Coupled elliptic systems depending on the gradient with nonlocal BCs in exterior domains

We study the existence and multiplicity of positive radial solutions for a coupled elliptic system in exterior domains where the nonlinearities depend on the gradients and the boundary conditions are nonlocal. We use a new cone to establish the existence of solutions by means of fixed point index theory.

Our system (1.1) is quite general: (1) the nonlinearities f i depend on the functions u and v and their gradients; no monotonicity hypotheses are supposed. (2) the boundary conditions are nonlocal and represent feedback mechanisms. They have been deeply studied for ordinary differential equations, for example, in [11,24,25,41,42]). To search solutions of the elliptic PDE w = g |x| f w, |∇w| with some boundary conditions, a topological approach is associating, by using standard transformations, an integral operator of the form

G(t, s)g r(s) f w(s), w (s) ds.
It is straightforward, in the local problems, to find the Green's function G by integration and by using the BCs. However, let us remark that, in the nonlocal problems, this is a long and technical calculation, often resulting in a sum of terms of different signs.
Here, as in [41], we treat the nonlocal problem as the perturbation of the simpler local problem. In such a way, we handle the positivity properties of the simpler Green's function of the local problem.
Often, the associated integral operator is studied in the cone of nonnegative functions in the space C 1 [0, 1] or in a weighed space of differential functions as in [3]. In our case and in particular when seeking for multiple solutions, it is suitable to work in a smaller cone: we will introduce a new cone in which we will use Harnack-type inequalities.
Moreover, since we are interested in positive solutions, the functionals α i and β i must satisfy some positivity conditions; we will not suppose this in the whole space, but we choose to include the requirement in the definition of the cone.
We show that, under suitable conditions on the nonlinear terms, the fixed point index is 0 on certain open bounded subsets of the cone and 1 on the others; the choice of these subsets allows us to have more freedom on the conditions of the growth of the nonlinearities. These conditions relate the upper and lower bounds of the nonlinearities f i on stripes and some constants, depending on the kernel of the integral operator and on the nonlocal BCs that are easily estimable as we show in an example.

The associate integral operator
Consider in R n , n ≥ 3, the equation Since we are interested in radial solutions w = w(r), r = |x|, following [6], we rewrite (2.1) as By using the transformation Consider in R n the system of boundary value problems Set u(t) = u(r(t)) and v(t) = v(r(t)). Thus with system (2.3) we associate the system of ODEs where d i = r 0 2-nd i and g i (t) = We study the existence of positive solutions of system (2.4) by means of the associated system of perturbed Hammerstein integral equations where γ i is the solution of the BVP w (t) = 0, w(0) = 1, c i w(1) + d i w (1) = 0, that is, and k i is the Green's function associated with the homogeneous problem in which In the following proposition, we resume the properties of the functions γ i , δ i and k i , which will be further useful.
For i = 1, 2 and fixed [a i , b i ] ⊂ (0, 1) such that a i + b i < 1, we consider the cones We will further assume that, for i = 1, 2, Note that the hypotheses on the nonlinearities f i are used in [3] and are satisfied, for example, if the functions f i are continuous and, with respect to the last two variables, decreasing or bounded.
We now prove that T leaves the cone K invariant and is completely continuous.

Theorem 2.2 The operator T maps K into K and is completely continuous.
Proof To prove that T leaves the cone K invariant, it suffices to prove that On the other hand, we have Now we prove that, for every (u 1 , u 2 ) ∈ K , We have Since For any t ∈ [0, 1], we can easily compute that and that k i (t,s) Then, for all t ∈ [0, 1], we have For t = 1 2 , we obtain Therefore we conclude that Taking the supremum on [0, 1], we obtain Since α i and β i are linear functionals, it follows that To prove the complete continuity of T, let us note that the continuity of f , k i , α i , and β i give the continuity of each T i and thus the continuity of T.
Let U be a bounded subset of K ; from (2.6) it follows that T(U) is bounded in K . Now we prove that T(U) is relatively compact in K . It is a standard argument based on the uniform continuity of the kernels k i on [0, 1] × [0, 1] and on the Ascoli-Arzelà theorem that T i (U) is relatively compact in C[0, 1]. Now let (u n , v n ) n∈N be a sequence in U. Then T i (u n , v n ) ⊂ K i . There exists (u n k , v n k ) k∈N such that (T 1 (u n k , v n k )) k∈N converges in C[0, 1]. Since T 2 (U) is relatively compact, there exists (u n kp , v n kp ) p∈N := (u n p , v n p ) p∈N ⊂ (u n k , v n k ) k∈N such that (T i (u n p , v n p )) p∈N → w i ∈ C[0, 1] for i = 1, 2. Since that is, ((T i (u n p , v n p )) ) p∈N is a Cauchy sequence in · ω for i = 1, 2. Then (T i (u n p , v n p )) p∈N is a Cauchy sequence in C 1 ω [0, 1], and so it converges to w i ∈ C 1 ω [0, 1]. The closedness of K implies that (w 1 , w 2 ) ∈ K , and therefore T(U) is relatively compact in K .
To use the fixed point index, we utilize the following sets in K for ρ 1 , ρ 2 > 0: Since w ω ≤ 4 w ∞ in K , we have w ≤ 4 w ∞ , and therefore K ρ 1 ,ρ 2 and V ρ 1 ,ρ 2 are open and bounded with respect to K . It is straightforward to verify that these sets satisfy the following properties: (P 2 ) (w 1 , w 2 ) ∈ ∂K ρ 1 ,ρ 2 if and only if (w 1 , w 2 ) ∈ K and for some i ∈ {1, 2} w i ∞ = ρ i and The following theorem follows from classical results about fixed point index (more details can be seen, for example, in [4,28]).
Then F has at least one fixed point x ∈ (Ω ∩ K) \ (Ω 1 ∩ K). Denoting by i K (F, U) the fixed point index of F in some U ⊂ X, we have The same result holds if i K (F, Ω ∩ K) = 0 and i K F, Ω 1 ∩ K = 1.

A system of elliptic PDE
We define the following sets: Then system (2.3) has at least one positive radial solution.
Proof Note that the choice of the numbers ρ i and s i ensures the compatibility of conditions (3.1) and (3.2). We want to show that i K (T, K ρ 1 ,ρ 2 ) = 1 and i K (T, V s 1 ,s 2 ) = 0, so that from Theorem 2.3 it follows that the completely continuous operator T has a fixed point in V s 1 ,s 2 \ K ρ 1 ,ρ 2 . Then system (2.3) admits a positive radial solution.
First, we claim that λ(u, v) = T(u, v) for all (u, v) ∈ ∂K ρ 1 ,ρ 2 and λ ≥ 1, which implies that the index of T is 1 on K ρ 1 ,ρ 2 . Suppose this is not true. Let λ ≥ 1, and let (u, v) ∈ ∂K ρ 1 ,ρ 2 be such that In view of (P 2 ), without loss in generality, let us suppose that u ∞ = ρ 1 . Then In a similar way, applying β 1 , we obtain Denoting we can write the previous conditions as Therefore we get that Taking the supremum on [0, 1] in the last inequality, it follows that which is a contradiction. Now we show that that the index of T is 0 on V s 1 ,s 2 . Consider l(t) = 1 for t ∈ [0, 1] and note that (l, l) ∈ K . Now we claim that (u, v) = T(u, v) + λ(l, l) for (u, v) ∈ ∂V s 1 ,s 2 and λ ≥ 0.
Applying the matrix (N 1 ) -1 to both sides of the last equality, we obtain Thus, as in the previous step, we have Then, for t ∈ [a 1 , b 1 ], we obtain Taking the minimum over [a 1 , b 1 ] gives By means of Theorem 3.1 and the fixed point index properties in Theorem 2.3 we can state results on the existence of multiple positive solutions for system (2.3). Here we enunciate a result on the existence of two positive solutions (see [32,33] for the conditions that ensure three or more positive results). we can consider in R 3 the system of BVPs 1 2 ]. By direct computation we obtain

Theorem 3.2 Suppose that there exist
Since in our example the mixed perturbed conditions state that c 1 = 2, d 1 = 4, c 2 = 3, and d 2 = 2, we easily compute C i and M i : Then the hypotheses of Theorem 3.2 are satisfied, and hence system (3.6) has at least two positive solutions.

Nonexistence results
We now show a nonexistence result for the system of elliptic equations (2.3) when the the functions f i have an enough "small" or "large" growth. For u > 0, taking the supremum for t ∈ [0, 1], we have u ∞ < u ∞ , a contradiction. Suppose that (4.2) holds and assume that there exists (u, v) ∈ K such that (u, v) = T(u, v) and (u, v) = (0, 0). Let, for example, u ∞ = 0; then σ := min t∈[a 1 ,b 1 ] u(t) > 0 since u ∈ K 1 . Thus, as in the proof of Theorem 3.1, we have, for t ∈ [a 1  For u > 0, taking the infimum for t ∈ [a 1 , b 1 ], we obtain σ > σ , a contradiction.