POSITIVE SYMMETRIC SOLUTIONS OF SINGULAR SEMIPOSITONE BOUNDARY VALUE PROBLEMS

Using the method of upper and lower solutions, we prove that the singular boundary value problem,


Introduction
We are interested here in the existence of positive solutions to the singular boundary value problem (BVP) where α > 0 is a given exponent and f : R → R is a continuous function that is bounded below.By a positive solution, we mean a suitable function u that satisfies u > 0 on (0, 1).
Since the lower bound on f may be negative, we allow f to satisfy the semipositone condition f (0) < 0 . ( By reflection across the origin, a positive solution u of (1) yields a positive symmetric solution w of the BVP −w ′′ = f (w) w −α in (−1, 1), w(−1) = 0 = w(1) .
Our interest in problem (1) stems from its relation to elliptic partial differential equations of the form where p > 1, Ω ⊂ R N is a bounded region with smooth boundary ∂Ω, and ∆ p is the p-Laplacian.More precisely, problem (1) is the one-dimensional analogue of problem (3) when p = 2, Ω is the unit ball, and one seeks radial solutions.Progress on this simplified case therefore provides some insight into what can happen in the more complicated higher-dimensional problem (3).Note that [8] establishes the existence of positive radial solutions of problem (3) for various exponents p and dimensions N when f is nonsingular; the present paper employs different methods to extend the results of [8] to the singular case.Furthermore, the analysis of positive solutions to BVPs is of significance due to the fact that when natural phenomena are modelled via BVPs, only positive solutions to the problem will make physical sense.
Since we rely on the well-known method of upper and lower solutions to obtain positive solutions of (1), we briefly recall the relevant facts for the reader's convenience.See Chapter 2 of [6], for example, for a much more general development of such techniques for two-point BVPs.For our purposes, a function u ∈ C 2 [0, 1] is called a lower solution of (1) if while a function u ∈ C 2 [0, 1] that satisfies the reversed inequalities is termed an upper solution of (1).The following result will be fundamental (cf.Theorem 1.3, p. 77 of [6]).
Theorem 1.1 Let u and u be lower and upper solutions, respectively, of (1) such that u ≤ u.Then there exists a solution u of (1) such that u ≤ u ≤ u.
To find a positive solution of a semipositone problem via the method of upper and lower solutions, it is well-known that the principal difficulty is identifying a positive lower solution.Doing so is the objective of the following section, while Section 3 uses its results to establish the existence of a positive solution of (1) for certain nonlinearities f .For recent works on positive symmetric solutions of two-point boundary value problems involving singular and/or semipositone nonlinearities, we refer the reader to [1] - [4], [7], [10] - [14] and the references therein.Contrary to these works, we adopt a different fixed-point formulation of our auxiliary problem to which the Schauder Fixed Point Theorem applies directly (cf.Lemma 2.1 below).Finally, we remark that much less is known about such boundary value problems in higher dimensions, but a recent preprint by Chhetri and Robinson [5] provides some interesting results in this direction.

Constructing Positive Lower Solutions
In this section, we consider the auxiliary BVP where the exponent α ∈ (0, 1) is given and the function σ : R → R is defined by for constants K > 0 and L < 0. We will see in Section 3 that a positive solution of ( 5) yields a positive subsolution of (1); we will solve (5) by concatenating solutions of two related problems.First, given any constant L < 0, we show in Section 2.1 that there exist ρ ∈ (0, 1) and v > 0 such that Having found ρ, Section 2.2 determines a corresponding K > 0 such that has a solution w ≥ 1 with w ′ (ρ) = v ′ (ρ).With v and w in hand, the function u defined by is a solution of (5).To complement these results, Section 2.5 shows that problem (6) cannot have a positive solution if α ≥ 1.Before solving (6), we note that a natural ansatz for its solution is One easily finds that this ansatz yields a solution of the differential equation in (6) when It follows that v(ρ) = 1 when , and this value of ρ belongs to (0, 1) if and only if c > 1, i.e., Thus, problem (6) only has a solution of the form (9) if |L| is sufficiently large.In contrast, we show that (6) has a solution for any L < 0; even when (10) holds, the solution of ( 6) produced below cannot be of the form (9). As a result, one expects to find multiple positive solutions of both problems ( 6) and ( 5), and it would be interesting to determine precisely how many positive solutions these problems have.

Existence for Problem (6)
Suppose that v > 0 is a decreasing solution of (6); since multiplying both sides of this equation by 2v ′ and integrating yields for an integration constant d to be specified.Since v ′ < 0, we have and integrating from r to 1 gives ds .
This preliminary calculation motivates the following definition: for a given constant This operator is clearly completely continuous.and hence that for any r ∈ [0, 1].Since L < 0 and 0 For such an M d , the bounds in (13) guarantee that the set D ⊂ C[0, 1] defined by is invariant under T d , i.e., T (D) ⊂ D. As D is clearly bounded, closed and convex, the Schauder Fixed Point Theorem [9] pp.25-26 applies and yields the following result.
Lemma 2.1 Let d > 1 be given, and suppose that M d > 0 satisfies (14).Then there A fixed point v ∈ D of T d is automatically decreasing, vanishes at r = 1, and is therefore positive on (0, 1).Since v ≥ 0 and d > 1, and we conclude that there must be a point ρ ∈ (0, 1) such that We thus obtain the following result.
Corollary 2.1 Let L < 0 and α ∈ (0, 1) be given.There is a point ρ ∈ (0, 1) such that the singular boundary value problem has a positive solution v.
EJQTDE Spec.Ed.I, 2009 No. 24 Proof.Direct calculations show that w ≡ 1 is a lower solution of ( 7), while the unique solution w K of is an upper solution of ( 7); note that It follows from Theorem 1.1 that there is a solution w of the problem such that 1 ≤ w ≤ w K .
The main result of this paper, Theorem 2.1, relies on finding a solution w of problem ( 7) with a prescribed slope at ρ.The following sequence of lemmas shows that this can be done.
Proof.Lemma 2.2 guarantees that, for any K > 0, problem (7) has a decreasing solution w satisfying 1 ≤ w ≤ w K .Consequently, and the calculations used earlier to define the operator T d show that It follows that decreasing K yields a solution w whose slope at ρ is as small (in absolute value) as desired.
Proof.Let w K 1 denote the function defined by (17); we know that from which we obtain It then follows by direct calculation that w K 1 will be a lower solution of (7) if K > 0 satisfies Since w K provides an upper solution as in the proof of Lemma 2.2, there exists a solution w of ( 7) such that w K 1 ≤ w ≤ w K .In particular, By taking K 1 sufficiently large, choosing a K that satisfies (19), and then using identity (18), we see that there exists a K such that (7) has a solution with arbitrarily large slope at ρ.
Proof.Using the two previous lemmas, there exist positive constants K < K and corresponding solutions w ≤ w of problem (7) such that w ′ (ρ) < m < w ′ (ρ).Let K be any constant between K and K; since w and w are distinct lower and upper solutions, respectively, of problem (7) with coefficient K, there must exist a solution w K of this problem that lies between w and w.Since we can use w K as an upper or lower solution for problem (7) with other coefficients, we thus obtain a family of solutions To see that w ′ K (ρ) varies continuously with K, fix a constant K * between K and K, let K n be an increasing sequence that converges to K * , let w n ∈ F denote the solution corresponding to K n , and let w * ∈ F be the solution corresponding to K * .The set of functions { w n } is clearly uniformly bounded (by w K * (0)), and the calculations leading to (11) and (18) show that these functions are equicontinuous.The Arzelà-Ascoli Theorem therefore guarantees that some subsequence converges uniformly; relabeling as necessary, we find that the functions w n converge uniformly to some function w.Combining these convergence results with a fixed point characterization of problem (7) (obtained by, e.g., proceeding as in Section 2.1) shows that w solves problem (7) with coefficient K * .w and w * are therefore the unique positive solutions of the initial value problem with v 0 = w(0) and v 0 = w * (0), respectively.The uniqueness of trajectories and the fact that w(ρ) = w * (ρ) = 1 force w(0) = w * (0), and it then follows from (18) that both w K (0) and w ′ K (ρ) depend continuously on the parameter K. Having verified this continuous dependence on K, the remainder of the proof is a direct application of the Intermediate Value Theorem.
If α = 1 and v is a positive solution of (6), then Multiplying both sides by 2v ′ and integrating now gives for some integration constant d.As r → 1, v again approaches 0 and we obtain the same contradiction as in the case α > 1.

Applications
As indicated earlier, a positive solution of problem (5) will be a positive subsolution of (1) for appropriate nonlinearities f .Under additional assumptions on f , it is easy to find a larger supersolution and thereby obtain a positive solution of (1), as illustrated in our final result.
Theorem 3.1 Let L < 0 be given, and let K > 0 be a corresponding value such that (5) has a positive solution ψ.Let M be the maximum of ψ, suppose that the constants a and b satisfy 0 < a < b α+1 , and let f : R → R be a continuous function such that EJQTDE Spec.Ed.I, 2009 No. 24