2010 Mathematics Subject Classification: 35J65, 35K60, 35B05, 35R05.Existence of classical C2()TC() solutions of non-cooperative weakly coupled systems of elliptic second-order PDE is proved via the method of sub- and super-solutions

Boyadzhiev, G.

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Pliska Stud. Math. Bulgar. 21 (2012), 315–320STUDIA MATHEMATICABULGARICAEXISTENCE THEOREMS FOR NON-COOPERATIVEELLIPTIC SYSTEMSG. BoyadzhievAbstract. Existence of classicalC2(Ω)⋂C(Ω) solutions of non-cooperativeweakly coupled systems of elliptic second-order PDE is proved via the methodof sub- and super-solutions.1. Introduction. Let Ω ∈ Rn be a bounded domain with smooth boundary∂Ω. In this paper are considered weakly coupled linear elliptic systems of the form(1) LMu = f(x) in Ωand boundary data(2) u(x) = g(x) on ∂Ω,where LM = L + M , L is a matrix operator with null off-diagonal elementsL = diag (L1, L2, . . . , LN ), and matrix M = {mki(x)}Nk,i=1. Scalar operatorsLkuk = −n∑i,j=1Dj(akij(x)Diuk)+n∑i=1bki (x)Diuk + ckuk in Ωare supposed uniformly elliptic ones for k = 1, 2, . . . , N , i.e. there are constantsλ,Λ > 0 such thatλ |ξ|2 ≤n∑i,j=1akij(x)ξiξj ≤ Λ |ξ|22010 Mathematics Subject Classification: 35J65, 35K60, 35B05, 35R05.Key words: Elliptic systems, non-cooperative, existence, sub- and super-solution.316 G. Boyadzhievfor every k and any ξ = (ξ1, . . . , ξn) ∈ Rn.Right-hand side f(x) is supposed a bounded vector-function, that is(∗) |f l(x)| ≤ C in Ωfor every l = 1, . . . , N , where C is a positive constant.Coefficients ck and mik in (1) are supposed continuous in Ω, and akij(x),bki (x) ∈ C1(Ω) ∩ C(Ω). Assume in addition that for every k = 1, . . . , N(3)n∑i=1 n∑j=1Djakij(x) + bki (x)2, |ck| ≤ bholds for x ∈ Ω, where b is a positive constant.Hereafter by f−(x) = min(f(x), 0) and f+(x) = max(f(x), 0) are denotedthe non-negative and, respectively, the non-positive part of the function f. Thesame convention is valid for matrixes as well. For instance, we denote byM+ thenon-negative part of M , i.e. M+ = {m+ij(x)}Ni,j=1.In this paper is employed the method of sub- and super-solutions in orderto prove the existence of a classical C2(Ω)⋂C(Ω) solution of problem (1). Akey-point of the method is the validity of the comparison principle. Unlike thecooperative systems, for non-cooperative ones there is no complete theory forthe validity of the comparison principle. In [1] are given some sufficient con-ditions such that the comparison principle holds, which are recalled in section“Comparison principle for non-cooperative linear elliptic systems” below.We consider linear systems only for the sake of simplicity. The results holdas well for quasi-linear weakly coupled elliptic systemsQl(u) = −dival(x, ul,Dul) + F l(x, u1, . . . , uN ,Dul) = f l(x) in Ωul(x) = gl(x) on ∂Ωfor l = 1, . . . , N, where the coefficients al(x, u, p), F l(x, u, p), f l(x), gl(x) aresupposed to be at least measurable functions with respect to the x variable andlocally Lipschitz continuous on u and p.2. Comparison principle for non-cooperative linear elliptic sys-tems. Let us recall the following Theorem (Theorem 3 in [1]):Existence theorems for non-cooperative elliptic systems 317Theorem 1. Let (1) be a weakly coupled elliptic system with irreducible co-operative part of L∗M−. Then the comparison principle holds for the classicalsolutions of system (1) if there is x0 ∈ Ω such that(4) λ+N∑k=1m+kj(x0) > 0 for j = 1 . . . , Nand(5) λ+m+jj(x) ≥ 0 for every x ∈ Ω and j = 1 . . . , Nwhere λ is the principal eigenvalue of the operator LM− in Ω.The same result holds if the cooperative part of L∗M−has structure withJordan cells on the main diagonal and zeroes otherwise (Theorem 4 in [1]).Theorem 2. Assume m−ij ≡ 0 for i 6= j and (2) is satisfied. Then thecomparison principle holds for the classical C2(Ω)⋂C(Ω) solutions of system(1) if there is x0 ∈ Ω such that(6) λj +N∑k=1m+kj(x0) > 0 for every j = 1 . . . , N, and(7) λj +m+jj(x) ≥ 0 for every x ∈ Ω and j = 1 . . . , N,where λj is the principal eigenvalue of L˜j = Lj +m−jj in Ω.Theorem 2 is formulated for diagonal matrix M−, but the statement is validwith obvious modification if M− has Jordan cells on the main diagonal.Finally (Theorem 5 in [1]), in case that the cooperative partM− is triangular,we haveTheorem 3. Assume the cooperative part M− of system (1) is triangular,i.e. m−ij = 0 for i = 1, . . . , N , j > i. Then the comparison principle holds for theclassical C2(Ω)⋂C(Ω) solutions of system (1), if there is ε > 0 such that(8) λj − (1− δ1j)ε+N∑k=1m+kj(x0) > 0for j = 1 . . . , N and some x0 ∈ Ω and(9) λj − (1− δ1j)ε+m+jj(x) ≥ 0 for every x ∈ Ω and j = 1 . . . , N,where λj is the principal eigenvalue of the operator Lj +m−jj.318 G. Boyadzhiev3. Existence of classical solution. The first step of the method is ex-istence of super- and sub-solution of system (1), (2). It is easy to check thatconstant-vector (M, . . . ,M) is a super-solution for any constant M such that(10)n∑i=1mki(x) ≥CM,where C is the upper bound |f l(x)| (see (∗)).Theorem 4. Suppose conditions (4), (5); (6), (7) or (8), (9) hold for system(1), (2), according to the structure of matrix M , as well as (10). Assume v(x) isa classical super-solution and w(x) is a a classical sub-solution of (1), (2). Thenthere exists a classical solution u(x) of the problem (1), (2) with null boundarydata.Since the system (1) is a linear one, we assume in the following proof withoutloss of generality that g(x) = 0.S k e t c h o f t h e p r o o f. Let denoteF k(x, u1, . . . , uN ) =n∑i=1mki(x)ui + ckuk1. Consider the sequence of vector - functions u0, u1, . . . , ul, . . . , where u0 =w(x) and ul ∈ H10 (Ω) defines ul+1 by induction as a solution of the problem(11) −N∑i,j=1Di(akij(x)Djukl+1) +N∑i=1bki (x)Dukl+1 + σukl+1 == fk(x)− F k(x, u1k, . . . , uNk ) + σukl in Ωwith null boundary conditions(12) ukl+1(x) = 0 on ∂Ωfor every k = 1, . . . , N .Let denote the left-hand side of (11) by Ak(x, u, σ), and the right-hand side– by Bk(x, u, σ), k = 1, . . . , N .The problem (11), (12) is reducible system and in fact decomposes to Nindependent equations. Then Theorem 8.3 in [3] (page 348) is applicable, hencethese equations are solvable in C2,α(Ω) and(13) ‖ukl ‖Cβ(Ω) < c,Existence theorems for non-cooperative elliptic systems 319(14)∥∥∥∥∂ukl∂xi∥∥∥∥Cβ(Ω< c1 for every i = 1, . . . , n, γ = 1, . . . ,m.Furthermore ul0 ≤ ul1 ≤ · · · ≤ ukl+1 ≤ · · · by the comparison principle.The proof of ul0 ≤ ul1 is trivial since ul0 is a sub-solution of (1), (2).3. Obviously the inequality ul+1(x) ≤ v(x) holds for every ul+1, since v(x) isa super-solution of the same system (1), (2).4. The sequence of vector-functions {uk} is monotonously increasing andbounded from above in Ω. Therefore there is a function u such that uk(x)→ u(x)point-wise in Ω. Furthermore, (13) yields {uk} is uniformly equicontinuous inΩ and {uk} < const, since ukl (x) is Holder continuous and therefore |ukl (x) −ukl (x0)| ≤ c(|x − x0|β) for every l = 1, . . . , N . By Arzela–Ascoli compactnesscriterion there is a sub-sequence {ukj} that converges uniformly to u ∈ C(Ω).For convenience we denote {ukj} by {uk}.Since u ∈ C(Ω) and all functions {ukj} satisfy the null boundary conditions,then u satisfies the boundary conditions as well.The functions uk are Holder continuous with the same Holder constant, there-fore u is Holder continuous as well with the same Holder constant, i.e. u ∈ Cβ(Ω).Since ul+1(x) is monotone and u(x) is continuous, then {(uk)2} → u2 in Ω.Then the Dominated Convergence Theorem (Theorem 5 at p.648 in [2]) yieldsuk → u(x) in (L2(Ω))N .5. Analogously to the previous step, (14) yields {Diuk} is uniformly equicon-tinuous in Ω and {Diuk} < const. According to Arzela–Ascoli compactnesscriterion there is sub-sequence {Diukj} that converges uniformly to Diu ∈ C(Ω).For convenience we denote {ukj} by {uk}.6. For every 0 < η(x) = (η1(x), . . . , ηN (x)) ∈ (H10 (Ω))N∫Ω N∑i,j=1akij(x)Djukl+1Diηk(x) +N∑i=1bki (x)Dukl+1ηk(x) + σukl+1ηk(x) dx ==∫Ω(fk(x)− F k(x, u1k, . . . , uNk ) + σukl )ηk(x)dxholds and for k →∞ we obtain320 G. Boyadzhiev∫Ω N∑i,j=1akij(x)DjukDiηk(x) +N∑i=1bki (x)Dukηk(x) dx ==∫Ω(fk(x)− F k(x, u1, . . . , uN ))ηk(x)dxthat is u(x) is solution of (1), (2). REFERENCES[1] G. Boyadzhiev. Comparison principle for non-cooperative elliptic systems.Nonlinear Anal. 69, 11 (2008), 3838–3848.[2] L. C. Evans. Partial Differential Equations. Graduate Studies in Mathe-matics, vol. 19. Providence, RI, AMS, 1998.[3] O. A. Ladyzhenskaya, N. Ural’tseva. Linear and Quasilinear Equationsof Eliptic Type. Moscow, Nauka, 1964 (in Russian).Institute of Mathematics and InformaticsBulgarian Academy of SciencesAcad. G. Bonchev Str., Bl. 81113 Sofia, Bulgariae-mail: gpb@math.bas.bg

Existence Theorems for Non-Cooperative Elliptic Systems

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Existence Theorems for Non-Cooperative Elliptic Systems

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