1 + ɛ 1/p ξ α 4.10 η β η ξ i α L w 1 + ɛ 1/q η β+1 1 + ɛ 1/p ξ i ] α w L Np/Np 1+p Np/N p

8 H. M. SERAG, E. EL-ZAHRANI EJDE-2005/85 It is well nown [23], that ξ ξ p c{[ ξ p 2 ξ ξ p 2 ξ ]ξ ξ } α/2 { ξ p + ξ p } 1 α/2 ξ, ξ 4.11 where α = p if 1 p 2 and α = 2 if p > 2. From 4.11 and the continuous imbedding of D 1,p in L Np/N p, we get w w p [ ax [ + bx ξ p 2 1 + ɛ 1/p ξ p 1 ξ p 2 ξ 1 + ξ 1/p ξ p 1 ] L Np/Np 1+p η β η 1 + ɛ 1/q η β+1 ξ α 1 + ɛ 1/p ξ α η β η ξ i α L 1 + ɛ 1/q η β+1 1 + ɛ 1/p ξ i ] α Np/Np 1+p w w Applying lemma 4.1, we obtain w w p 1 D 0 as + in D 1,p which 1,p implies w w strongly as + in D 1,p. Similarly we can prove that z z as + in D 1,p, So T is completely continuous operator. Since is a convex, bounded, closed in: D 1,p D 1,q, we can apply Schauder s Fixed Point Theorem and obtain the existence of a fixed point for T, which gives the existence of solution U ɛ =: u ɛ, v ɛ of S ɛ. Now, we can prove the existence of solution for system 1.3. Theorem 4.3. If 2.2 2.4 are satisfied, then for any f L Np 1+p, g Nq L Nq 1+q, system 1.3 has a nonnegative solution U = u, v in the space: D 1,p D 1,q. Proof. This proof is done in three steps. Step 1. First we prove that U ɛ =: ɛ 1/p u ɛ, ɛ 1/q v ɛ is bounded in D 1,p D 1,q. Multiplying the first equation of 4.1 by ɛu ɛ and integrating over, we obtain ɛ 1/p u ɛ p ax ɛ 1/p u ɛ + bx ɛ 1/p u ɛ + ɛ 1/p ɛ 1/p u ɛ f ax N/p ɛ 1/p u ɛ Np/N p + bx Np/Np 1+p ɛ 1/p u ɛ Np/N p + ɛ 1/p ɛ 1/p u ɛ Np/N p f NpNp 1+p M ɛ 1/p u ɛ Np/N p M ɛ 1/p u ɛ D 1,p so ɛ 1/p u ɛ p 1 D 1,p M which implies U ɛ =: ɛ 1/p u ɛ is bounded in D 1,p. Similarly for ɛ 1/q v ɛ. Np

EJDE-2005/85 MAXIMUM PRINCIPLE AND EXISTENCE OF POSITIVE SOLUTIONS 9 Step 2. U ɛ =: ɛ 1/p u ɛ, ɛ 1/q v ɛ converges to 0,0 strongly in D 1/p D 1/q. From step 1, U ɛ =: ɛ 1/p u ɛ, ɛ 1/q v ɛ converges wealy to u, v in D 1,p D 1,q and strongly in L Np N p L Nq N q. Multiplying the first equation of 4.1, by ɛ 1/p, we get p ɛ 1/p u ɛ = ax ɛ1/p u ɛ p 2 ɛ 1/p u ɛ +bxɛ1/q v ɛ β ɛ 1/q v ɛ ɛ 1/p u ɛ α 1 + ɛ 1/p u ɛ p 1 1 + ɛ 1/q v ɛ β+1 1 + ɛ 1/p u ɛ α +fɛ1/p Again using Lemma 4.1, we have ax ɛ1/p u ɛ p 2 ɛ 1/p u ɛ 1 + ɛ 1/p u ɛ p 1 and similarly ax u p 2 u 1 + u p 1 Np strongly in L Np 1+p bx ɛ1/q v ɛ β ɛ 1/q v ɛ ɛ 1/p u ɛ α 1 + ɛ 1/q v ɛ β+1 1 + ɛ 1/p u ɛ bx v β v α 1 + v β+1 Np strongly in L Np 1+p. Using a classical result in [21], we have So u α 1 + u α p ɛ 1/p u ɛ p u strongly in L Np Np 1+p. p u = ax u p 2 u 1 + u p 1 + bx v β v 1 + v β+1 u α 1 + u α 4.12 Multiplying this equality by u and integrating over, then applying 2.4 we get λ a p 1 ax u p u p dx v β+1 u α+1 Using Hölder inequality and 2.3, as in the proof of Theorem 3.1, we deduce λ a p 1 α+1/p ax u p β+1 α+1 dx v q β+1 α+1 4.13 Similarly, from the second equation of 4.1, we have λ d q 1 β+1/q dx v q β+1 α+1 ax u p β+1 α+1 4.14 Multiplying 4.13 by 4.14, we obtain λ d q 1 β+1/q λ a p 1 α+1/p 1 dx v q ax u p β+1 α+1 0 From conditions 3.4 and 3.5, we have u = v = 0, which implies that u, v 0. We show that u = v = 0. Multiplying 4.12 by u, we get λ d q 1 β+1/q λ a p 1 α+1/p 1 dx v q ax u p β+1 α+1 0 which implies that u = v = 0, and step 2 is complete. Step 3. u ɛ, v ɛ is bounded in D 1,p D 1,q : Assume that u ɛ D 1,p or v ɛ D 1,q Set t ɛ = max u ɛ D 1,p, v ɛ D 1,q and z ɛ = u ɛ t 1/p ɛ, w ɛ = v ɛ t 1/q ɛ.

10 H. M. SERAG, E. EL-ZAHRANI EJDE-2005/85 Dividing the first equation of 4.1 by t 1/p ɛ and the second by t 1/q ɛ, we have z ɛ p 2 z ɛ p z ɛ = ax 1 + ɛ 1/p u ɛ p 1 + bx w ɛ β w ɛ z ɛ α 1 + ɛ 1/q v ɛ β+1 1 + ɛ 1/p u ɛ + α ft 1/p ɛ w ɛ q 2 w ɛ q w ɛ = dx 1 + ɛ 1/q v ɛ q 1 + cx z ɛ α z ɛ w ɛ β 1 + ɛ 1/p u ɛ β+1 1 + ɛ 1/q v ɛ + α gt 1/q ɛ As above, we can prove that z ɛ, w ɛ z, w strongly in D 1,p D 1,q ; and taing the limit, we obtain p z = axψ p z + bx w β z α w q w = dxψ q w + cx w β z α z and we deduce w = z = 0. Since there exists a sequence ɛ n n N such that either z ɛn = 1 or w ɛn = 1 we obtain a contradiction. Hence u ɛ, v ɛ is bounded in D 1,p D 1,q, we can extract a subsequence denoted u ɛ, v ɛ which converges to u 0, v 0 wealy in D 1,p D 1,q as ɛ 0. By using similar procedure as above, we can prove that that u ɛ, v ɛ converges strongly to u 0, v 0 in D 1,p D 1,q. Indeed, since ɛ 1/p u ɛ x, ɛ 1/q v ɛ x 0, 0 a.e. on, then, as in [6], we have u ɛ x p 2 u ɛ x ax 1 + ɛ 1/p u ɛ x p 1 ax u 0x p 2 u 0 x a.e. in as ɛ 0, u ɛ p 2 u ɛ ax 1 + ɛ 1/p u ɛ p 1 M u ɛ p 1 Mh p 1 L p, v ɛ β v ɛ u ɛ α bx bx v 1 + ɛ 1/q v ɛ β+1 1 + ɛ 1/p 0 β v 0 u 0 α a.e. in as ɛ 0, u ɛ αi v ɛ β v ɛ u ɛ α bx 1 + ɛ 1/q v ɛ β+1 1 + ɛ 1/p u ɛ M 2h α l β+1 L p α Hence from the Dominated Convergence Theorem and Lemma 4.1, we obtain [ u ɛ p 2 u ɛ ax u R 1 + ɛ N 1/p u ɛ p 1 0 p 2 u 0 p ] 1/p 0, [ v ɛ q 2 u ɛ dx v R 1 + ɛ N 1/q v ɛ p 1 0 p 2 v 0 q ] 1/q 0, v ɛ β v ɛ u ɛ α bx R 1 + ɛ N 1/q v ɛ β+1 1 + ɛ 1/p u ɛ Big v 0 β v α 0 u 0 α p 1/p 0, u ɛ α u ɛ v ɛ β cx 1 + ɛ 1/p u ɛ α+1 1 + ɛ 1/q v ɛ u 0 α u β 0 v 0 β q 1/q 0, as ɛ 0. Therefore, passing to the limit, u ɛ, v ɛ u 0, v 0 we obtain from 4.1, Hence u 0, v 0 satisfies 1.3. p u 0 = ax u 0 p 2 u 0 + bx v 0 β u 0 α v 0 + f q v 0 = dx v 0 q 2 v 0 + cx u 0 α v 0 β u 0 + g We remar that if α = β = 0 and p = q = 2, we obtain the results presented in [18, 19].

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12 H. M. SERAG, E. EL-ZAHRANI EJDE-2005/85 [25] Zeidler, E.; Nonlinear Functional Analysis and its Applications Vol. III, Variational Methods and Optimization, Springer Verlag, Berlin 1990. Hassan M. Serag Mathematics Department, Faculty of Science, Al-Azhar University, Nasr City 11884, Cairo, Egypt E-mail address: serraghm@yahoo.com Eada A. El-Zahrani Mathematics Department, Faculty of Science for Girls, Dammam, P. O. Box 838, Pincode 31113, Saudi Arabia E-mail address: eada00@hotmail.com