RADIAL SOLUTIONS FOR A NONLOCAL BOUNDARY VALUE PROBLEM

of (1.1). This may be seen as the stationary problem corresponding to a class of nonlocal evolution (parabolic) boundary value problems related to relevant phenomena in engineering and physics. The literature dealing with such problems has been growing in the last decade. The reader may find some hints on the motivation for the study of this mathematical model, for example, in the paper by Bebernes and Lacey [1]. For more recent developments, see [2] and the references therein.


Introduction
Let us consider the following nonlocal BVP in a ball U = B(0, R) of R n : (1) where f and g are continuous functions. For simplicity we shall take R = 1. We want to study the existence of positive radial solutions of (1)- (2). This may be seen as the stationary problem corresponding to a class of nonlocal evolution (parabolic) boundary value problems related to relevant phenomena in Engineering and Physics. The literature dealing with such problems has been growing in the last decade.
The reader may find some hints on the motivation for the study of this mathematical model e.g. in the paper by Bebernes and Lacey [1]. For more recent developments, see [2] and the references therein.
Here we are considering a nonlocal term inserted in the right-hand side of the equation. Note, however, that it is also of interest to study boundary value problems where the nonlocal expression appears in a boundary condition. We refer the reader to the recent paper by Yang [14] and its references.
When dealing with a nonlinear term with rather general dependence on the nonlocal functional as in (1) new difficulties arise with respect to the treatment of standard boundary value problems. Differences of behaviour which are met in general elliptic and parabolic problems are already present in simple models as those we shall analyse in this paper. For instance, the use of the powerful lower and upper solutions method (good accounts of which can be consulted in the monographs of Pao [11] and De Coster and Habets [4]) is limited by the absence of general maximum principles. Even for linear problems with nonlocal terms the issue of positivity is far from trivial and may require a detailed study via the analysis of the Green's operator, as in Freitas and Sweers [7].
The purpose of this paper is twofold. First, we want to improve a quite recent result of P. Fijalkowski and B. Przeradski [6]: these authors have obtained existence of positive radial solutions of (1)-(2) by using a Krasnoselski's fixed point theorem in cones; the main assumption is that f may grow at most like Au + B, the bound on A being computed by means of a Green's function. By using a similar theoretical background, together with the consideration of the eigenvalues of the underlying linear problem, we show that an improvement of that bound is possible. This is done in Theorem 3.2. Second, while remaining in the same simple general setting, we shall handle (1)-(2) from the point of view of the upper and lower solution method. We establish a nonlocal maximum principle (Lemma 4.6) and we use it as a device to obtain a monotone approximation scheme for the radial solutions of (1)- (2) in presence of lower and upper solutions (Theorem 4.10). We follow an idea used by D. Jiang, W. Gao, A. Wan [10] in studying a fourth order periodic problem.
Note that we could use similar methods to consider the case where U = B(0, 1)\B(0, ρ), with 0 < ρ < 1. Similar results could then be reached. We remark also that for special classes of functions f and g different approaches are needed. For instance, in [8] variational methods have been used to study existence and multiplicity when f (u, v) = g(u)/v p (p > 0) and g behaves as an exponencial function.
The authors wish to thank the referee for carefully reading the manuscript and hints to improve its final form.

Some auxiliary results
It is well known that the existence of a solution for some boundary value problems is equivalent to the existence of a fixed point of a certain operator. For our purpose we need to consider a second order ordinary differential equation of the form with boundary conditions where f is a continuous function in [0, 1] × R and p ∈ C[0, 1] is positive and increasing in ]0, 1].
If p > 0 in [0, 1], it is well known that the problem is fully regular, having a standard reduction to a fixed point problem: where T is the linear operator that takes v ∈ C[0, 1] into the unique solution u of In addition we can write explicitly where G(t, s) is the Green's function associated to the problem. The Green's function is continuous in [0, 1] × [0, 1] , so T is a completely continuous linear operator in C[0, 1]. We are interested in the case where p(t) > 0 in ]0, 1] only. Under certain assumptions we still have a continuous Green's function for the linear problem (6). The reader can find a more general approach in [9], but for completeness we include here a simple version which is sufficient for our purpose: with boundary conditions (5). Integrating both sides we get Integrating again, we obtain where It is trivial to see that T v(1) = 0 and if we differentiate the expression for T v(t) we obtain and thus Remark 2.2. The continuous functions p(t) = t n with n > 0, satisfy the assumptions of the lemma.
The following fixed point theorem of M. Krasnoselskii will be used in the next section (see Deimling [3]). Theorem 2.3. Let P be a cone in a Banach space and S : P → P a completely continuous operator. If there exist positive constants r < R such that (compression case) Sx ≥ x , for all x ∈ P such that x = r, Sx ≤ x , for all x ∈ P such that x = R, then S has a fixed point x in P such that r < x < R.
3 Nonlinearities with linear growth in u: a positive solution Let f : R + × R → R + and g : R + → R be continuous functions. The radial solutions v of the problem (1)-(2) solve the ordinary differential equation which is equivalent to − r n−1 v (r) = r n−1 f v(r), ω n 1 0 s n−1 g(v(s)) ds , with boundary conditions lim where ω n is the measure of the unit sphere in R n . The homogeneous equation −v − (n − 1)v /r = 0, with the boundary conditions (9), has only the trivial solution, and therefore there exists a Green's function associated to the linear problem. In fact, the Green's function may be written according to lemma 2.1 (see also [6]) (ii) and for n = 2, G(r, t) = −t ln (max (r, t)) .

Hence the boundary value problem (8)-(9) is equivalent to the integral equation
In C [0, 1], the Banach space of continuous functions in [0, 1] with the usual norm, let P be the cone of the non negative functions. The radial solutions of (1)-(2) are exactly the fixed points of the completely continuous operator S : P → P , defined by In [6], the following theorem is proved: Then the problem (1)-(2) has a positive radial solution.
We will show that the estimate on the constant A in the previous result can be improved.
Consider the problem (8)-(9) and the associated eigenvalue problem: We have To find the eigenvalues, it is useful to consider the auxiliar initial value problem: The solution v to this problem is well defined in [0, +∞[, oscillates, and has zeros {ξ n | n ∈ N} such that 0 < ξ 1 < ξ 2 < . . . → +∞, with ξ n+1 − ξ n → π (see [13]). Define u(r) = v(βr). Then Using (13) we have It is obvious that u (0) = 0, so it remains to find β such that u(1) = 0. As u(1) = v(β), we get β = ξ n for some n ∈ N, hence β = ξ n and, therefore, the eigenvalues of (12) are Let us identify the zeros of the unique solution of (13). We have and the last equation has the form which is easily reduced to a Bessel equation (cf. [12]). Using the new independent variable we obtain the transformed equation whose solutions are well known, and thus we get: where c 1 , c 2 are constants and J i , K i are Bessel functions of order i, of the first and the second kind respectively. Taking into consideration the boundary conditions, the constant c 2 must be zero in both cases (otherwise we would have lim r→0 For our boundary value problem we know that γ −1 = 2n (see [6]). If we compare √ 2n with ξ 1 -the zeros of these Bessel functions are well known -we can see that √ 2n < ξ 1 and hence, γ −1 < λ 1 (first eigenvalue of (12)).
For instance, for n = 2 or n = 4 we have By adapting the approach of [6] we shall prove the following improved version of theorem 3.1: Theorem 3.2. Let f : R + × R → R + and g : R + → R be continuous functions, and λ 1 defined as above.
Suppose there exist constants A, B ∈ R such that 0 ≤ A < λ 1 , and for all v ≥ 0 and y ∈ R.
Let φ be an eigenfunction associated with the first eigenvalue λ 1 . We have Since our computation above shows that we may assume that in the Banach space Then, as stated before, we can write problem (8)- (9) as Let T denote the operator introduced in section 2, with p(s) = s n−1 . This operator acts in Taking the least upper bound in the left hand side of last inequality, we obtain This estimate, which is the main reason to work in the functional space X, will be used in the proof of theorem 3.2 in a crucial way. is continuous. Moreover it takes bounded sets into bounded sets. Now let us consider the following decomposition of T : where 1] : u(1) = 0 , i 2 , i 3 are imbeddings, and T * is the operator T acting between those two spaces.
The operator (T * ) −1 takes u into −u − (n−1) r u ; it is obviously linear continuous and bijective and, therefore, using the Open Map Theorem, we get that T * is continuous. The imbedding i 2 is a well known completely continuous operator and using L'Hospital's rule we can prove that i 3 is also continuous. Since S = i 3 i 2 T * i 1 the conclusion of the lemma is now straightforward.

A simple computation yields
Defining Ω 2 = {v ∈ X | v X < T B X /(1 − A /λ 1 )} with A < A < λ 1 , then for v ∈ P ∩ ∂Ω 2 we have (using the positivity of T and the estimate (16)) Applying Krasnoselskii's fixed point Theorem 2.3 (compression version) we find a fixed point of S, and therefore a positive radial solution of (1)-(2).
In both theorems above, as mentioned in [6], the condition on f does not depend on the second variable, and, therefore, nothing is restraining the behaviour of g. The arguments used there are also valid for the same problem with f (v(r), α(v)), for any continuous functional α in X.
A similar procedure allows us to us prove a result in the spirit of the one considered in [6] where g is restrained, but the condition on f is weakened:

Lower and upper solutions and monotone approximation
We will now apply the lower and upper solution method to find solutions of the boundary value problem (8)- (9). We should point that in [11] (p.695) a monotone method approach using lower and upper solutions is applied to an epidemic problem with diffusion. The problem considered in there is a second order system of two PDE with a nonlocal term, under assumptions related to those we use below (in particular a Lipschitz condition) and where uniqueness is obtained as well.
We will use two different types of conditions concerning the given functions f and g, and construct monotone convergent sequences to solutions of the problem.
Concerning the convergence of the sequences, as the cone of positive functions in C[0, 1] is normal (since 0 ≤ u ≤ v implies u ≤ v ), we can use the standard argument ( [15], p.283), which gives the convergence of this iteration method to fixed points of Φ, and these are exactly the smallest and largest fixed points in [α 0 , β 0 ] ⊂ C[0, 1]. This solution is the limit of a monotone sequence constructed as in the statement of the theorem.
Let us now try another approach using the lower and upper solutions method, where we drop a part of the monotonicity assumptions. Proof. Suppose by contradiction that there exists a function u 0 that satisfies the assumptions above and is negative at some point.
For a given function u(r) ∈ C[0, 1], consider the boundary value problem with v (0) = 0 = v(1). Using the operator L defined in the beginning of this section, this equation is equivalent to the fixed point equation in Remark 4.7. Using a comparison method as the one in the proof of lemma 4.6, we get and therefore Φ u is a contraction mapping.
Proof. The computation used here is similar to another one used in [10]. We have L (α 1 − α 0 ) ≥ f α 0 , ω n Applying this Lemma in the following iterations, we prove that α 0 ≤ α 1 ≤ · · · ≤ α n ≤ · · · ≤ β n ≤ · · · ≤ β 1 ≤ β 0 as in the proof of Theorem 4.4. Concerning the convergence of the sequences, there is a slight difference from the usual method, because in each iteration we use a different operator. But, as α n+1 (r) = L −1 0 f uα n r), ω n 1 0 s n−1 g (α n+1 (s)) ds + λα n and α n ∞ ≤ max ( α 0 ∞ , β 0 ∞ ), we have that α n+1 C 1 is bounded, and, therefore, using Arzela-Ascoli Theorem, there exists a convergent subsequence of α n . Considering the monotonicity of α n , we get the conclusion by the standard argument.  and there exists k > 0 such that f (k, ω n g(k)/n) < 0. Suppose in addition that f and g satisfy the assumptions of Theorem 4.10.
Then there exists a positive solution of (18)-(19). This solution may be approximated by monotone sequences. In fact, a simple calculation shows that for > 0 small enough, φ is a positive lower solution of (18)-(19). The constant k is clearly an upper solution. The statement follows.