EXISTENCE OF SOLUTIONS OF THE DIRICHLET PROBLEM FOR AN INFINITE SYSTEM OF NONLINEAR DIFFERENTIAL-FUNCTIONAL EQUATIONS OF ELLIPTIC TYPE

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1 Opuscula Mathematica Vol. 25 No Tomasz. Zabawa EXITENCE OF OLUTION OF THE DIRICHLET PROBLEM FOR AN INFINITE YTEM OF NONLINEAR DIFFERENTIAL-FUNCTIONAL EQUATION OF ELLIPTIC TYPE Abstract. The Dirichlet problem for an infinite weakly coupled system of semilinear differential-functional equations of elliptic type is considered. It is shown the existence of solutions to this problem. The result is based on Chaplygin s method of lower upper functions. Keywords: infinite systems, elliptic differential-functional equations, monotone iterative technique, Chaplygin s method, Dirichlet problem. Mathematics ubject Classification: Primary 35J45, econdary 35J65, 35R10, 47H INTRODUCTION Let be an infinite set. Let G R m be an open bounded domain with C 2+α boundary (α (0, 1)). Let B() be the Banach space of all bounded functions w : R, w(i) =w i (i ) with the norm w B() := sup w i. i In B() there is a partly order w w defined as w i w i for every i. Elements of B() will be denoted by (w i ) i,too. Let C(Ḡ) be the space of all continuous functions v : Ḡ R with the norm v C( Ḡ) := max v(x). x Ḡ In this space v ṽ means that v(x) ṽ(x) for every x Ḡ. ByCl+α (Ḡ), where l =0, 1, 2,.. α (0, 1), we denote the space of all continuous functions in Ḡ whose 333

2 334 Tomasz. Zabawa derivatives of order less or equal l exist are Hölder continuous with exponent α in G (see [5], pp ). By H l,p (G) we denote the obolev space of all functions whose weak derivatives of order l are included in L p (G) (see [1], pp ). A notation g C l+α ( G) (resp. g H l,p l+α ( G)) means that there exists a function g C (Ḡ) (resp. g H l,p (G) C(Ḡ)) such that g(x) =g(x) for every x G. In these spaces norms are defined as g C l+α ( G) := inf g Cl+α (Ḡ) : g Cl+α (Ḡ) : x G : g(x) =g(x)} g H 2,p ( G) := inf g H 2,p (G) : g H2,p (G) C(Ḡ) : x G : g(x) =g(x)}. We denote z =(z i ) i C (Ḡ) if z : Ḡ B() zi : Ḡ R (i ) is a continuous function sup i z i C( <. ThespaceC Ḡ) (Ḡ) is a Banach space with the norm z C (Ḡ) := sup z i C( Ḡ) i the partly order z z defined as z i (x) z i (x) for every x Ḡ, i. The space C l+α (Ḡ) is a space of all functions (zi ) i such that z i l+α C (Ḡ) for every i sup i z i C l+α <. In this space the norm is defined as (Ḡ) z C l+α (Ḡ) =supz i C l+α. (Ḡ) i We will write that z = (z i ) i L p (G) if zi L p (G) for every i sup i z i Lp (G) <. A notation z =(zi ) i H l,p (G) means that zi H l,p (G) for every i sup i z i H l,p <. In these spaces the norms are defined as (G) z L p (G) =supz i Lp (G) i z H l,p (G) =supz i. H l,p (G) i We consider the Dirichlet problem for an infinite weakly coupled system of semilinear differential-functional equations of the following form L i [u i ](x) =f i (x, u(x),u), for x G, i (1) u i (x) =h i (x), for x G, i, (2) where m m L i [u i ](x) := a i jk(x)u i x jx k (x)+ b i j(x)u i x j (x), j,k=1 j=1

3 Existence of olutions of the Dirichlet Problem (... ) 335 which are strongly uniformly elliptic in Ḡ, f i : Ḡ B() C (Ḡ) (x, y, z) f i (x, y, z) R for every i. The notation f(x, u(x),u) means that the dependence of f on the second variable is a function-type dependence f(x,u(x), ) is a functional-type dependence. A function u is said to be regular in Ḡ if u C(Ḡ) C2 (G). A function u is said to be a classical (regular) solution of the problem (1), (2) in Ḡ if u is regular in Ḡ fulfills the system of equations (1) in G with the condition (2). A function u is said to be a weak solution of the problem (1), (2) in Ḡ if u L2 (G) such that L i [u i ] L 2 (G) L i [u i ](x)ξ(x) dx = f i (x, u(x),u)ξ(x) dx G for every i for any test function ξ C 0 (Ḡ). We would like to find assumptions which guarantee existence of the classical solutions of the problem (1), (2) G u: Ḡ B(). Regular functions u 0 = u 0 (x) v 0 = v 0 (x) in Ḡ satisfying the infinite systems of inequalities: L i [u i 0](x) f i (x, u 0 (x),u 0 ) for x G, i, (3) u i 0(x) h i (x) for x G,i, L i [v0](x) i f i (x, v 0 (x),v 0 ) for x G, i, (4) v0(x) i h i (x) for x G,i are called a lower an upper function for the problem (1), (2), respectively. If u 0 v 0,wedefine K := (x, y, z): x Ḡ, y [m 0,M 0 ],z u 0,v 0 }, where m 0 := (m i 0) i, m i 0 := min x Ḡ u i 0(x), M 0 := (M i 0) i, M i 0 := max x Ḡ v i 0(x) u 0,v 0 := ζ C (Ḡ): u 0(x) ζ(x) v 0 (x) for x Ḡ}. Assumptions. We make the following assumptions. (a) L is a strongly uniformly elliptic operator in Ḡ, i.e., there exists a constant µ>0 such that m m a i jk(x)ξ j ξ k µ ξj 2, i, j,k=1 for all ξ =(ξ 1,...,ξ m ) R m, x G. j=1

4 336 Tomasz. Zabawa (b) The functions a i jk, bi j for i, j, k =1,...,m are functions of class C0+α (Ḡ) a i jk (x) =ai kj (x) for every x Ḡ. (c) h i C 2+α ( G) for every i sup i h i <. C 2+α ( G) (d) There exists at least one ordered pair u 0, v 0 of a lower an upper function for the problem (1), (2) in Ḡ such that u 0 (x) v 0 (x) for x Ḡ. (e) f(,y,z) C 0+α (Ḡ) for y B(), z C(Ḡ). (f) For every i, x Ḡ, y, ỹ B(), z C(Ḡ) f i (x, y, z) f i (x, ỹ, z) Lf y ỹ B(), where L f > 0 is a constant independent of i f i (x, y 1,...,y i 1,y i,y i+1,...,z) f i (x, y 1,...,y i 1, ỹ i,y i+1,...,z) k i y i ỹ i, where k i > 0 is a constant there exists k < such that k i k for every i. (g) f(x,, ) is a continuous function for every x Ḡ. (h) f i (x,,z) is a quasi-increasing function for every i, x Ḡ, z C(Ḡ) i.e., for every i for arbitrary y, ỹ B() if y j ỹ j for all j such that j i y i = ỹ i,thenf i (x, y, z) f i (x, ỹ, z) for x Ḡ, z C(Ḡ). (i) f i (x, y, ) is an increasing function for every i, x Ḡ, y B(). 2. AUXILIARY REULT From the assumption (f) we have k := (k i ) i B(). Let β =(β i ) i C 0+α (Ḡ). We define the operator P : C 0+α (Ḡ) β γ C2+α (Ḡ), where γ =(γ i ) i is the solution (supposedly unique) of the following problem (L i k i I)[γ i ](x) =f i (x, β(x),β)+k i β i (x) for x G, i, γ i (x) =h i (x) for x G, i. (5) We remark that the problem (5) is a system of separate problems with only one equation, so P[β] is a collection of solutions of these problems.

5 Existence of olutions of the Dirichlet Problem (... ) 337 Lemma 1. The operator P : C 0+α (Ḡ) C2+α (Ḡ) is a continuous bounded operator. If the operator P maps C 0+α (Ḡ) into C0+α(Ḡ), thenitisacompactoperator. Proof. Let β C 0+α (Ḡ), so β i (x) β i ( x) H β x j x j α, R m where H β > 0 is some constant independent of i x R m We define the operator such that for every i F =(F i ) i : C 0+α (Ḡ) β δ C0+α (Ḡ) F i [β](x) =δ i (x) :=f i (x, β(x),β)+k i β i (x). =( m j=1 x2 j )1/2. For arbitrary i we have: δ i (x) δ i ( x) = f i (x, β(x),β)+k i β i (x) f i ( x, β( x),β) k i β i ( x) f i (x, β(x),β) f i ( x, β(x),β) + f i ( x, β(x),β) f i ( x, β( x),β) + + k i β i (x) β i ( x) (H f + L f H β + kh β ) x x α R, m where H f + L f H β + kh β is some constant independent of i. By the properties of f, we see that the operator F is a continuous bounded operator. Now, we have our problem for arbitrary i in the following form (L i k i I)[γ i ](x) =δ i (x) for x G, (6) γ i (x) =h i (x) for x G, which satisfies the assumptions of the chauder theorem ([7], p. 115), so the problem (6) for every i has a unique solution γ i 2+α C (Ḡ) the following estimate γ i ( C C 2+α δ i + (Ḡ) C 0+α h i ) (Ḡ) C 2+α (7) ( G) holds, where C>0 is independent of δ, h i. Let us introduce the operator G =(G i ) i : C 0+α (Ḡ) δ γ C2+α (Ḡ). The function where γ i = G i [δ i ]=G i 1[h i ]+G i 2[δ i ], G i 1 : C 2+α (Ḡ) hi G i 1[h i 2+α ] C (Ḡ)

6 338 Tomasz. Zabawa G i 1[h i ] is the unique solution of the problem (6) with δ i (x) =0in Ḡ, G i 2 : C 0+α (Ḡ) δi G i 2[δ i 2+α ] C (Ḡ) G i 2[δ i ] is the unique solution of the problem (6) with h i (x) =0on G. The operator G i is a continuous operator because the operator G i 1 is independent of δ i, G i 2 is a continuous operator (from (7)) with respect to δ i.by(7), we have γ i = C 2+α G i F i [β i ] ( C (Ḡ) C 2+α δ i + (Ḡ) C 2+α h i ) (Ḡ) C 2+α, ( G) where C>0 is independent of δ, h i. Thus the operator G F is a continuous bounded operator. ince G C 2+α, the imbedding operator I: C 2+α (Ḡ) C0+α (Ḡ) is a compact operator ([1], p. 11). o P = I G F is a compact operator. Next, let us consider the operator P as a operator mapping L p (G). Lemma 2. The operator P is a compact operator mapping L p (G) into Lp (G). Proof. We define δ the operator F such in the proof of Lemma 1 but on an element of L p (G). The operator F: L p (G) Lp (G) F is a continuous bounded operator by arguing as [6] (Th. 2.1, Th. 2.2 Th. 2.3, pp ) [9] (Th.19.1,p.204). Now, we have our problem for arbitrary i in the following form (L i k i I)[γ i ](x) =δ i (x) for x G, (8) γ i (x) =h i (x) for x G, which satisfies the assumptions of Agmon Douglis Nirenberg theorem for arbitrary i, so the problem (8) has a unique weak solution γ i H 2,p (G) the following estimate γ i ( H 2,p (G) C δ i Lp (G) + h i ) H 2,p (9) ( G) holds, where C>0 is independent of δ, h i. Let us introduce the operator G =(G i ) i : L p (G) δ γ H2,p (G). The function where γ i = G i [δ i ]=G i 1[h i ]+G i 2[δ i ], G i 1 : H 2,p (G) h i G i 1[h i ] H 2,p (G)

7 Existence of olutions of the Dirichlet Problem (... ) 339 G i 1[h i ] is the unique weak solution of the problem (8) withδ i (x) =0in Ḡ, G i 2 : L p (G) δ i G i 2[δ i ] H 2,p (G) G i 2[δ i ] is the unique solution of the problem (8) withh i (x) =0on G. The operator G i is a continuous operator because the operator G i 1 is independent of δ i, G i 2 is a continuous operator (from (9)) with respect to δ i. Also we know that γ i H 2,p (G) = G i F i [β i ] H 2,p (G) C ( δ i L p (G) + h i H 2,p ( G) where C>0 is independent of δ, h i. Thus the operator G F is a continuous bounded operator. ince G C 2+α, the imbedding operator I: H 2,p (G) Lp (G) is a compact operator ([1], p. 97), P = I G F is also a compact operator. Now, we prove next some properties of the operator P. Lemma 3. The operator P is an increasing operator. Proof. Let β(x) β(x) in Ḡ, so for all i, βi (x) β i (x) in Ḡ. Let γ := P[β] γ := P[ β]. For arbitrary i (L i k i I)[ γ i γ ) i ](x) = ( ) = f (x, i β(x), β f i (x, β(x),β)+k i βi (x) β i for x G, (x) ( γ i γ i )(x) =0, for x G. By assumption (h), (i) (f), (L i k i I)[ γ i γ i ](x) [ ( f i x, β 1 (x),...,β i 1 (x), β ) ] i (x),β i+1 (x),...,β f i (x, β(x),β) + ( ) + k i βi (x) β i (x) ), 0. o for every i (L i k i I)[ γ i γ i ](x) 0 for x G, γ i (x) γ i (x) =0 for x G. By the maximum principle ([8], p. 64) γ i (x) γ i (x) 0 in Ḡ. We have for all i γ i (x) γ i (x), so γ(x) γ(x) in Ḡ.

8 340 Tomasz. Zabawa Lemma 4. If β is an upper (resp. a lower) function for the problem (1), (2) in Ḡ, then P[β] β (resp. P[β] β) inḡ P[β] is an upper (resp. a lower) function for problem (1), (2) in Ḡ. Proof. Let γ = P[β]. By (5) we have for every i (L i k i I)[γ i β i ](x) = (L i k i I)[γ i ](x)+(l i k i I)[β i ](x) = = f i (x, β(x),β)+k i β i (x)+l i [β i ](x) k i β i (x) =f i (x, β(x),β)+l i [β i ](x) from (4) o for every i f i (x, β(x),β)+l i [β i ](x) 0 (γ i β i )(x) =h i (x) β i (x) 0. (L i k i I)[γ i β i ](x) 0 for x G, (γ i β i )(x) 0 for x G. Now,byusing the maximum principle ([8], th. 6, p. 64) separately for every i γ i (x) β i (x) 0 in Ḡ. o γ(x) β(x) in Ḡ. From Lemma 1 it follows that γ C 2+α (G) from (5) the assumption (f), we get for every i L i [γ i ](x) f i (x, γ(x),γ)= (L i k i I)[γ i ](x) f i (x, γ(x),γ) k i γ i (x) = = f i (x, β(x),β)+k i β i (x) f i (x, γ(x),γ) k i γ i (x) ( f i (x, γ 1 (x),...,γ i 1 (x),β i (x),γ i+1 (x),...,γ) f i (x, γ(x),γ) ) + + k i (β i (x) γ i (x)) 0 in Ḡ, so it is a upper the function for problem (1), (2) in Ḡ. 3. MAIN REULT Theorem. If the assumptions (a) (i) hold, then the problem (1), (2) has at least one classical solution u such that u u 0,v 0. Proof. By induction, we define two sequences of functions u n } n=0 v n } n=0 by setting: u 1 = P[u 0 ], u n = P[u n 1 ], v 1 = P[v 0 ], v n = P[v n 1 ].

9 Existence of olutions of the Dirichlet Problem (... ) 341 Because u 0 v 0 are regular functions, these sequences are well defined by Lemma 1. The sequence u n } n=0 is increasing v n } n=0 is decreasing by Lemma 4: by induction: u 1 (x) =P[u 0 ](x) u 0 (x) in Ḡ, v 1 (x) =P[v 0 ](x) v 0 (x) in Ḡ, u n (x) =P[u n 1 ](x) u n 1 (x) in Ḡ, n =1, 2,... v n (x) =P[v n 1 ](x) v n 1 (x) in Ḡ, n =1, 2,... ince the operator P is increasing by the assumption (d) we have consequently by induction Therefore u 1 (x) =P[u 0 ](x) P[v 0 ](x) =v 1 (x) in Ḡ u n (x) v n (x) in Ḡ. u 0 (x) u 1 (x) u n (x) v n (x) v 1 (x) v 0 (x). The sequences u n } n=0 v n } n=0 are monotone bounded, so they have pointwise limits we can define: u(x) := lim n u n(x), v(x) := lim n v n(x) for every x Ḡ. The functions u n } n=0 v n } n=0 are functions of class L p (G). Let be p (m, ) (we need this assumption to can use a imbedding theorem). Because u n } n=0 v n } n=0 are bounded functions in L p (G) P is an increasing compact operator in L p (G) (from Lemma 2), Pu n} Pv n } are converging sequences in L p (G) u(x) = lim P[u n](x) = lim P[ P[u n 1 ] ] (x) =P[u](x), n n v(x) = lim P[v n](x) = lim P[ P[v n 1 ] ] (x) =P[ v](x). n n ince u, v L p (G) : by the Agmon Douglis Nirenberg theorem we have u = P[u], v = P[ v] (10) u, v H 2,p (G).

10 342 Tomasz. Zabawa 2,p Because the obolev space H (Ḡ) for p>mis continuously imbedding in C0+α (Ḡ) u i C C u i, 0+α (Ḡ) H 2,p (G) where C is independed of i ([1], p ), we get u, v C 0+α (Ḡ). (11) Applaying the chauder theorem to (10) separately for every s for (11) we get From the proof we know that u, v C 2+α (Ḡ). u 0 (x) u(x) v(x) v 0 (x). Corollary. The solutions u, v are minimal maximal solution of the problem (1), (2) in u 0,v 0. Proof. If w is a solution of the problem (1), (2) then w(x) =P[w](x) u 0 (x) w(x) v 0 (x). BecauseP is an increasing operator, we have u 1 (x) =P[u 0 ](x) P[w](x) =w(x) =P[w](x) P[v 0 ](x) =v 1 (x) by induction we get Thus REFERENCE u n (x) w(x) v n (x). u(x) = lim n u n(x) w(x) lim n v n(x) = v(x). [1] Adams R.A.: obolev paces. New York, Academic Press [2] Agmon., Douglis A., Nirenberg L.: Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary condition. I. Comm. Pure Appl. Math. 12 (1959), [3] Appell J., Zabreiko P.: Nonlinear uperposition Operators. Cambrige, Cambrige University Press [4] Brzychczy.: Existence of solution of nonlinear Dirichlet problem for differentialfunctional equations of elliptic type. Ann. Polon. Math. 58 (1993), [5] Gilbarg D., Trudinger N..: Eliptic Partial Differential Equations of econd Order. Berlin, Heidelberg, New York, Tokyo, pringer [6] Krasnosel skii M.A.: Topological Methods in the Theory of Nonlinear Integral Equations. Oxford, Pergamon Press 1963.

11 Existence of olutions of the Dirichlet Problem (... ) 343 [7] Ladyzhenskaya O. A., Ural tseva N. N.: Linear Quasilinear Elliptic Equations. New York, London, Academic Press [8] Protter M.H., Weinberger H. F.: Maximum Principles in Differential Equations. New York, pringer [9] Vainberg M. M.: Variational Methods for the tudy of Nonlinear Operators. an Francisco, Holden-Day Tomasz. Zabawa zabawa@mat.agh.edu.pl AGH University of cience Technology, Faculty of Applied Mathematics al. Mickiewicza 30, Kraków, Pol Received: October 18, 2004.