Eigenvalues of elliptic functional differential systems via a Birkhoff--Kellogg type theorem

Motivated by recent interest on Kirchhoff-type equations, in this short note we utilize a classical, yet very powerful, tool of nonlinear functional analysis in order to investigate the existence of positive eigenvalues of systems of elliptic functional differential equations. An example is presented to illustrate the theory.


Introduction
A well known result in nonlinear analysis is the Birkhoff-Kellogg invariant-direction Theorem [5]. In the case of an infinite-dimensional normed linear space V this theorem reads as follows. The invariant direction Theorem has been object of deep studies in the past, with applications and extensions in several directions, we refer the reader to [3,9,13,15,16,17,18,23,28,29,34,40] and references therein. In particular, we highlight that [15,28,34] provide interesting applications to the existence of eigenvalues and eigenfunctions of elliptic boundary value problems.
Here we make use of the following Birkhoff-Kellogg type result, which is set in cones. This is a special case of a result due to Krasnosel'skiȋ and Ladyženskiȋ [27], see also [28,Theorem 5.5]. Before stating the result, we recall that a cone C of a real Banach space (X, ) is a closed set with C + C ⊂ C, µC ⊂ C for all µ ≥ 0 and C ∩ (−C) = {0}, and we introduce the following notation: Theorem 1.2. Let (X, ) be a real Banach space, let T : C r → C be compact and suppose that inf Then there exist λ 0 ∈ (0, +∞) and x 0 ∈ ∂C r such that x 0 = λ 0 T x 0 .
By means of Theorem 1.2 we discuss the solvability, with respect to the parameter λ, of the following system of second order elliptic functional differential equations subject to functional boundary conditions (BCs) where Ω ⊂ R n is a bounded domain with a sufficiently smooth boundary, L i is a strongly uniformly elliptic operator, B i is a first order boundary operator, u = (u 1 , . . . , u n ), Du = (∇u 1 , . . . , ∇u n ), f i are continuous functions, ζ i are sufficiently regular functions, w i and h i are suitable compact functionals. The class of systems occurring in (1.1) is fairly general and allows us to deal with nonlocal problems of Kirchhoff-type. This is a very active area of research, a typical example of a Kirchhoff-type problem is which has been investigated by Ma in his survey [36]. An extension to systems of the BVP (1.2) has been considered by Figueiredo and Suárez [14], namely The approach employed in [14] is the sub-supersolution method. The system (1.3) has been studied also by the sub-supersolution method in [8,35,6,7], while variational methods were employed in [19,32,38,49]. Note that there has been also interest in Kirchhoff-type problems with gradient terms appearing within the nonlinearities, we mention the recent papers by Alves and Boudjeriou [1], Yan and co-authors [47], Chen [11] and references therein.
The framework of (1.1) allows us to deal with non-homogenous BCs of functional type. In the case of nonlocal elliptic equations, non-homogeneous BCs have been investigated by Wang and An [44], Morbach and Corrẽa [37] and by the author [24]. The formulation of the functionals occurring in (1.1) allows us to consider multi-point or integral BCs. There exists a wide literature on this topic, we refer the reader to the reviews [10,12,33,39,43,46] and the papers [20,21,25,26,41,42,45].
Here we discuss, under fairly general conditions, the existence of positive eigenvalues with corresponding non-negative eigenfunctions for the system (1.1) and illustrate how these results can be applied in the case of nonlocal elliptic systems. Our results are new and complement previous results of the author [24], by allowing the presence of gradient terms within the nonlinearities and the functionals. The results also complement the ones in [4], by considering more general nonlocal elliptic systems.

Eigenvalues and eigenfunctions
In what follows, for everyμ ∈ (0, 1) we denote by Cμ(Ω) the space of allμ-Hölder continuous functions g : Ω → R and, for every k ∈ N, we denote by C k+μ (Ω) the space of all functions g ∈ C k (Ω) such that all the partial derivatives of g of order k areμ-Hölder continuous in Ω (for more details see [2, Examples 1.13 and 1.14]).
(3) B i is a boundary operator given by where ν is an outward pointing and nowhere tangent vector field on ∂Ω of class C 1+μ (not necessarily a unit vector field), ∂u ∂ν is the directional derivative of u with respect to ν, b i : ∂Ω → R is of class C 1+μ and moreover one of the following conditions holds: It is known that, under the previous conditions (see [2], Section 4 of Chapter 1), a strong maximum principle holds, given g ∈ Cμ(Ω), the BVP admits a unique classical solution u ∈ C 2+μ (Ω) and, moreover, given ζ i ∈ C 2−δ i +μ (∂Ω) the BVP also admits a unique solution γ i ∈ C 2+μ (Ω).
In order to investigate the solvability of the system (1.1), we make use of the cone of non-negative functionsP = C(Ω, R + ). The solution operator associated to the BVP (2.1), K i : Cμ(Ω) → C 2+μ (Ω), defined as K i g = u is linear and continuous. It is also known (see [2], Section 4 of Chapter 1) that K i can be extended uniquely to a continuous, linear and compact operator (that we denote again by the same name) K i : C(Ω) → C 1 (Ω) that leaves the coneP invariant, that is K i (P ) ⊂P .
We utilize the space C 1 (Ω, R n ), endowed with the norm where z ∞ = max x∈Ω |z(x)|, and consider the cone P = C 1 (Ω, R n + ). We rewrite the elliptic system (1.1) as a fixed point problem, by considering the operators T, Γ : C 1 (Ω, R n ) → C 1 (Ω, R n ) given by where K i is the above mentioned extension of the solution operator associated to (2.1), γ i ∈ C 2+μ (Ω) is the unique solution of the BVP (2.2) and Definition 2.1. We say that λ is an eigenvalue of the system (1.1) if there exists u ∈ C 1 (Ω) with u 1 > 0 such that the pair (u, λ) satisfies the operator equation If the pair (u, λ) satisfies (2.4) we say that u is an eigenfunction of the system (1.1) corresponding to the eigenvalue λ. If, furthermore, the components of u are non-negative, we say that u is a non-negative eigenfunction of the system (1.1).
Proof. Due to the assumptions above, the operator T + Γ maps P ρ into P and is compact (by construction, the map F is continuous and bounded and Γ is a finite rank operator). Take u ∈ ∂P ρ , then for every x ∈Ω we have Taking the supremum for x ∈Ω in (2.9) we obtain (2.10) Note that the RHS of (2.10) does not depend on the particular u chosen. Therefore we have inf u∈∂Pρ T u + Γu ≥ φ i 0 ,ρ > 0, and the result follows by Theorem 1.2.
Remark 2.3. Note that we have chosen to use inequalities in (2.5)-(2.8); this is due that, in applications, it is often easier and somewhat more efficient to use estimates on the nonlienarieties involved. Furthermore note that, in our reasoning, what really matters is that some positivity occurs in one component of the system, either in the nonlinearity f i or in the functional h i .
The following Corollary provides a sufficient condition for the existence of an unbounded set of eigenfunctions for the system (1.1). Corollary 2.4. In addition to the hypotheses of Theorem 2.2, assume that ρ can be chosen arbitrarily in (0, +∞). Then for every ρ there exists a non-negative eigenfunction u ρ ∈ ∂P ρ of the system (1.1) to which corresponds a λ ρ ∈ (0, +∞).
We now show the applicability of the above results in the context of systems of nonlocal elliptic equations with functional BCs.
Thus we can apply Corollary 2.4, obtaining a uncountably many pairs (u ρ , λ ρ ) of non-negative eigenfunctions and positive eigenvalues for the system (2.11).