Lecture I.: (Simple) Basic Variational and Topological Methods (an introductory lecture for graduate students and postdocs)

Transcription

1 Lecture I.: (Simple) Basic Variational and Topological Methods (an introductory lecture for graduate students and postdocs) Lecture II.: Regular and Singular Systems with the p- and q-laplacians (an advanced lecture, suitable also for graduate students and postdocs) Lecture III.: Special Solution Methods for Singular Systems (an advanced lecture) Peter Takáč Universität Rostock D-1855 Rostock, Germany

2 forschung/anganalysis/index.html Mabel Cuesta, Université du Littoral (ULCO) / U.L.B. Bruxelles Jacques Giacomoni, Université de Pau et des Pays de l Adour (UPPA) Ian Schindler, Université Toulouse 1 Capitole (CeReMath) Jacqueline Fleckinger-Pellé, Université Toulouse 1 Capitole (CeReMath) Partially supported by the academic exchange program PROCOPE (France Germany)

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5 Lecture I.: (Simple) Basic Variational and Topological Methods (an introductory lecture for graduate students and postdocs) Peter Takáč Motivation from Mathematical Biology: Gierer-Meinhardt reaction-diffusion system The (p 1)-homogeneous quasilinear elliptic operator u p u def = div( u p 2 u), called the p-laplacian, is defined for u W 1,p (Ω) with values pu W 1,p (Ω), the dual space of W 1,p (Ω), where 1 p + 1 p = 1. p is called the monotone p-laplacian.

6 System of two quasilinear elliptic equations: (1) p u = λ a(x) u α 1 v β 1 1 v + f(x) in Ω; q v = µ b(x) v α 2 u β 2 1 u + g(x) in Ω; u = v = on Ω, where Ω is a bounded domain in R N whose boundary Ω is a connected C 2 -manifold, ( Ω not necessarily connected???), x = (x 1,..., x N ) is a generic point in Ω, p, q (1, ) are given numbers, f, g L (Ω) are given functions, a, b L (Ω) are given functions satisfying a def = essinf x Ω a(x) >, b def = essinf x Ω b(x) >, and α i, β i are constants with α i and β i > for i = 1,2.

7 If f and g on Ω, we may compare system (1) with a homogeneous nonlinear eigenvalue problem for the unknown pair of parameters (λ, µ) R + R + = (, )2 associated with the unknown pair of nonnegative eigenfunctions u W 1,p (Ω) and v W1,q (Ω). Computing the partial derivatives f u, f v g u, g v = ( λα1 a u α 1 2 u v β 1 1 v, λβ 1 a u α 1 v β 1 1 ) µβ 2 b v α 2 u β 2 1, µα 2 b v α 1 2 v u β 1 1 u we observe that the system is cooperative (regular) if, e.g. α 1 < p 1, α 2 < q 1, β 1 >, β 2 >, and competitive (singular) if, e.g. α 1 < p 1, α 2 < q 1, β 1 <, β 2 <.

8 Given any v C (Ω), i.e. v : Ω R + is a continuous function with v = on Ω, let us consider the first equation in system (1), p u = λ a(x) u α 1 v β 1 1 v + f(x) in Ω; u = on Ω, i.e., for u > and v > in Ω, (2) p u = m(x)(u + ) a +f(x) in Ω; u = on Ω, where a (, p 1) is a constant and m L loc (Ω) L1 (Ω) is a weight function. Owing to a < p 1, Problem (2) is subhomogeneous. Assume m L (Ω). A nonnegative solution u W 1,p (Ω) is obtained as a nonnegative critical point (the global minimizer) of the energy functional (3) E(u) = 1 p Ω u p dx f(x) udx Ω 1 a + 1 Ω (u+ ) a+1 m(x)dx

9 for u W 1,p (Ω). It is easy to see that this functional is coercive on W 1,p (Ω); by the Sobolev embedding W 1,p (Ω) La+1 (Ω) provided 1 < a < p 1. An alternative way for obtaining a nonnegative solution u W 1,p (Ω) is the following iteration procedure: Let u W 1,p (Ω) be the unique solution to (4) p u = f(x) in Ω; u = on Ω. We construct recursively a monotone increasing sequence of nonnegative functions u u 1 u 2... in W 1,p (Ω), such that (5) { p u n = m(x) u a n 1 + f(x) in Ω; u n = on Ω, for n = 1,2,3,.... The inequality u n 1 u n a.e. in Ω implies u n u n+1 a.e. in Ω, by the weak comparison principle (see, e.g., Tolksdorf (1983)):

10 If f, g L (Ω) are given and u, v W 1,p (Ω) satisfy { p u = f(x) in Ω; u Ω = ; (6) p v = g(x) in Ω; v Ω =, then f g a.e. in Ω = u v a.e. in Ω. Moreover, the weak comparison principle implies: u n u (m) n W 1,p (Ω) is a monotone increasing function of the weight m L (Ω), m a.e. in Ω, that is, if m, m L (Ω), then we have m m a.e. in Ω = u (m) n u ( m) n a.e. in Ω. The convergence of the monotone increasing sequence u n ր u, say, in L p (Ω) as n ր is easily obtained from the Lebesgue monotone convergence theorem. The W 1,p (Ω)-norms ( = the Lp (Ω)-norms) are bounded, by Hölder s inequality,

11 u n p W 1,p (Ω) = Ω u n p dx = Ω m(x) ua n 1 u ndx m L (Ω) u n 1 a L p (Ω) u n L p/(p a) (Ω) C m L (Ω) u n 1 a L p (Ω) u n p dx Ω where C R + is a Sobolev constant depending only on < a < p 1 and Ω. Hence, for n = 1,2,..., ( ) 1/p u n p 1 W 1,p (Ω) C m L (Ω) u n 1 a L p (Ω) C m L (Ω) u n a L p (Ω). The convergence of the sequence u n u in W 1,p (Ω) as n is then obtained from the convergence u n u L p (Ω), the weak convergence of a subsequence u n u in W 1,p (Ω), and the convergence u n W 1,p (Ω) u W 1,p (Ω) combined with the uniform convexity of W 1,p (Ω). As a particular result, u u (m) W 1,p (Ω) has the following properties:

12 Proposition. Let Ω R N be a bounded domain whose boundary Ω is a compact C 2 -manifold, 1 < p <, and < a < p 1. Assume m L (Ω). Then the Dirichlet problem (2) has a unique nonnegative solution u W 1,p (Ω), u = u(m), where the function u (m) W 1,p (Ω) has been constructed by the iteration procedure (4) and (5) described above. Furthermore, we have either u (m) = a.e. in Ω, or else u (m) (7) u (m) > in Ω and < on Ω. ν Finally, given any two weights m, m L (Ω), we have m m a.e. in Ω = u (m) u ( m) a.e. in Ω. ((2)) p u = m(x)(u + ) a + f(x) in Ω; u = on Ω. A closely related result has been obtained in Fleckinger, Hernández, Takáč, and de Thélin (1998).

13 The strong comparison principle (see, e.g., Tolksdorf (1983), Guedda and Véron (1989), and Cuesta and Takáč (1998)): Theorem. If f, g L (Ω) and u, v W 1,p (Ω) satisfy ((6)) { p u = f(x) in Ω; u Ω = ; p v = g(x) in Ω; v Ω =, then f g a.e. in Ω with f g in Ω = (8) u < v in Ω and u ν > v ν on Ω. Of course, ν : Ω R N denotes the outer unit normal vector field on the boundary of Ω.

14 Moreover, the strong comparison principle implies: u n u (m) n W 1,p (Ω) is a strictly monotone increasing function of the weight m L (Ω), m a.e. in Ω, that is, if m, m L (Ω), then we have u (m) n m m a.e. in Ω with m m in Ω = < u ( m) n in Ω and u (m) n ν u( m) n > ν on Ω. This is still an OPEN PROBLEM; we will discuss it in the next lecture.

15 Lecture II.: Regular and Singular Systems with the p- and q-laplacians (an advanced lecture, suitable also for graduate students and postdocs) Peter Takáč The OPEN PROBLEM: Strong Comparison Principle. Theorem. If f, g L (Ω) and u, v W 1,p (Ω) satisfy { p u = f(x) in Ω; u Ω = ; ((6)) p v = g(x) in Ω; v Ω =, then f g a.e. in Ω with f g in Ω = ((8)) u < v in Ω and u ν > v ν on Ω.

16 Verified in some special cases: 1. General solutions in an interval Ω = ( R, R) or radially symmetric solutions in a ball Ω = B R () (Cuesta and Takáč (1998)); 2. Positive solutions in a domain Ω R N whose boundary Ω is a connected compact C 2 -manifold (Cuesta and Takáč (1998)); 3. Positive solutions in a domain Ω R N with a condition on the set {x Ω : f(x) = g(x)} (M. Lucia and S. Prashanth (23)); 4. General solutions in a domain Ω R N with a condition on the set {x Ω : f(x) = g(x)} (M. Guedda and L. Véron (1989)). 5. General solutions in a domain Ω R N with a condition on the set {x Ω : u(x) = v(x) = } (Damascelli, Pacella, and Ramaswamy (1999)).

17 COUNTEREXAMPLE to Strong Comparison Principle: Let Ω = {x R N : x < 1}, p > 2, and λ <. Set r = x ; hence, r 1. Recall p = p p 1 > 1. For every number θ satisfying p < θ <, we define (9) (1) u θ (x) def = 1 r θ and f θ (x) def = [(p 1)(θ 1) 1 + N] θ p 1 r (p 1)(θ 1) 1 λ(1 r θ ) p 1 for x Ω. Obviously, u θ C 1 (Ω) and f θ C (Ω) satisfy u θ > in Ω and f θ > in Ω, together with ψ p ( u θ ) u θ p 2 u θ [C 1 (Ω)] N and (11) p u θ λu p 1 θ = f θ (x) in Ω; u θ = on Ω. We observe that u θ () = 1, and u θ1 (x) < u θ2 (x) whenever < r < 1 and p < θ 1 < θ 2 <.

18 We claim that, given any two numbers θ 1 and θ 2 satisfying p p 2 θ 1 < θ 2 <, there exists a constant λ p λ p (θ 1, θ 2 ) > such that also f θ1 (x) < f θ2 (x) for < r 1 and every λ with λ p λ <. Consequently, the SCP is violated for the boundary value problem (11). To prove our claim, it suffices to verify that there exists a constant λ p λ p (θ 1, θ 2 ) > with the following property: For all θ and λ satisfying θ 1 θ θ 2 and λ p λ <, we have (12) θ f θ (x) > for < r 1. Notice that the partial derivative θ f θ f θ / θ exists, (p 1) 1 θ f θ = [p(θ 1) + N]θ p 2 r (p 1)(θ 1) 1 + [p(θ 1) + N θ]θ p 1 r (p 1)(θ 1) 1 ln r + λ(1 r θ ) p 2 r θ ln r for x Ω, and thus θ f θ C (Ω). Inequality (12) is verified by a direct calculation (ln r < ).

19 System of two quasilinear elliptic equations: (13) p u = λ a(x) u α 1 v β 1 1 v in Ω; q v = µ b(x) v α 2 u β 2 1 u in Ω; u = v = on Ω, where Ω is a bounded domain in R N whose boundary Ω is a connected C 2 -manifold, ( Ω not necessarily connected???), x = (x 1,..., x N ) is a generic point in Ω, p, q (1, ) are given numbers, a, b L (Ω) are given functions satisfying def a = essinf a(x) >, x Ω b def = essinf x Ω b(x) >, and α i, β i are constants with α i and β i > for i = 1,2. The (p 1)-homogeneous quasilinear elliptic operator u p u def = div( u p 2 u), called the p-laplacian, is defined for u W 1,p (Ω) with values pu W 1,p (Ω), the dual space of W 1,p (Ω), where 1 p + 1 p = 1.

20 We view system (13) as a homogeneous nonlinear eigenvalue problem for the unknown pair of parameters (λ, µ) R + R + = (, )2 associated with the unknown pair of nonnegative eigenfunctions u W 1,p (Ω) and v W1,q (Ω). We will refer to such a couple (λ, µ) as a principal eigenvalue of system (13). Notice that system (13) is neither variational nor of a Hamiltonian type, in general, except for the cases when either f v g f u or u g v where we have denoted by f(x, u, v) and g(x, u, v), respectively, the right-hand side of the first and second equations in (13). Computing the partial derivatives f v, g u f u, g v = ( λβ1 a u α 1 v β 1 1, µβ 2 b v α 2 u β 2 1 λα 1 a u α 1 2 u v β 1 1 v, µα 2 b v α 1 2 v u β 1 1 u )

21 we observe that the former case occurs if and only if α 1 = β 2 1, α 2 = β 1 1, and λβ 1 a(x) = µβ 2 b(x) for a.e. x Ω, whereas the latter case occurs if and only if either α 1 1 = β 2, α 2 1 = β 1, and λα 1 a(x) = µα 2 b(x) for a.e. x Ω, or α 1 = α 2 =. The special superhomogeneous (often termed superlinear ) case α 1 = α 2 = and β 1 β 2 > (p 1)(q 1) was treated in Clément, Manásevich, and Mitidieri (Comm. P.D.E., 1993). Systems with variational and Hamiltonian structures are studied in de Thélin and Vélin (1991, 1993), de Figueiredo (28), for instance. Our method does not require any variational or Hamiltonian structure for system (13).

22 Variational structure: For any p, q (1, ) and u W 1,p (Ω), v W1,q (Ω), E(u, v) def = 1 p where whence Ω u p dx + 1 q Ω v q dx + E(u, v)dx, Ω E u = f(u, v) and E = g(u, v), v f v = g u. Hamiltonian structure: For p = q = 2 and u, v W 1,2 (Ω), H(u, v) def = where whence Ω u v dx + Ω H(u, v)dx, H v = f(u, v) and H u f u = g v. = g(u, v),

23 We wish to apply a simplified version of a Kreĭn-Rutman theorem for homogeneous nonlinear mappings due to Takáč (Nonl. Anal., 1996). Given any f L (Ω), we denote by T p (f) u W 1,p (Ω) the unique weak solution of the boundary value problem (14) p u = f(x) in Ω; u = on Ω. It is well-known (DiBenedetto, Lieberman, Tolksdorf) that u C 1,β (Ω) for some β (,1). We denote X = [C 1(Ω)]2, X + = {(f, g) X : f and g in Ω}, and X + is the topological interior of X + in X. Finally, define the map S : X X by S(u, v) def = (ũ, ṽ) with ũ = T p ( a u α 1 v β 1 1 v ), ṽ = T q ( b v α 2 u β 2 1 u ) for (u, v) X.

24 Homogeneity of S is obtained from that of S 2 : u ṽ v = ṽ ũ : [C 1 (Ω)] [C 1 (Ω)] [C 1 (Ω)]. A curve of principal eigenvalues Lemma 1. (i) (u, v) X + \ {} is a weak solution of (13) for some (λ, µ) (R + )2 (u, v) X + and S(u, v) = (λ 1/(p 1) u, µ 1/(q 1) v). (ii) For all ρ, σ R + and for all (u, v) X + we have S(ρu, σv) = ( (ρ α 1σ β 1) 1/(p 1) S 1 (u, v), (ρ β 2σ α 2) 1/(q 1) S 2 (u, v) ). (iii) If (u, v) X + solves (13) with (λ, µ ) in place of (λ, µ), then for any ρ, σ >, the pair (ρu, σv) solves (13) with (λ, µ) satisfying (15) λ = ρ p 1 α 1σ β 1λ, µ = σ q 1 α 2ρ β 2µ.

25 1. The case when S is homogeneous (ρ α 1σ β 1) 1/(p 1) = ρ, (ρ β 2σ α 2) 1/(q 1) = σ. For ρ = σ R + = (, ) this means α 1 + β 1 = p 1 and α 2 + β 2 = q 1. Look for principal eigenvalues of the map S via the Kreĭn-Rutman Theorem: Theorem 1. Assume (16) α 1 + β 1 = p 1 and α 2 + β 2 = q 1. Then there exists Λ > and a couple (u 1, v 1 ) X + such that system (13) possesses a positive weak solution (u, v) X + associated to some (λ, µ) (R + )2 if and only if (17) λ 1 β 1 µ 1 β 2 = Λ Moreover, (u, v) = c(u 1, v 1 ) for some constant c >.

26 2. The case when S 2 is homogeneous We arrive at an alternative condition to Case 1: α 1 = α 2 = and β 1 β 2 = (p 1)(q 1). Another application Kreĭn-Rutman Theorem Let us assume α 1 < p 1 and α 2 < q 1. We introduce a new mapping T : X + X + defined by T(u, v) def = (J 1 (v), J 2 (u)) where, for (u, v) X, J 1 (v) = ũ is the unique (weak) solution ũ of { p ũ = a(x) ũ (18) α 1v β 1 in Ω; ũ = on Ω, and J 2 (u) = ṽ is the unique (weak) solution ṽ of { q ṽ = b(x) ṽ (19) α 2u β 2 in Ω; ṽ = on Ω. Existence is obtained by classical minimization while uniqueness follows from a convexity argument.

27 The regularity results combined with the strong maximum principle (Tolksdorf (1983), Vázquez (1984)) imply that the pair (ũ, ṽ) belongs to X +. Alternatively, the strict subhomogeneity may be used. Let us consider the mapping T 2. Notice that T 2 (u, v) = (J 1 J 2 (u), J 2 J 1 (v)) for any (u, v) X. Hence, we can decouple T 2 and look for eigenvalues of each component. We see now that the condition (2) below for the homogeneity of J 1 J 2 and J 2 J 1 is less restrictive that the one found earlier for S 2. We will need the following result: Lemma 2. Let us denote V = C 1 (Ω). Then the mapping J i : V + V + is nondecreasing for i = 1,2.

28 Lemma 3. (i) u, v V + \ {} is a weak solution of (13) for some (λ, µ) (R + )2 u, v V + and J 1 (v) = λp 1 α 1 1 u, J 2 (u) = µ 1 q 1 α 2 v. (ii) For any ρ, σ > we have β 1 β 2 (J 1 J 2 )(ρu) = ρ(p 1 α 1 )(q 1 α 2 ) (J 1 J 2 )(u), β 1 β 2 (J 2 J 1 )(σv) = σ(p 1 α 1 )(q 1 α 2 ) (J 2 J 1 )(v). Sub- / Superhomogeneity Condition: β 1 β 2 (p 1 α 1 )(q 1 α 2 ) 1. Homogeneity Condition: = 1, i.e., (2) β 1 β 2 = (p 1 α 1 )(q 1 α 2 ).

29 The following main result holds: Theorem 2. Assume α 1 < p 1, α 2 < q 1 and the homogeneity condition ((2)) β 1 β 2 = (p 1 α 1 )(q 1 α 2 ). Then there exist Λ > and (u, v ) X + such that system (13) possesses a positive weak solution (u, v) X + associated to some (λ, µ) (R + )2 if and only if (21) λ 1 β1 (p 1 α 1 ) µ Moreover, (u, v) = (ρu, ρµ 1/β 2v ) with some positive constant ρ. 1 β2 (q 1 α 2 ) = Λ.

30 Non-homogeneous systems (22) p u = λa(x) u α 1 v β 1 1 v + f(x) in Ω; q v = µb(x) v α 2 u β 2 1 u + g(x) in Ω; u = v = on Ω. where f, g L (Ω) are given functions. Our aim is to study the solvability of system (22) in the following three cases: (a) λ, µ > and (λ, µ) below or to the left of the eigenvalue curve C 1, (b) (λ, µ) C 1, and (c) λ, µ > and (λ, µ) above or to the right of, but close to the eigenvalue curve C 1. We recall that the eigenvalue curve C 1 has been defined in (21) with Λ def = Λ β 2 /(q 1 α 2 ) 1.

31 When (λ, µ) lies below or to the left of C 1 In order to assure the existence of C 1 we assume the more general condition (2) jointly with β 1, β 2 >, α 1 < p 1, and α 2 < q 1. Theorem 3. Let f, g L (Ω), f, g, and let λ >, µ > be such that (23) λ 1 β1 (p 1 α 1 ) µ 1 β2 (q 1 α 2 ) < Λ. Then system (22) has a unique weak solution (u, v) X +. If, moreover, α 1 = α 2 = and f + g then there is a unique solution in X. Corollary 3. Consider f, g L (Ω) and let λ >, µ > be such that (23) holds. Then system (22) possesses at least one solution.

32 The case when (λ, µ) C 1 Theorem 4. Let f, g L (Ω), f, g, f + g, and let λ >, µ > be such that λ 1 β1 (p 1 α 1 ) µ 1 β2 (q 1 α 2 ) = Λ. Then system (22) has no solution in X +. If, moreover, α 1 = α 2 = then there is no solution in X. When (λ, µ) lies above or to the right of C 1 An antimaximum principle for systems Theorem 5. Let α 1 = α 2 =, β 1 β 2 = (p 1)(q 1), and (λ 1, µ 1 ) C 1. Consider two functions f, g L (Ω) with f + g. Then there exists δ > such that, for all pairs (λ, µ) R 2 with λ 1 < λ < λ 1 + δ and µ 1 < µ < µ 1 + δ, every (weak) solution (u, v) of system (22) satisfies u and v in Ω, i.e., u, v V +.

33 (22) p u = λa(x) u α 1= v β 1 1 v + f(x) in Ω; q v = µb(x) v α 2= u β 2 1 u + g(x) in Ω; u = v = on Ω. where f, g L (Ω) are given functions. The homogeneity condition (2) β 1 β 2 = (p 1 α 1 )(q 1 α 2 ) = (p 1)(q 1).

34 A singular quasilinear system (Gierer-Meinhardt system in morphogenesis) (24) p u = λ a(x) u a 1 v b 1 + f(x) in Ω; q v = µ b(x) v a 2 u b 2 + g(x) in Ω; u = v = on Ω, where Ω is a bounded domain in R N whose boundary Ω is a C 2 -manifold, Ω not necessarily connected, x = (x 1,..., x N ) is a generic point in Ω, p, q (1, ) are given numbers, f, g L (Ω) are given functions, a, b L (Ω) are given functions satisfying a def = essinf x Ω a(x) >, b def = essinf x Ω and a i, b i are constants with a i and b i > for i = 1,2. b(x) >,

35 Lecture III.: Special Solution Methods for Singular Systems (an advanced lecture) Peter Takáč A singular quasilinear system (Gierer-Meinhardt system in morphogenesis)

36 ((24)) p u = λ a(x) u a 1 v b 1 + f(x) in Ω; q v = µ b(x) v a 2 u b 2 + g(x) in Ω; u = v = on Ω, where Ω is a bounded domain in R N whose boundary Ω is a C 2 -manifold, Ω not necessarily connected, x = (x 1,..., x N ) is a generic point in Ω, p, q (1, ) are given numbers, f, g L (Ω) are given functions, a, b L (Ω) are given functions satisfying a def = essinf x Ω a(x) >, b def = essinf x Ω and a i, b i are constants with a i and b i > for i = 1,2. b(x) >,

37 Application of Schauder s Fixed-point Theorem Let us assume a 1, a 2 >. We introduce a mapping T : X + X + defined by T(u, v) def = (J 1 (v), J 2 (u)) where, for (u, v) X +, J 1 (v) = ũ is the unique positive (weak) solution ũ of { p ũ = a(x) ũ (25) a 1v b 1 + f(x) in Ω; ũ = on Ω, and J 2 (u) = ṽ is the unique positive (weak) solution ṽ of { q ṽ = b(x) ṽ (26) a 2u b 2 + g(x) in Ω; ṽ = on Ω. Clearly, both these problems are strictly subhomogeneous. Again, existence is obtained by classical minimization while uniqueness follows from a convexity argument for the following Dirichlet problem with a weight:

38 Given 1 < r <, δ >, κ, and a function K L 1 loc (Ω), we look for a distributional solution u W 1,r (Ω) to the equation in W 1,r (Ω), r = r 1 r, (27) r u = K(x)(u + κ) δ + f(x) in Ω. First, we state a new regularity result for any positive weak solution to equation (27) ( J. Giacomoni, I. Schindler, and P. Takáč (212) ): Theorem 1. Let u be a positive weak solution to (27) with κ = and K(x) const d(x) ω for x Ω, where ω > is a constant. (i) If < δ + ω < 1 then u C 1,α (Ω) for some < α < 1. G+S+T (27) (ii) If 1 δ + ω < 2 1 δ r then u C,α (Ω) for some < α < 1. G+S+T (212)

39 Proof. Hardy s inequality for W 1,r (Ω) and equivalent Campanato norms in Hölder spaces. ((27)) r u = K(x)(u + κ) δ + f(x) in W 1, r r 1(Ω). Theorem 2. Let 1 < r < and κ < 1, and let K : Ω R + be a function in L 1 loc (Ω) with the following additional integrability property: (I) There is a function ϕ W 1,r (Ω) such that ϕ > a.e. in Ω and (28) K(x)(ϕ + κ) 1 δ dx < if δ 1, Ω (29) Ω K(x)[ ln(ϕ + κ)] + dx < if δ = 1. Then eq. (27) possesses a unique positive weak solution u κ W 1,r (Ω). Moreover, < κ κ < 1 = u κ u κ a.e. in Ω.

40 The regularity results combined with the strong maximum principle (Tolksdorf (1983), Vázquez (1984)) imply that the pair (ũ, ṽ) belongs to X +. Alternatively, the strict subhomogeneity may be used. Let us to consider the mapping T 2. Notice that for any (u, v) X, T 2 (u, v) = (J 1 J 2 (u), J 2 J 1 (v)).

41 Theorem 3. Let a 1, b 1, a 2, b 2 > satisfy (p 1 + a 1 )(q 1 + a 2 ) > b 1 b 2. (Subhomogeneity) Let us abbreviate (3) (31) and assume that p(q 1 + a γ = 2 ) qb 1, (p 1 + a 1 )(q 1 + a 2 ) b 1 b 2 q(p 1 + a θ = 1 ) pb 2, (p 1 + a 1 )(q 1 + a 2 ) b 1 b 2 (32) 1 1 p < γ < 1 and 1 1 q < θ < 1. Then system (24) possesses a unique positive weak solution pair (u, v) in W 1,p (Ω) W1,q (Ω) that satisfies the following inequalities with a constant C > : (Schauder s Fixed-point Theorem) (33) C 1 φ γ 1,p u C φγ 1,p and C 1 φ θ 1,q v C φθ 1,q hold in Ω.

42 The normalized positive eigenfunction φ 1,p for the principal eigenvalue λ 1,p of p : (34) p φ 1,p = λ 1,p φ 1,p p 2 φ 1,p in Ω; φ 1,p = on Ω. φ 1,p W 1,p (Ω) is normalized by φ 1,p > a.e. in Ω and Ω (φ 1,p) p dx = 1. φ 1,p behaves like the distance function d(x) def = dist(x, Ω) for x Ω near the boundary Ω, by the strong maximum principle (Vázquez (1984)). For γ > we have (φ 1,p ) γ W 1,p (Ω) γ > 1 p = 1 1 p. Moreover, with c = (1 γ)(p 1), (35) p (φ γ 1,p ) { = γ p 1 λ 1,p φ γ(p 1) } 1,p + c φ1,p c 1 φ 1,p p = γ p 1 φ c 1 1,p { λ1,p φ p 1,p + c φ 1,p p},

43 For < γ < 1, p (φ γ 1,p ) φ (1 γ)(p 1) 1 1,p in Ω. Definition. A pair of positive functions (u, v) W 1,p (Ω) W1,q (Ω) is a pair of subsolutions to system (24) (i.e., a subsolution pair) if and only if the following inequalities hold in the sense of distributions (i.e., Radon measures) in W 1, p p 1(Ω) and W 1, q q 1(Ω), respectively: (P) p u 1 u a 1v b 1 q v 1 v a 2u b 2 in Ω; u Ω =, u > in Ω, in Ω; v Ω =, v > in Ω. Reversed inequalities: a pair of positive functions (u, v) W 1,p (Ω) W1,q (Ω) is a pair of supersolutions to system (24) (i.e., a supersolution pair) if and only... (P).

44 Remark. It follows from the weak comparison principle ineq-s (P) = u J 1 (v) and v J 2 (u) a.e. in Ω, ineq-s (P) = u J 1 (v) and v J 2 (u) a.e. in Ω. We have introduced the notions of a pair of sub- and supersolutions to system (24) based on the inequalities u J 1 J 2 (u) and u J 1 J 2 (u), respectively, for the mapping J 1 J 2, while ignoring the reversed inequalities v J 2 J 1 (v) and v J 2 J 1 (v) for J 2 J 1. Existence of a pair of sub- and supersolutions, invariance of the conical shells Four (4) alternatives are of interest. We consider the following one: Alternative 1. We search for positive solutions (u, v) W 1,p (Ω) W1,q (Ω) to problem (24) by making the Ansatz that the function u (v, respectively)

45 should behave like a fixed (positive) power of the distance function d(x) for x Ω near Ω, u φ γ 1,p and v φθ 1,q near Ω, for some constants γ, θ (,1). Both φ 1,p and φ 1,q belong to C 1,β (Ω) for some < β < 1 and, by the strong maximum principle (Vázquez (1984)), φ 1,p d(x) and φ 1,q d(x) near Ω. In our search for a pair of sub- and supersolutions to system ( (24) we choose the following special forms: m1 φ γ 1,p, M 2φ θ ) 1,q for a pair of subsolutions and ( M1 φ γ 1,p, m 2φ θ 1,q ) for a pair of supersolutions to system (24), where < m 1 M 1 and < m 2 M 2 are additional constants to be determined together with γ, θ (,1). Of course, we need to choose these constants in such a way that the conical shell (36) C def = { (u, v) C (Ω) C (Ω) : m 1 φ γ 1,p u M 1φ γ 1,p and m 2φ θ 1,q v M 2φ θ 1,q = [ m 1 φ γ 1,p, M 1φ γ 1,p] [ m2 φ θ 1,q, M 2φ θ 1,q ] }

46 be invariant under the operator T : (u, v) T(u, v) def = (J (37) 1 (v), J 2 (u)) : C W 1,p (Ω) W1,q (Ω) defined before Theorem 1 by eqs. (25) and (26), i.e., we need T(C) C. A careful reader will find out that also the weaker condition T 2 (C) C would suffice; this, however, is more difficult to verify directly without knowing that T (C) C. Hence, we wish to fulfill the following four inequalities: (38) T 1 (M 2 φ θ 1,q ) m 1φ γ 1,p and T 2(m 1 φ γ 1,p ) M 2φ θ 1,q, (39) T 2 (M 1 φ γ 1,p ) m 2φ θ 1,q and T 1(m 2 φ θ 1,q ) M 1φ γ 1,p. In ( order to verify them, it suffices to show that m1 φ γ 1,p, M 2φ θ ) 1,q is a pair of subsolutions and ( M1 φ γ 1,p, m 2φ θ ) 1,q is a pair of supersolutions to system (24), provided all constants γ, θ (,1), < m 1 M 1 <, and < m 2 M 2 < are chosen

47 properly. Thus, all inequalities (38) and (39) will follow from the following four inequalities, respectively: (4) (41) (42) (43) p (m 1 φ γ 1,p ) 1 (m 1 φ γ 1,p )a 1(M 2 φ θ 1,q )b 1 q (M 2 φ θ 1,q ) 1 (M 2 φ θ 1,q )a 2(m 1 φ γ 1,p )b 2 q (m 2 φ θ 1,q ) 1 (m 2 φ θ 1,q )a 2(M 1 φ γ 1,p )b 2 p (M 1 φ γ 1,p ) 1 (M 1 φ γ 1,p )a 1(m 2 φ θ 1,q )b 1 in Ω, in Ω, in Ω, in Ω, supplemented by the Dirichlet boundary conditions on Ω and the positivity in Ω, (44) φ 1,p Ω = φ 1,q Ω = φ 1,p >, φ 1,q > in Ω. and For < γ < 1, p (φ γ 1,p ) φ (1 γ)(p 1) 1 1,p in Ω.