Nonzero positive solutions of elliptic systems with gradient dependence and functional BCs

We discuss, by topological methods, the solvability of systems of second-order elliptic differential equations subject to functional boundary conditions under the presence of gradient terms in the nonlinearities. We prove the existence of non-negative solutions and provide a non-existence result. We present some examples to illustrate the applicability of the existence and non-existence results.


Introduction
In this paper we study the solvability of a system of second-order elliptic differential equations subject to functional boundary conditions (BCs for short). Namely, we investigate parametric systems of the type where m ≥ 1 is a fixed natural number, O ⊆ R n is a bounded and connected open set of class C 1,α for some α ∈ (0, 1), and λ k , η k , k = 1, . . . , m, are non-negative real parameters. Moreover L 1 , . . . , L m are uniformly elliptic, second-order linear partial differential operators (PDOs) in divergence form on O. That is, for k = 1, . . . , m, i,j (x) i,j is symmetric for every x ∈ O; • L k is uniformly elliptic in O, i.e., there exists Λ k > 0 such that i,j (x)ξ i ξ j ≤ Λ k ξ 2 for any x ∈ O and ξ ∈ R n \ {0} Date: December 6, 2019. 2010 Mathematics Subject Classification. Primary 35J47, secondary 35B09, 35J57, 35J60, 47H10. Key words and phrases. Positive solution, elliptic system, gradient terms, functional boundary condition, cone, fixed point index. 1 where ξ stands for the Euclidean norm of ξ ∈ R n ; • for every non-negative function ϕ ∈ C ∞ 0 (O, R) one has Furthermore, for every fixed k = 1, . . . , m we also assume that • f k is a real-valued continuous function defined on O × R m × R nm ; • h k is a real-valued continuous functional defined on the space C 1 (O, R m ); • ζ k ∈ C 1,α (O, R) and ζ k ≥ 0 on O.
The system (1.1) is quite general, and includes, for example, as a particular case a Dirichlet boundary value problem for elliptic systems with gradient dependence of the form Systems of nonlinear PDEs of this kind are widely studied in view of applications: in fact, the nonlinearities in (1.2) may depend also on the gradient of the solution, and thus represent convection terms. These problems, in general, are not easily dealt with by means of variational methods. Different approaches in the study of PDEs with gradient terms have been proposed: for example sub-and super-solutions, topological degree theory, mountain pass techniques. We mention, for instance, the pioneering works of Amann and Crandall [3], Brézis and Turner [4], Mawhin and Schmitt [22,23], Pokhozhaev [25] and the more recent contributions [1,7,9,10,13,26,27,29]. See also the very recent survey [8] and references therein.
In this paper we adopt a topological approach, based on the classical notion of fixed point index (see e.g. [16]) for the existence result, Theorem 3.3 below, whereas we prove a nonexistence result via an elementary argument. In some sense we follow a path established by Amman [2,3] and successfully used by many authors in different contexts. We point out that our approach applies not only to Dirichlet BCs but permits to consider (possibly nonlinear) functional BCs, including the special cases of linear (multi-point or integral ) BCs of the form Ωα Ωβ here, in (1.3),α ijk ,β ijkl are non-negative coefficients and ω i , τ i ∈ O while, in (1.4),α jk ,β jkl are non-negative continuous functions on O. In particular we observe that nonlinear, nonlocal BCs have seen recently attention in the framework of elliptic equations: we refer the reader to the papers [5,6,14,15,17,18,24] and references therein. We wish to point out that an advantage of our setting, with respect to the theory developed in [5,6,14,15,17,24], is the possibility to allow also gradient dependence within the functionals occurring in the BCs. This follows the approach used recently in [19,20] within the setting of ODEs.
Note that functional BCs that involve gradient terms may occur in applications. For example, consider a particular case of (1.1) for m = 1 and n = 2, namely where B is the Euclidean ball in R 2 centered at 0 with radius 1, · is the Euclidian norm and η i are non-negative coefficients. The BVP (1.5) can be used as a model for the steady states of the temperature of a heated disk of radius 1, where a controller located in the border of the disk adds or removes heat according to the value of the temperature and to its variation, both registered by a sensor located in the center of the disk. In the context of ODEs, a good reference for this kind of thermostat problems is the recent paper [28]. As already pointed out, a peculiarity of system (1.1) is the dependence on the gradient of the solutions, both in the nonlinearity and in the functionals occurring in the BCs, and this represents the main technical difficulty that we have to deal with in this paper. For this purpose, we have to perform a preliminary study of the Green's function of the partial differential operators which occur in (1.1). In Section 2 we collect some properties and estimates on Green's function, which are probably known to the experts in the field, nevertheless we include them for the sake of completeness. Roughly speaking, these estimates yield the a priori bounds needed to compute the fixed point index in suitable cones of non-negative functions.
Section 3 contains our main results, while the final Section 4 includes some examples illustrating our results. In particular, we fix m = 2 and n = 3, and, taking into account the parameters λ 1 , λ 2 , η 1 , η 2 , we provide existence and non-existence results in some concrete situations.

Preliminaries on divergence-form elliptic operators
In this Section we present, mostly without proof, several results concerning divergenceform operators which shall play a central rôle in the forthcoming sections. We refer the reader to, e.g., [11,12] for a detailed treatment of this topic. 3 To being with, let O ⊆ R n be a fixed open set and let L be a second-order linear PDO on O of the following divergence form: . . , b n ) and c = (c 1 , . . . , c n )). Throughout the sequel, we shall suppose that the following "structural assumptions" on O and L are satisfied: (H0) O is bounded, connected and of class C 1,α for some α ∈ (0, 1); (H1) the coefficient functions of L are Hölder-continuous of exponent α up to ∂O, i.e., for every i, j ∈ {1, . . . , n}; for every x ∈ O and every i, j ∈ {1, . . . , n}; (H3) L is uniformly elliptic in O, i.e., there exists Λ > 0 such that for any x ∈ O and any ξ ∈ R n ; It should be noticed that, since the coefficient functions of L are assumed to be just Höldercontinuous on O, it is not possible to compute Lu in a point-wise sense (even if u is smooth on O); for this reason, the following definition is plainly justified.
Definition 2.1. Let the assumptions (H0)-to-(H4) be in force, and let f ∈ L 2 (O). We say that a function u : O → R is a solution of the equation Given g ∈ W 1,2 (O), we say that u is a solution of the Poisson problem if u is a solution of (2.2) and, furthermore, u − g ∈ W 1,2 0 (O). Now, as a consequence of the "sign assumption" (H4) it is possible to prove that a suitable form of the Weak Maximum Principle holds for L (see, e.g., [12,Theorem 8.1]); from this, one can straightforwardly deduce Lemma 2.2 below (see [12,Corollary 8.2]), ensuring that the Poisson problem (2.3) possesses at most one solution. 2.1. The Poisson problem for L. A first group of results we aim to present is about existence and regularity of solutions for the Poisson problem (2.3) for L. In order to do this, we first introduce the following Banach spaces: Given f ∈ C 1 (O, R), it will be also convenient to define, with abuse of notation,  Throughout the sequel, we indicate by u f, g the unique solution in W 1,2 (O) of (2.3) (for fixed f ∈ L 2 (O) and g ∈ W 1,2 (O)), whose existence is guaranteed by Theorem 2.3. In the particular case when g ≡ 0, we simply write u f instead of u f, 0 . Remark 2.5. Let f 1 , f 2 ∈ L 2 (O) and, for i = 1, 2, let u i = u f i ∈ W 1,2 0 (O) be the unique solution of (2. 3) with f = f i (and g ≡ 0). Since, obviously, it holds that we conclude that the unique solution of (2. 3) with f = f 1 + f 2 and g ≡ 0 is u f 1 + u f 2 .
Since we aim to apply suitable fixed-point techniques to operators acting on spaces of C 1functions, we are interested in solving (2.3) for continuous f and regular g. In this context, the unique solution u f, g of (2.3) turns out to be much more regular that W 1,2 ; in fact, we have the following crucial result (for a proof, see [12,Thm.s 8.16,8.33 and 8.34]).
(ii) There exists a constant C > 0, only depending on n, Λ and O, such that Now, in view of Theorem 2.6-(i), we can define a linear operator as follows is the unique solution of (2.3) with g ≡ 0. We shall call G L the Green operator for L. By exploiting assertions (ii)-(iii) of Theorem 2.6, it is possible to deduce some continuous-compactness properties of G L which shall play a central rôle in the next sections; to be more precise, we have the following proposition.
Proposition 2.7. Let the assumptions (H0)-to-(H4) be in force, and let G L be the operator defined in (2.9). Then the following facts hold: Proof. (i) On account of Theorem 2.6-(ii), for every f ∈ C(O, R) one has (ii) Let {f j } j be a bounded sequence in C(O, R). On account of (2.10), we see that the ; as a consequence, a standard application of Arzel-Ascoli's Theorem implies the existence of u 0 , . . . , where {f j k } k is a suitable sub-sequence of {f j } j . By combining (a) and (b), we deduce that u 0 ∈ C 1 (O, R) and that ∇u 0 = (u 1 , . . . , u n ); moreover, one has

2.2.
Green's function for L. Now we have established Proposition 2.7, we turn to present a second group of results: this is about the existence of a Green's function for L allowing to obtain an integral representation formula for G L .
To begin with, we demonstrate the following key theorem.
Theorem 2.8. Let the assumptions (H0)-to-(H4) be in force, and let L be as in (2.1). There exists a function g L : Furthermore, g L enjoys the following properties: x ∈ O and every 1 ≤ p < n/(n − 1); (III) g L (y; ·) ∈ W 1,p 0 (O) for a.e. y ∈ O and every 1 ≤ p < n/(n − 1); (IV) there exists a constant c 1 > 0 such that, for a.e. x, y ∈ O, one has Finally, g L is unique in the following sense: ifg Throughout the sequel, we shall refer to the function g L in Theorem 2.8 as the Green's function for the operator G L (and related to the open set O).
Proof. We begin by proving the existence part of the theorem. In order to do this, we make pivotal use of several results established in the very recent paper [21].
Moreover, by [21, Theorem 6.10] we also have that where c 0 > 0 is a suitable constant. In view of these facts, to complete the demonstration we are left to prove assertion (iii) and the point-wise estimates in (2.13).
To this end, let us introduce the so-called (formal) adjoint L T of L: this is the linear differential operator defined on O in the following way (2.14) Clearly, L T takes the same divergence-form of L in (2.1) (with b and c interchanged); furthermore, due to the "symmetry" in assumption (H4), it is readily seen that L T satisfies the "structural assumptions" (H1)-to-(H4).
As a consequence, all the results established so far do apply to L T . In particular, for every (iv) for every fixed g ∈ C(O, R) one has On the other hand, since [21,Proposition 6.13] shows that from (iii) we infer that g L (y; ·) = G(·; y) ∈ W 1,p (O) for almost every y ∈ O and every exponent p ∈ [1, n/(n − 1)). This is exactly assertion (III). Finally, we prove the point-wise estimates in assertion (IV). First of all, since L satisfies assumptions (H1)-to-(H4), we are entitled to apply [21,Theorem 8.1], ensuring that where c ′ 1 > 0 is a suitable constant. Moreover, since also L T satisfies assumptions (H1)-to-(H4), another application of [21,Theorem 8.1] gives As for the uniqueness part of the theorem, let us suppose that there exists another functioñ Now, the space C ∞ 0 (O, R) being separable (with its usual LF-topology), there exists a count- We then define E := ∪ φ∈F E(φ). Since F is countable and E(φ) has zero-Lebesgue measure for every φ, we see that E has measure zero; moreover, for every This proves that, for every x ∈ O \ E, the distribution g L (·; x) −g(·; x) vanishes on F ; the latter being dense, we then conclude that This ends the proof.
Remark 2.9. The approach adopted for the proof of Theorem 2.8 shows the reason why we have assumed that d − div(b) ≥ 0 and d − div(c) ≥ 0 in the sense of distributions. In fact, under this assumption, all the mentioned results in [21] hold both for L and for its transpose L T ; in particular, this allows us to obtain point-wise estimates both for and ∇ y g L (y; x).
Remark 2.10. It is contained in the proof of Theorem 2.8 the following fact: if L is of the form (2.1) and if b ≡ c on O, then the Green's function for G L is symmetric, that is, In fact, if b ≡ c on O, then the adjoint operator L T coincides with L (see (2.14)); thus, following the notation in the proof of Theorem 2.8, we have Remark 2.11. By carefully scrutinizing the proofs of the existence results for g L contained in [21,Proposition 5.3], one can recognize that the following properties hold: (a) for a.e. x ∈ O and every ǫ > 0, we have g L (·; x) ∈ W 1,2 (O \ B(x, ǫ)); where L T is as in (2.14).
We now use the point-wise estimates in (2.12)-(2.13) to prove the following lemma.
for a.e. x ∈ O. (2.20) Here, ω n is the Lebesgue measure of the unit ball B(0, 1) ⊆ R n . Proof. We begin by proving (2.19). To this end we first notice that, if x ∈ O is arbitrary, then O ⊆ B(x, ρ); as a consequence, by crucially exploiting estimate (2.12) we get which is exactly the desired (2.19). As for the proof of (2.20), we argue essentially in the same way: by crucially exploiting the estimate (2.13) we get where ρ := diam(O) and1 denotes the constant function equal to 1 on O. As a consequence, We conclude this part of the Section by deducing from (2.11) an integral representation for the x i -derivatives of G L (f ). To this end we first observe that, if f ∈ C(O, R), Lemma 2.12 ensures that the following "potential-type" functions are well-defined: In fact, by estimate (2.20) in Lemma 2.12 we have (for i = 1, . . . , n) Moreover, from the above computation we also infer that (again for i = 1, . . . , n) We are then ready to prove the following Proposition.
Proposition 2.14. Let the assumptions (H0)-to-(H4) be in force, and let f ∈ C(O, R). Moreover, let i ∈ {1, . . . , n} be fixed, and let P (i) f be as in (2.21). Then, we have Proof. We first notice, since G L (f ) ∈ C 1,α (O, R), the identity (2.22) follows if we show that the L ∞ -function P (i) f is the weak derivative (in L 1 (O)) of G L (f ). To prove this fact, we argue as follows: firstly, if φ ∈ C ∞ 0 (O, R), by the estimate (2.19) in Lemma 2.12 we get we are then entitled to apply Fubini's Theorem, obtaining On the other hand, since the estimate (2.20) in Lemma 2.12 implies that another application of Fubini's Theorem is legitimate, and we get Due to the arbitrariness of φ ∈ C ∞ 0 (O, R), we then conclude that P (i) f is the weak derivative of G L (f ) in L 1 (O), and the proof is complete. To begin with, we remind the following theorem (see, e.g., [12,Theorem 8.6]). as a consequence, by the very definition of G L we infer that This proves that λ := 1/σ > 0 lays in the (point-wise) spectrum of G L (thought of as an operator from C(O, R) into itself), and thus r(G L ) > 0.
(ii) First of all, since C 1 (O, R) is continuously embedded in C(O, R), we straightforwardly derive from Proposition 2.7-(ii) that G L is compact from C(O, R) into itself; moreover, if we 13 denote by V 0 the convex cone in C(O, R) defined as we know from Proposition 2.7-(iii) that G L (V 0 ) ⊆ V 0 . Since, obviously, V 0 − V 0 is dense in C(O, R) and since, by statement (i), the spectral radius r(G L ) of G L is strictly positive, we are entitled to apply Krein-Rutman's Theorem, ensuring that r(G L ) is an eigenvalue of G L with positive eigenvector: this means that there exists u 0 ∈ V 0 \ {0} such that . Gathering together all these facts, we conclude that u 0 ∈ C 1,α (O, R) \ {0} and that u ≥ 0 on O, as desired.

Existence and non-existence results
In this Section we study the solvability of the following system of second order elliptic differential equations subject to functional BCs To be more precise, we suppose that (I) O is bounded, connected and of class C 1,α for some α ∈ (0, 1); (II) for every fixed k = 1, . . . , m, the differential operator L k satisfies assumptions (H1)to-(H3) introduced in Section 2, that is, ( * ) L k takes the divergence form (2.1), i.e., for any x ∈ O and ξ ∈ R n \ {0}; ( * ) for every non-negative function Furthermore, for every fixed k = 1, . . . , m we also assume that (VI) λ k , η k are non-negative real parameters.
Throughout the sequel, if u 1 , . . . , u m are real-valued functions defined on O, we set If, in addition, Furthermore, since the function ζ k belongs to C 1,α (O, R) (see assumption (V)), there exists a unique solution γ k ∈ C 1,α (O, R) of the Dirichlet problem We then denote by G k the Green's operator G L k for L k defined in (2.9), and we indicate by g k the Green's function g L k for the operator G k defined through Theorem 2.8. We remind that, if f ∈ C(O, R) is arbitrary fixed, G k (f) is the unique solution in C 1,α (O, R) of the Poisson problem (3.2); moreover, we have the representation formulas holding true for a.e. x ∈ O and any i = 1, . . . , n (see Theorem 2.8 and Proposition 2.14). Finally, according to Proposition 2.17, we denote by r k = r(G k ) > 0 the spectral radius of the operator G k (thought of as an operator from C 1 (O, R) into itself) and we fix once and for all a function ϕ k ∈ C 1,α (O, R) \ {0} such that (setting µ k := 1/r k ) Now that we have properly introduced all the "mathematical objects" appearing in the problem (3.1), it is opportune to define what we mean by a solution of this problem.
To this end, we first fix some notation. For every index k ∈ {1, . . . , m}, we denote by F k the so-called superposition (Nemytskii) operator associated with f k , that is, Moreover, we consider the operators T , Γ : We can now give the definition of solution of the problem (3.1).
Definition 3.1. We say that a function u ∈ C 1 (O, R m ) is a weak solution of the system (3.1) if u is a fixed point of the operator T + Γ, that is, If, in addition, the components of u are non-negative and u j ≡ 0 for some j, we say that u is a nonzero positive solution of the system (3.1).
For our existence result, we make use of the following proposition that states the main properties of the classical fixed point index, for more details see [2,16]. In what follows the closure and the boundary of subsets of a coneP are understood to be relative toP . We can now state a result regarding the existence of positive solutions for the system (3.1). In the sequel, we will consider on the space R s (where s will be either m, n or mn) the following maximum norm where R ρ = {v ∈ R n : |v| ≤ ρ} (for t > 0); we also introduce, with abuse of notation, the sets and the following inequalities are satisfied: (c) 3 for any l = 1, . . . , n we have λ k M k G k,l + η k H k ∂ x l γ k ∞ ≤ ρ k , where (3.11) G k,l := sup x∈O O |∂ x l g k (y; x)| dy (see Lemma 2.12).
Then the system (3.1) has a non-zero positive weak solution u ∈ C 1 (O, R m ) such that Proof. For the sake of readability, we split the proof into different steps.
Step II: We now prove that A : P (̺) → P is compact. To this end, let {u j } j∈N be a bounded sequence in P (̺), and let k ∈ {1, . . . , m} be fixed. Since h k is non-negative and bounded on P (̺) (see assumption (a) 2 ), the sequence {h k [u j ]} j is bounded in (0, ∞); as a consequence, there exists θ 0 ∈ [0, ∞) such that (up to a sub-sequence) On the other hand, since {u j } j ⊆ P (̺) and since f k is continuous on O × I(̺) × R(̺) (see assumption (a) 1 ), we have (using the notation in (3.10)) As a consequence, since the operator G k is compact (as an operator from C(O, R) into C 1 (O, R), see Proposition 2.7-(ii)), it is possible to find a function w k ∈ C 1 (O, R) such that (again by possibly passing to a sub-sequence) Gathering together (3.15), (3.16) and (3.6), we infer that (up to a suitable sub-sequence) lim j→∞ A(u j ) = λ k w k + η k γ k θ 0 k=1,...,m =: u in C 1 (O, R m ). 18 Finally, since {A(u j )} j ⊆ P (by Step I) and since P is closed, we conclude that u ∈ P ; this proves the compactness of A (as an operator from P (̺) to P ).
To proceed further, we consider the set P 0 ⊆ C 1 (O, R m ) defined as follows: where I 0 and B 0 are as in assumption (b). Now, if the operator A = T + Γ has a fixed point u 0 ∈ ∂P 0 ∪ ∂P (̺) (where the boundaries are both relative to P ), then u 0 is a solution of problem (3.1) satisfying (3.12), and the theorem is proved. If, instead, A is fixed-point free on ∂P 0 ∪ ∂P (̺), the fixed-point indexes i P (A, int(P 0 ) ∩ P ) and i P (A, int(P (̺)) ∩ P ) are well-defined. Assuming this last possibility, we consider the following steps.
Step III: In this step we prove the following fact: Since u ∈ ∂P (̺), there exists an index k ∈ {1, . . . , m} such that either We then distinguish these two cases.
• ∇u k ∞ = ρ k . In this case, by the very definition of · ∞ , there exists l ∈ {1, . . . , n} such that ∂ x l u k ∞ = ρ k . Moreover, by Proposition 2.14 we have for a.e. x ∈ O. By means of this representation formula, we then obtain As a consequence, by taking the supremum for x ∈ O in (3.20) (and by reminding that ∂ x l u k ∞ = ρ k ), from assumption (c) 3 we infer that which is clearly a contradiction (as σ > 1).
This completes the demonstration of (3.18).
Step IV: In this last step we prove the following fact: According to Proposition 3.2-(i), to prove (3.21) it suffices to show that there exists a suitable function e ∈ P \ {0} satisfying the property (3.22) A(u) + σe = u for every u ∈ ∂P 0 and every σ > 0.
Since u ∈ ∂P 0 ⊆ P 0 ⊆ P (̺) (by definition of P 0 , see assumption (b)), we know from Step I that A(u) ∈ P ; as a consequence, if k 0 is as in assumption (b), we have Furthermore, by exploiting once again assumption (b) we get Gathering together all these facts, for every x ∈ O we have (3.23) and Proposition 2.7-(iii) By iterating the above argument, for every x ∈ O we get but this is contradiction with the boundedness of u k 0 ∈ C 1 (O, R) (as ϕ k 0 ≡ 0).
We are now ready to conclude the proof of the theorem: in fact, by combining (3.17), (3.21) and Proposition 3.2-(iii), we infer the existence of a fixed point u 0 ∈ int(P (̺)) ∩ P \ P 0 of A = T + Γ; thus, u 0 is a solution of (3.1) satisfying (3.12).
As a consequence, the operator Γ maps On the other hand, since the operators G 1 , . . . , G m map C(O, R) into C 1,α (O, R), we also have that Gathering together all these facts, we conclude that any weak solution of (3.1) (i.e., any fixed point of A = T + Γ in C 1 (O, R m )) is actually of class C 1,α on O.
An elementary argument yields the following non-existence result.
Theorem 3.5. Let the assumptions (I)-to-(IV) be in force. Moreover, let us suppose that there exists a finite sequence ̺ = {ρ k } m k=1 ⊆ (0, ∞) such that, for every k = 1, . . . , m, the following conditions hold: (a) f k is continuous on O × I(̺) × R(̺), and there exist τ k ∈ (0, +∞) such that (c) the following inequality holds: Then the system (3.1) has at most the zero solution in P (̺).
Example 4.2. On Euclidean space R 3 , let us consider the following BVP (4.10) where B is the Euclidean ball with centre 0 and radius 1 and we adopt the same notation of Example 4.1.
Furthermore, it is straightforward to check that all the structural assumptions (I)-to-(VI) listed at the beginning of Section 3 are satisfied (for every α ∈ (0, 1)). We now aim to show that, in this case, assumptions (a)-to-(c) in statement of Theorem 3.5 are fulfilled.
Assumption (b). First of all, it is very easy to check that both h 1 and h 2 are continuous and non-negative when restricted to the cone P (̺) ⊆ C 1 (B, R); moreover, since the condition u = (u 1 , u 2 ) ∈ P (̺) implies that 0 ≤ u 1 , u 2 ≤ 1, we get (4.14) and this proves that h 1 fulfills assumption (b) (with ξ 1 = (4π)/3). Finally, by exploiting the very definition of · ∞ , we have and thus also h 2 satisfies assumption (b) (with ξ 2 = 1).