# A p-Laplace equation with nonlocal boundary condition in a perforated-like domain.

1 Introduction

Let [OMEGA] be a bounded open set of [R.sup.n], n [greater than or equal to] 2, 0 [member of][OMEGA]. We consider the radially symmetric solutions of the following p-Laplace equation in a perforated-like domain

-div([[absolute value of [nabla]u].sup.p-2][nabla]u)= h(x)f (x,u,[absolute value of [nabla]u]), x [member of] [OMEGA]\{0}, (1.1)

subject to the nonlocal boundary condition

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.2)

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.3)

where [OMEGA]\{0} can be considered as the limit of [OMEGA]\[B.sub.[epsilon], [B.sub.[epsilon] is a ball centered at the origin with radius [epsilon] small enough, p > 2, [alpha][greater than or equal to] 0, f, g and h are given functions and v denotes the unit outward normal to the boundary [partial derivative][OMEGA]. In order to discuss the radially symmetric solutions, we assume that [OMEGA] is the unit ball B, h(x), g(x) and f(x,u,[absolute value of [nabla]u]) are radially symmetric, namely,

h(x) = h([[absolute value of x]), g(x) = g([[absolute value of x]), f(x,u,[absolute value of [nabla]u]) = f([[absolute value of x],u,[absolute value of [nabla]u]).

Let r = [absolute value of x], then by a direct calculation, we can rewrite the problem (1.1)-(1.3) as

([r.sup.n-1] [[phi].sub.p](u'))'+ [r.sup.n-1]h(r)f(r,u,[absolute value of u']) = 0, r [member of] (0,1), (1.4) [[phi].sub.p](u'(1))= [[integral].sup.1.sub.0] [s.sup.n-1] [[phi].sub.p](u'(s))dg(s), (1.5) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.6)

where [[phi].sub.p](s) = [[absolute value of s].sup.p-2]s. Let q = p/(p - 1), then we have [[phi].sup.-1.sub.p](s) = [[phi].sub.q](s) for any s [member of r] R.

Nonlocal boundary value problem often occurs in the study of the electro-chemistry, the thermal conduction problem, the semiconductor problem, etc., see [1]-[3]. This class of problems were first considered by Bitsadze [4] in the early 1960s. From then on, more and more workers take their notice of these problems, such as Il'in and Moiseev [5], Karakostas and Tsamatos [6]-[[beta]], etc. Until now, the problems with nonlocal boundary value condition, as well as with the multi-point boundary value condition, also attract many authors to pay attention to, see [9]-[15], and the references cited therein. To the best of our knowledge, most works we mentioned above are focus on the discussion of the existence of the solutions, however, the works of studying the uniqueness of positive solutions for p-Laplacian are rather few in the literature.

In this paper, we study the existence, the uniqueness and some other properties of the radially symmetric solutions of the nonlocal boundary value problem (1.1)-(1.3), in which we extend the function f(u) in [13] to the more general case f(x,u, [absolute value of [nabla]u]). Since f is dependent on the first-order partial derivatives of u(x), we have to find the variational relationship between u(r) and u'(r) under different conditions. And a fixed point result, called the nonlinear alternative of Leray-Schauder, which can be found in [16], would be used to obtain the existence of solutions.

The paper is organized as follows. In Section 2, we introduce some necessary preliminaries and give the statement of our main results. The proofs of the main results will be given in Section 3.

2 Preliminary and Statement of the Main Result

We firstly present the assumptions.

(H1) f(r,s,t) is a continuous and positive function defined on [0,1] x R x R, which is strictly decreasing with respect to s for each fixed (r,t) [member of] [0,1] x [0,+[infinity]) and t for each fixed (r,s) [member of] [0,1] x [0,+[infinity]), respectively;

(H2) h(r) is a positive and continuous function on [0,1];

(H3) g(r) is a nondecreasing function on [0,1] with 0 = g(0) < g(1) < 1.

Now we introduce the definition of the solution.

Definition 2.1. A function u(r) is said to be a solution of the equation (1.4), ifu(r)[member of] C([0,1]) [intersection] [C.sup.1] ((0,1]), u(r) [greater than or equal to] 0 on [0,1], and the integral equality

[[integral].sup1.sub.0]([r.sup.n-1][[phi].sub.p](u')[phi]'(r)-[r.sup.n-1]h(r)f(r,u,[absolute value of u'])[phi](r)) dr = 0

holds for any [phi][member of][C.sup.[infinity].0]((0,1)).

Remark 2.1. Let u(r) be a solution of the equation (1.4), then u satisfies the equation (1.4) in (0,1).

Proof. According to Definition 2.1, we have

([r.sup.n-1][[phi].sub.p] (u'))' = -[r.sup.n-1]h(r)f (r, u, [absolute value of u'])

in (0, 1) in the sense of distribution. Furthermore, by virtue of the assumptions

(H1) and (H2), we obtain

([r.sup.n-1][[phi].sub.p](uu))'[member of] C((0,1)),

hence,

[r.sup.n-1][[phi].sub.p](u') [member of] [C.sup.1]((0,1)),

which implies that u satisfies the equation (1.4) in (0,1). The proof is complete.

Next, we can derive the following properties of the solution u for the nonlocal boundary value problem (1.4)-(1.6), by using the assumptions above.

Proposition 2.1. Let u(r) be a solution of the nonlocal boundary value problem (1.4)(1.6). Then

(i) u'(r) > 0, r [member of] (0,1);

(ii) u(r) [member of] [C.sup.2] ((0,1));

(iii) u"(r) < 0, r [member of] (0,1];

(iv) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Proof. (i) From the equation (1.4), we can see that

([r.sup.n-1][[phi].sup.p](u'))' = -[r.sup.n-1]h(r)f(r, u, [absolute value of u']) < 0, r [member of] (0,1), which implies that

[r.sup.n-1][[phi].sup.p](u'))' > [[phi].sub.p](u'(1)), r [member of] (0,1).

If u'(1) = 0, from the above inequality, we have

0 = [[phi].sub.p](u'(1))= [[integral].sup.1.sub.0] [s.sup.n-1][[phi].sub.p](u'(1)) dg(s) > 0,

which is a contradiction. If u'(1) < 0, however, we see that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

which contradicts the nonlocal boundary value condition (1.5). Thus u'(1) > 0 and [[phi].sub.p](u'(r)) > 0, r [member of] (0,1), that is,

u'(r) > 0, r [member of] (0, 1].

(ii) Recalling Remark 2.1, we have

[r.sup.n-1][[phi].sup.p](u')[member of] [C.sup.1] ((0,1)).

Since u'(r) > 0, r [member of] (0,1], it follows

[r.sup.n-1][[phi].sup.p](u')= [r.sup.n-1])[(u').sup.p-1] [member of] [C.sup.1] ((0,1)).

Note that n [greater than or equal to] 2 and p > 2, we have

u(r) [member of] [C.sup.2] ((0, 1)) .

(iii) The equation (1.4) implies that for any r [member of] (0, 1],

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

(iv) Integrating the equation (1.4) from r to 1, it yields

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

uniformly in r [member of] (0, 1). Since

[r.sup.n-1][[phi].sub.p](u'(r)) > [[phi].sub.p](u'(1)) > 0,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is existent and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Summing up, we complete the proof of Proposition 2.1.

Define a normed linear space X, which is the set of all real-valued functions defined in C([0,1]) [intersection] [C.sup.1]((0,1]) with the norm

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

We can prove that X is a Banach space. Obviously, Proposition 2.1 indicates that for any solution u of the nonlocal boundary value problem (1.4)-(1.6), u[member of]X.

The main results in this paper are the following two theorems.

Theorem 2.1. Assume p > n and (H1)-(H3) hold. For any [alpha] [greater than or equal to] 0, the nonlocal boundary value problem (1.4)-(1.6) has a unique solution [u.sub.[alpha]](r)[member of]X. Furthermore, if 0 [less than or equal to] [[alpha].sub.1] < [[alpha].sub.2], then [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] has no local maximum value point in (0,1).

Theorem 2.2. Assume p [less than or equal to] n and (H1)-(H3) hold. For any [alpha] [greater than or equal to] 0, the nonlocal boundary value problem (1.4)-(1.6) has no solution.

According to Theorem 2.1, we have y(r) < max{y(0), y(1)} . Therefore,

Remark 2.2. If p > n and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] are two solutions of the problem (1.4)-(1.6) with [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Then we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

3 Proofs of the Main Results

In this section, we give the proofs of the main results. In order to study the existence and uniqueness of solution of the nonlocal boundary value problem (1.4)-(1.6), we should first consider the following approximate problem, where we assume (H1)-(H3) hold true, and [alpha] [greater than or equal to] 0, [beta] > 0,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

By applying some similar methods of the proof of Proposition 2.1, we can also obtain the following properties of the solutions for the boundary value problem (3.1).

Remark 3.1. Assume u(r) is a solution of the problem (3.1), then

(i) u'(r) > 0, r [member of] (0, 1];

(ii) u(r) [member of] [C.sup.2] ((0, 1));

(iii) u"(r) < 0, r [member of] (0,1];

(iv) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

In what follows, we shall show that the problem (3.1) admits one and only one solution in C([0,1]) [intersection] [C.sup.1] ((0,1]). We need the following lemmas.

Lemma 3.1. Assume [u.sub.1](r) and [u.sub.2](r) are two solutions ofthe problem (3.1). If there exist a point b [member of] (0,1], such that [u.sub.1] (b) = [u.sub.2] (b), then for any r [member of] [0, b], we have

[u.sub.1] (r) = [u.sub.2] (r), r [member of] [0, b].

Proof. We give the proof by contradiction. Without loss of generality, suppose that there exist a point [r.sub.0] [member of] (0, b) such that [u.sub.1] ([r.sub.0]) < [u.sub.2] ([r.sub.0]). Note that [u.sub.1](0) = [u.sub.2](0) and [u.sub.1](b) = [u.sub.2](b), we might as well take

[b.sub.1] = inf r; 0 [less than or equal to] r < [r.sub.0], [u.sub.1](s) < [u.sub.2](s), s [member of] (r,[r.sub.0])

and

[b.sub.2] = sup r; [r.sub.0] < r [less than or equal to] b, [u.sub.1](s) < [u.sub.2](s), s [member of] ([r.sub.0], r).

Then for any r [member of] ([b.sub.1], [b.sub.2]), we have [u.sub.1](r) < [u.sub.2](r) and [u.sub.1]([b.sub.1]) = [u.sub.2]([b.sub.1]), [u.sub.1] ([b.sub.2]) = [u.sub.2]([b.sub.2]). Obviously, [u.sub.1](r) - [u.sub.2](r) must have a minimal value point c [member of] ([b.sub.1], [b.sub.2]), such that [u'.sub.1] (c) = [u.sub.2](c), [u".sub.1(c)] > [u".sub.2]' (c). A simple calculation for the first equation of the problem (3.1) shows that the solution u of the problem (3.1) satisfies

(n - 1)[r.sup.n-2][(u').sup.p-1] + (p - 1)[r.sup.n-1])[(u').sup.p-2]u" + [r.sup.n-1]h(r)f (r,u,u') = 0.

Therefore, for the two solutions [u.sub.1](r) and [u.sub.2](r), we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

By virtue of (H1) and (H2), we see that f(c, [u.sub.1] (c), [u.sub.1] (c)) > f(c, [u.sub.2](c), [u.sub.2](c)) and h(c) > 0, hence we conclude that [u".sub.1](c) < [u".sub.2](c), which is a contradiction. The proof is complete.

Lemma 3.2. Assume [u.sub.1](r) and [u.sub.2](r) are two solutions of the problem (3.1) with boundary value conditions

[u.sub.1] (0) = [[alpha].sub.1] > 0, [u'.sub.1] (1) = [[beta].sub.1] > 0;

[u.sub.2] (0) = [[alpha].sub.2] > 0, [u'.sub.2] (1) = [[beta].sub.2] > 0,

respectively. If there exists a point b [member of] (0,1) such that [u.sub.1](b) < [u.sub.2](b), [u'.sub.1] (b) < ['u.sub.2] (b), then for any r [member of] (b, 1], we have [u.sub.1](r) < [u.sub.2] (r) and [u'sub.1](r) < [u1.sub.2] (r).

Proof. Suppose to the contrary, that is, there exists a point [r.sub.0] [member of] (b, 1] such that [u'.sub.1]([r.sub.0]) > [u'.sub.2] ([r.sub.0]), then we take

[r.sup.*] = inf {r; b < r [less than or equal to] [r.sub.0], [u'.sub.1](s) > [u'.sub.2](s), s [member of] (b, r]}.

Obviously, [r.sup.*] is exist, hence we have

[u.sub.1](r) < [u.sub.2](r), r [member of] [b, [r.sup.*]],

[u'.sub.1](r) < [u'.sub.2](r), r [member of] [b, [r.sup.*]],

and

[u'.sub.1] ([r.sup.*]) = [u'.sub.2] ([r.sup.*]).

Then on the point [r.sup.*], by utilizing a similar method of the proof of Lemma 3.1, we obtain [u".sub.1] ([r.sup.*]) < [u".sub.2] ([r.sup.*]), which implies that [r.sup.*] is not a stationary point of [u'.sub.1](r) - [u'.sub.2](r). Clearly, it is a contradiction since [u'.sub.1]([r.sup.*]) = [u'.sub.2]([r.sup.*]). Thus we have

[u'.sub.1](r) < [u'.sub.2](r), r [member of] [b,1].

Since [u.sub.1](b) < [u.sub.2](b), we further have

[u.sub.1] (r) < [u.sub.2](r), r [member of] [b, 1].

The proof is complete.

Lemma 3.3. For any fixed [alpha] [greater than or equal to] 0 and [beta] > 0, the problem (3.1) admits at most one solution.

Proof. We give the proof by contradiction. Without loss of generality, assume that [u.sub.1] (r) and [u.sub.2](r) are two solutions of the problem (3.1), and there exists one point b [member of](0,1], such that [u.sub.1](b) < [u.sub.2](b). By a simple analysis, we see that there must exist another point [r.sub.0] [member of] (0, b] such that [u.sub.1] ([r.sub.0]) < [u.sub.2] ([r.sub.0]) and [u'.sub.1] ([r.sub.0]) < [u'.sub.2] ([r.sub.0]). Recalling Lemma 3.2, we can conclude that

[u.sub.1](r) < [u.sub.2](r) and [u1.sub.1](r) < [u1.sub.2](r), for any r [member of] [[r.sub.0], 1).

Then from (H1) we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The contradiction implies that the lemma is proved.

We use the following fixed point theorem, which can be found in [16], to obtain the solution of the problem (3.1).

Lemma 3.4. Assume U is a relatively open subset of a convex set K in a Banach space X. Let G: [bar.U] [right arrow] K bea compact map, p [member of] U, and N[lambda] (u) = N(u, [lambda]):[bar.U] x [0,1] [right arrow] K a family of compact maps (i.e., N ([bar.U] x [0,1]) is contained in a compact subset of K and N: [bar.U] x [0,1] [right arrow] K is continuous) with [N.sub.1] = G and [N.sub.0] = p, the constant map to p. Then either

(i) G has a fixed point in [bar.U]; or

(ii) There is a point u [member of] [partial derivative]U and [lambda][member of] (0,1) such that u = [N.sub.[lambda]]u.

Lemma 3.5. If p > n, then for any fixed [alpha] [greater than or equal to] 0 and [beta] > 0, the problem (3.1) has a solution u(r) [member of] C([0,1]) [intersection] [C.sup.1] ((0,1]).

Proof. We first consider the following problem for any fixed parameter [lambda] [member of] (0,1),

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3.2)

Obviously, since Remark 3.1 also holds for the problem (3.2), we can see that u [member of] X for any solution u of the problem (3.2). Therefore, solving the problem (3.2) is equivalent to finding a nonnegative solution u(r) [member of] X with u(r) satisfying

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3.3)

Since p > n, it follows that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

is integrable on (0,1), then the right side of (3.3) is reasonable in [0,1]. Define the operator [N.sub.[lambda]] : K [right arrow] K by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3.4)

where K = {u [member of] X; u(0) = [alpha], u' (1) = [beta]} .

In what follows, we will show that the operator [N.sub.1] has a fixed point in X. The proof will be given in several steps.

Step 1: We shall show that there is a constant M*, independent of [lambda], such that [parallel]u[parallel] [less than or equal to] M* for any solution u(r) of the problem (3.2) for each [lambda] [member of] (0,1).[parallel]u[parallel] [less than or equal to] M* for any solution u(r) of the problem (3.2) for each [lambda] [member of] (0,1).

Let u(r) be a solution of the problem (3.2). According to the equation (3.3), we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where

[M.sub.1] > [[integral].sup.1.sub.0] [r.sup.n-1] h (r) f (r,0,0) dr + [[beta].sup.p-1]

is a constant. Thus there is a constant M*, independent of A, such that [parallel]u[parallel] < M* for any solution u(r) of the problem (3.2) for each [lambda] [member of] (0, 1).

Step 2: It is easy to see that [N.sub.[lambda]] is continuous for any fixed [lambda]. We will show that [N.sub.[lambda]] is even completely continuous for fixed [lambda] by Arzela-Ascoli theorem.

Let [OMEGA] be a bounded subset of K, i.e., [parallel]u[parallel] < C for all u [member of] [OMEGA]. Here C > 0 is a constant. Firstly, following the proof in Step 1, we can see that [N.sub.[lambda]][OMEGA] is closed and bounded, and there exist the following two inequalities

[absolute value of ([N.sub.[lambda]]u)(r)] [less than or equal to] [alpha] + [[integral].sup.1.sub.0] [[phi].sub.q] (1/[s.sup.n-1][M.sub.2]) ds (3.5)

and

[absolute value of ([N.sub.[lambda]](u)] [less than or equal to][[phi].sub.q](M2), (3.6)

where

[M.sub.2] = sup{[absolute value of f(r,u,0)];0[less than or equal to]r [less than or equal to] 1, -C [less than or equal to] u [less than or equal to]C}[[integral.sup.1.sub.0][t.sup.n-1]h(t)dt[[beta].sup.p-1]

We next show the equicontinuity of [N.sub.[lambda]][OMEGA] on [0,1]. For u [member of][OMEGA] and [r.sub.1], [r.sub.2][member of] [0,1], we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3.7)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3.8)

The equicontinuity of [N.sub.[lambda]][OMEGA] on [0,1] now follows from the inequality (3.7) and the equation (3.8). Therefore, the Arzela-Ascoli theorem implies that [N.sub.[lambda]] is completely continuous.

Step 3: We shall show that N([bar.U] x [0,1]) is contained in a compact subset of K, where

U = {u [member of] K; [parallel]u[parallel] [less than or equal to] M* + 1}, ([N.sub.0]u)(r)=[alpha] + [beta][[integral].sup.r.sub.0][[phi].sub.q]([1/[s.sup.n-1]])ds.

Let N([u.sub.n], [[lambda].sub.n]) be any sequence in N([bar.U] x [0,1]). Then it is easy to see that N([u.sub.n], [[lambda].sub.n]) is uniformly bounded and equicontinuous on [0, 1] since the inequality (3.5)-(3.7) and the equation (3.8) does not depend on the fixed [lambda], thus the Arzela-Ascoli theorem again yields the result. By Lemma 3.4, we deduce that [N.sub.1] has a fixed point, i.e., the problem (3.1) has a solution u(r) [member of] C([0,1]) [intersection] [C.sup.1] ((0,1]).

Summing up, we complete the proof of Lemma 3.5.

Lemma 3.3 and Lemma 3.5 implies that the problem (3.1) has an unique solution. According to this, in what follows, by using the shooting method, precisely speaking, by selecting [beta] > 0 suitably in the problem (3.1) such that the nonlocal boundary value condition (1.5) holds, we will show that the problem (1.4)-(1.6) admits a unique solution. Before going further, we need the following lemmas.

Lemma 3.6. If [u.sub.1] (r), [u.sub.2] (r) are two solutions of the problem (3.1) with [u'.sub.1](1) = [[beta].sub.1], [u'.sub.2](1) = [[beta].sub.2], and [[beta].sub.1] > [[beta].sub.2] > 0, then

[u.sub.1] (r) [greater than or equal to] [u.sub.2](r), r [member of] (0,1].

Proof. If the lemma were not true, there must exist one point [r.sub.0] [member of] (0, 1) such that [u.sub.1] ([r.sub.0]) < [u.sub.2]([r.sub.0]). Since [u.sub.1] (0) = [u.sub.2](0), it is easy to see that there exists another point [r.sub.1] [member of] (0, [r.sub.0]), such that [u.sub.1] ([r.sub.1]) < [u.sub.2]([r.sub.1]), [u'.sub.1] ([r.sub.1]) < [u'.sub.2]([r.sub.1]) by considering the continuity of the solution. According to Lemma 3.2, we conclude that [u'.sub.1] (1) < [u'.sub.2](l), namely, [[beta].sub.1] < [[beta].sub.2], which is a contradiction. The proof is complete. ?

Lemma 3.7. If [u.sub.1] (r), [u.sub.2] (r) are two solutions of the problem (3.1) with [u.sub.1](1) = [[beta].sub.1], [u.sub.2](1) = [[beta].sub.2], and [[beta].sub.1] > [[beta].sub.2] > 0, then

[u'.sub.1] (r) [greater than or equal to] [u'.sub.2](r), r [member of] (0,1).

Proof. If the lemma were not true, there exists a point [r.sub.0] [member of] (0,1) such that [u'.sub.1]([r.sub.0]) < [u'.sub.2]([r.sub.0]). According to Lemma 3.6, we see that [u.sub.1]([r.sub.0]) [greater than or equal to][u.sub.2]([r.sub.0]). We will show that [u'.sub.1]([r.sub.0]) > [u'.sub.2]([r.sub.0]). Otherwise, we have [u.sub.1]([r.sub.0]) = [u.sub.2]([r.sub.0]). Since that [u'.sub.1] ([r.sub.0]) < [u'.sub.2]([r.sub.0]) and [u.sub.1] (0) = [u.sub.2](0), therefore, there must exist one point b [member of] (0, [r.sub.0]) such that [u.sub.1](b) > [u.sub.2](b) and [u'.sub.1] (b) > [u'.sub.2](b) by considering the continuity of the solution. According to Lemma 3.2, we have [u.sub.1]([r.sub.0]) > [u.sub.2]([r.sub.0]) which is a contradiction. Since [u.sub.1](0) = [u.sub.2](0), [u.sub.2](r) - [u.sub.1](r) must have a minimal value point c [member of] (0, [r.sub.0]), hence we have [u.sub.1](c) > [u.sub.2](c), [u'.sub.1](c) = [u'.sub.2](c). By utilizing a similar method in Lemma 3.1, we see that [u".sub.1](c) > [u".sub.2](c), clearly, which contradicts the fact that c is a minimal value point of [u.sub.2](r) - [u.sub.1](r) . Therefore the proof is complete.

Combining Lemma 3.6 and Lemma 3.7, we obtain the following conclusion.

Corollary 3.1. Let [u.sub.1](r), [u.sub.2](r) be two solutions of the problem (3.1) with [u'.sub.1](1) = [beta], [u'.sub.2](1) = [[beta].sub.2], and [[beta].sub.1] > [[beta].sub.2] > 0. Then there exists a point b [member of] [0,1) such that

[u.sub.1](r) = [u.sub.2](r), 0 [less than or equal to] r [less than or equal to] b, (3.9)

[u.sub.1](r) > [u.sub.2](r), b < r [less than or equal to] 1, (3.10)

[u.sub.1](r) > [u'.sub.2](r), b < r [less than or equal to] 1. (3.11)

Proof. Let

b = sup {r; 0 [less than or equal to] r < 1, [u.sub.1] (r) = [u.sub.2](r)}.

By virtue of [u.sub.1](0) = [u.sub.2](0) and Lemma 3.1, we have 0 [less than or equal to] b < 1. Furthermore, together with Lemma 3.6, we also have

[u.sub.1] (r) = [u.sub.2](r), 0 [less than or equal to] r [less than or equal to] b, [u.sub.1] (r) > [u.sub.2](r), b < r [less than or equal to] 1.

So, it suffices to consider the inequality (3.11). If it were not true, then according to Lemma 3.7, there exists a point [r.sub.0] [member of] (b, 1) such that [u'.sub.1]([r.sub.0]) = [u'.sub.2] ([r.sub.0]). By utilizing a similar method to the proof of Lemma 3.1, we see that [u".sub.1] ([r.sub.0]) > [u".sub.2] ([r.sub.0]), which implies that the point [r.sub.0] is a minimal value point of [u.sub.2](r) - [u.sub.1](r) . Hence, there must exist a point [r.sub.1][member of](b,[r.sub.0]) such that [u'.sub.1]([r.sub.1]) < [u'.sub.2]([r.sub.1]), which contradicts Lemma 3.7. The proof is complete.

Combining with the above lemmas, we are now in a position to get the existence and uniqueness of solution of the problem (1.4)-(1.6) by the shooting methods. Define

k([beta]) = [[phi].sub.p]([u'.sub.[beta]](1))[[integral].sup.1.sub.0][s.sup.n-1][[phi].sub.p]([u'.sub.[beta]](s)) dg(s), [beta] > 0,

where [u.sub.[beta]] is the solution of the problem (3.1) with boundary value condition [u'.sub.[beta]](1) = [beta]. Then the desired unique solution of the problem (1.4)-(1.6) will be obtained by selecting an unique constant [beta]* > 0, such that k([beta]*) = 0. First of all, we need to establish the strict monotonicity and continuity of the function k([beta]) with respect to [beta].

Lemma 3.8. k([beta]) is continuous and strictly monotonous with respect to [beta] > 0.

Proof. Let [[beta].sub.1] > [[beta].sub.2] > 0. According to Corollary 3.1 and the first equation of the problem (3.1), we derive

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] are the solutions of the problem (3.1) with [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. By the assumption (H1), it is easy to see that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Thus

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

namely,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

By the above inequality, we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Furthermore, we can see the continuity of k([beta]) when [beta] > 0 from the following inequality

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Hence, the proof of Lemma 3.8 is complete.

Now, by virtue of the above established lemmas about the approximate problem (3.1), we are going to prove the main results in this paper.

Proof of Theorem 2.1. A simple calculation for the first equation of the problem (3.1) implies that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Noticing the assumption (H1)-(H3) and Fubini's Theorem, we see that, when [beta] > 1,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

On the other hand, when [beta] [less than or equal to] 1, let [delta] be a positive constant, which is small enough, and g([delta]) > 0. From the proof of Lemma 3.5, we know that if u is a solution of the problem (3.1), then there exists a constant M* > 0 such that [parallel]u[parallel] < M*, that is,

u(r) [less than or equal to] M* and u'(r) [less than or equal to] M* /[delta], r [member of] [[delta],1].

Applying the strict monotonicity of f and the positivity of f and h, we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3.13)

Combining the inequality (3.12) with the inequality (3.13), it follows

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Then recalling the strict monotonicity and continuity of k([beta]), there is one and only one [beta]* > 0, such that k([beta]*) = 0, and [u.sub.[beta]*] is the unique positive solution of nonlocal boundary value problem (1.4)-(1.6).

Finally, we consider the property of the solution of the problem (1.4)-(1.6). Suppose that there exists a point [r.sub.0] [member of] (0,1) such that y([r.sub.0]) is the local maximum value of y(r). Without loss of generality, we assume that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], then we see that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Following the proof of Lemma 3.1, however, we can obtain [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], which is a contradiction.

Summing up, we complete the Proof of Theorem 2.1.

Proof of Theorem 2.2. Suppose that the nonlocal boundary value problem (1.4)-(1.6) admits a solution u [member of] C([0,1]) [intersection] [C.sup.1((0,1])]. Then by a simple calculation to the equation (1.4), we see that u'(r) satisfies

u'(r) = [r.sup.-(n-1)/(p-1)][[phi].sub.q]([integral].sup.1.sub.r][s.sup.n-1]h(s)f(s, u, u')ds + [[phi].sub.p] (u' (1))), r [member of] (0,1].

Obviously, u'(r) is not integrable since p [less than or equal to] n, which contradicts the definition of u. Hence, we complete the proof of Theorem 2.2. ?

Received by the editors in June 2012 - In revised form in January 2013.

Communicated by P. Godin.

Acknowledgement The authors would like to express their deep thanks to the referees for their valuable suggestions for the revision of the manuscript. The authors also would like to thank Professor Jingxue Yin and Professor Chunpeng Wang, under whose guidance this paper was completed.

References

[1] N. I. Ionkin, Solution of a boundary value problem in heat conduction theory with nonlocal boundary conditions, Differential Equations, 13(1977), 294-304.

[2] Y. S. Choi, K. Y. Chan, A parabolic equation with nonlocal boundary conditions arising from electrochemistry, Nonlinear Anal., 18(4)(1992), 317-331.

[3] Z. Cui, Z. Yang, Roles of weight functions to a nonlinear porous medium equation with nonlocal source and nonlocal boundary condition, J. Math. Anal. Appl, 342(1)(2008), 559-570.

[4] A. V. Bitsadze, On the theory of nonlocal boundary value problems, Soviet Math. Dokl. 30(1964), 8-10.

[5] V. Il'in, E. Moiseev, Nonlocal boundary value problems of the second kind for a Sturm-Liouville operator, Differential Equations, 23(1987), 979-987.

[6] G. L. Karakostas, P. C. Tsamatos, Multiple positive solutions for a nonlocal boundary value problem with response function quiet at zero, Electronic J. Differential Equations, 2001(13)(2001), 1-10.

[7] G. L. Karakostas, P. C. Tsamatos, Sufficient conditions for the existence of nonnegative solutions of a nonlocal boundary value problem, Appl. Math. Lett., 15(13)(2002), 401-407.

[8] G. L. Karakostas, P. C. Tsamatos, Multiple positive solutions of some Fredholm integral equations arisen from nonlocal boundary-value problems, Electron. J. Differential Equations, 30(2002), 1-17.

[9] A. Boucherif, Second-order boundary value problems with integral boundary conditions, Nonlinear Anal., 70(2008), 364-371.

[10] M. Feng, D. Ji, W. Ge, Positive solutions for a class of boundary-value problem with integral boundary conditions in Banach spaces, J. Comput. Appl. Math., 222(2008), 351-363.

[11] H. Pang, W. Ge, M. Tian, Solvability of nonlocal boundary value problems for ordinary differential equation of higher order with a p-Laplacian, Comput. Math. Appl, 56(2008), 127-142.

[12] J. R. L. Webb, G. Infante, Positive solutions of nonlocal boundary value problems involving integral conditions, Nonlinear differ. equ. appl., 15(2008), 45-67.

[13] J. Yin, Y. Ke, C. Wang, Radially symmetric solutions of the p-Laplacian in perforated-like domain with nonlocal boundary condition, Nonlinear Anal., 60(2005), 11[beta]3-1196.

[14] G. Infante, M. Zima, Positive solutions of multi-point boundary value problems at resonance, Nonlinear Anal., 69(2008), 245[beta]-2465.

[15] J. R. Graef, J. R. L. Webb, Third order boundary value problems with nonlocal boundary conditions, Nonlinear Anal., 71(2009), 1542-1551.

[16] A. Granas, J. Dugundji, Fixed Point Theory, New York: Springer-Verlag, 2003.

* This work is partially supported by NSFC, partially supported by Ph.D. Specialities of Educational Department of China, partially supported by the Fundamental Research Funds for the Central Universities and the Research Funds of Renmin University of China (2010030171).

Hailong Ye

Yuanyuan Ke ([dagger])

School of Mathematical Sciences, South China Normal University, Guangzhou 510631, P.R. China

School of Information, Renmin University of China, Beijing 100872, P.R. China

Email: ke_yy@163.com.

([dagger]) Corresponding author.

Let [OMEGA] be a bounded open set of [R.sup.n], n [greater than or equal to] 2, 0 [member of][OMEGA]. We consider the radially symmetric solutions of the following p-Laplace equation in a perforated-like domain

-div([[absolute value of [nabla]u].sup.p-2][nabla]u)= h(x)f (x,u,[absolute value of [nabla]u]), x [member of] [OMEGA]\{0}, (1.1)

subject to the nonlocal boundary condition

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.2)

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.3)

where [OMEGA]\{0} can be considered as the limit of [OMEGA]\[B.sub.[epsilon], [B.sub.[epsilon] is a ball centered at the origin with radius [epsilon] small enough, p > 2, [alpha][greater than or equal to] 0, f, g and h are given functions and v denotes the unit outward normal to the boundary [partial derivative][OMEGA]. In order to discuss the radially symmetric solutions, we assume that [OMEGA] is the unit ball B, h(x), g(x) and f(x,u,[absolute value of [nabla]u]) are radially symmetric, namely,

h(x) = h([[absolute value of x]), g(x) = g([[absolute value of x]), f(x,u,[absolute value of [nabla]u]) = f([[absolute value of x],u,[absolute value of [nabla]u]).

Let r = [absolute value of x], then by a direct calculation, we can rewrite the problem (1.1)-(1.3) as

([r.sup.n-1] [[phi].sub.p](u'))'+ [r.sup.n-1]h(r)f(r,u,[absolute value of u']) = 0, r [member of] (0,1), (1.4) [[phi].sub.p](u'(1))= [[integral].sup.1.sub.0] [s.sup.n-1] [[phi].sub.p](u'(s))dg(s), (1.5) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.6)

where [[phi].sub.p](s) = [[absolute value of s].sup.p-2]s. Let q = p/(p - 1), then we have [[phi].sup.-1.sub.p](s) = [[phi].sub.q](s) for any s [member of r] R.

Nonlocal boundary value problem often occurs in the study of the electro-chemistry, the thermal conduction problem, the semiconductor problem, etc., see [1]-[3]. This class of problems were first considered by Bitsadze [4] in the early 1960s. From then on, more and more workers take their notice of these problems, such as Il'in and Moiseev [5], Karakostas and Tsamatos [6]-[[beta]], etc. Until now, the problems with nonlocal boundary value condition, as well as with the multi-point boundary value condition, also attract many authors to pay attention to, see [9]-[15], and the references cited therein. To the best of our knowledge, most works we mentioned above are focus on the discussion of the existence of the solutions, however, the works of studying the uniqueness of positive solutions for p-Laplacian are rather few in the literature.

In this paper, we study the existence, the uniqueness and some other properties of the radially symmetric solutions of the nonlocal boundary value problem (1.1)-(1.3), in which we extend the function f(u) in [13] to the more general case f(x,u, [absolute value of [nabla]u]). Since f is dependent on the first-order partial derivatives of u(x), we have to find the variational relationship between u(r) and u'(r) under different conditions. And a fixed point result, called the nonlinear alternative of Leray-Schauder, which can be found in [16], would be used to obtain the existence of solutions.

The paper is organized as follows. In Section 2, we introduce some necessary preliminaries and give the statement of our main results. The proofs of the main results will be given in Section 3.

2 Preliminary and Statement of the Main Result

We firstly present the assumptions.

(H1) f(r,s,t) is a continuous and positive function defined on [0,1] x R x R, which is strictly decreasing with respect to s for each fixed (r,t) [member of] [0,1] x [0,+[infinity]) and t for each fixed (r,s) [member of] [0,1] x [0,+[infinity]), respectively;

(H2) h(r) is a positive and continuous function on [0,1];

(H3) g(r) is a nondecreasing function on [0,1] with 0 = g(0) < g(1) < 1.

Now we introduce the definition of the solution.

Definition 2.1. A function u(r) is said to be a solution of the equation (1.4), ifu(r)[member of] C([0,1]) [intersection] [C.sup.1] ((0,1]), u(r) [greater than or equal to] 0 on [0,1], and the integral equality

[[integral].sup1.sub.0]([r.sup.n-1][[phi].sub.p](u')[phi]'(r)-[r.sup.n-1]h(r)f(r,u,[absolute value of u'])[phi](r)) dr = 0

holds for any [phi][member of][C.sup.[infinity].0]((0,1)).

Remark 2.1. Let u(r) be a solution of the equation (1.4), then u satisfies the equation (1.4) in (0,1).

Proof. According to Definition 2.1, we have

([r.sup.n-1][[phi].sub.p] (u'))' = -[r.sup.n-1]h(r)f (r, u, [absolute value of u'])

in (0, 1) in the sense of distribution. Furthermore, by virtue of the assumptions

(H1) and (H2), we obtain

([r.sup.n-1][[phi].sub.p](uu))'[member of] C((0,1)),

hence,

[r.sup.n-1][[phi].sub.p](u') [member of] [C.sup.1]((0,1)),

which implies that u satisfies the equation (1.4) in (0,1). The proof is complete.

Next, we can derive the following properties of the solution u for the nonlocal boundary value problem (1.4)-(1.6), by using the assumptions above.

Proposition 2.1. Let u(r) be a solution of the nonlocal boundary value problem (1.4)(1.6). Then

(i) u'(r) > 0, r [member of] (0,1);

(ii) u(r) [member of] [C.sup.2] ((0,1));

(iii) u"(r) < 0, r [member of] (0,1];

(iv) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Proof. (i) From the equation (1.4), we can see that

([r.sup.n-1][[phi].sup.p](u'))' = -[r.sup.n-1]h(r)f(r, u, [absolute value of u']) < 0, r [member of] (0,1), which implies that

[r.sup.n-1][[phi].sup.p](u'))' > [[phi].sub.p](u'(1)), r [member of] (0,1).

If u'(1) = 0, from the above inequality, we have

0 = [[phi].sub.p](u'(1))= [[integral].sup.1.sub.0] [s.sup.n-1][[phi].sub.p](u'(1)) dg(s) > 0,

which is a contradiction. If u'(1) < 0, however, we see that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

which contradicts the nonlocal boundary value condition (1.5). Thus u'(1) > 0 and [[phi].sub.p](u'(r)) > 0, r [member of] (0,1), that is,

u'(r) > 0, r [member of] (0, 1].

(ii) Recalling Remark 2.1, we have

[r.sup.n-1][[phi].sup.p](u')[member of] [C.sup.1] ((0,1)).

Since u'(r) > 0, r [member of] (0,1], it follows

[r.sup.n-1][[phi].sup.p](u')= [r.sup.n-1])[(u').sup.p-1] [member of] [C.sup.1] ((0,1)).

Note that n [greater than or equal to] 2 and p > 2, we have

u(r) [member of] [C.sup.2] ((0, 1)) .

(iii) The equation (1.4) implies that for any r [member of] (0, 1],

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

(iv) Integrating the equation (1.4) from r to 1, it yields

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

uniformly in r [member of] (0, 1). Since

[r.sup.n-1][[phi].sub.p](u'(r)) > [[phi].sub.p](u'(1)) > 0,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is existent and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Summing up, we complete the proof of Proposition 2.1.

Define a normed linear space X, which is the set of all real-valued functions defined in C([0,1]) [intersection] [C.sup.1]((0,1]) with the norm

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

We can prove that X is a Banach space. Obviously, Proposition 2.1 indicates that for any solution u of the nonlocal boundary value problem (1.4)-(1.6), u[member of]X.

The main results in this paper are the following two theorems.

Theorem 2.1. Assume p > n and (H1)-(H3) hold. For any [alpha] [greater than or equal to] 0, the nonlocal boundary value problem (1.4)-(1.6) has a unique solution [u.sub.[alpha]](r)[member of]X. Furthermore, if 0 [less than or equal to] [[alpha].sub.1] < [[alpha].sub.2], then [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] has no local maximum value point in (0,1).

Theorem 2.2. Assume p [less than or equal to] n and (H1)-(H3) hold. For any [alpha] [greater than or equal to] 0, the nonlocal boundary value problem (1.4)-(1.6) has no solution.

According to Theorem 2.1, we have y(r) < max{y(0), y(1)} . Therefore,

Remark 2.2. If p > n and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] are two solutions of the problem (1.4)-(1.6) with [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Then we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

3 Proofs of the Main Results

In this section, we give the proofs of the main results. In order to study the existence and uniqueness of solution of the nonlocal boundary value problem (1.4)-(1.6), we should first consider the following approximate problem, where we assume (H1)-(H3) hold true, and [alpha] [greater than or equal to] 0, [beta] > 0,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

By applying some similar methods of the proof of Proposition 2.1, we can also obtain the following properties of the solutions for the boundary value problem (3.1).

Remark 3.1. Assume u(r) is a solution of the problem (3.1), then

(i) u'(r) > 0, r [member of] (0, 1];

(ii) u(r) [member of] [C.sup.2] ((0, 1));

(iii) u"(r) < 0, r [member of] (0,1];

(iv) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

In what follows, we shall show that the problem (3.1) admits one and only one solution in C([0,1]) [intersection] [C.sup.1] ((0,1]). We need the following lemmas.

Lemma 3.1. Assume [u.sub.1](r) and [u.sub.2](r) are two solutions ofthe problem (3.1). If there exist a point b [member of] (0,1], such that [u.sub.1] (b) = [u.sub.2] (b), then for any r [member of] [0, b], we have

[u.sub.1] (r) = [u.sub.2] (r), r [member of] [0, b].

Proof. We give the proof by contradiction. Without loss of generality, suppose that there exist a point [r.sub.0] [member of] (0, b) such that [u.sub.1] ([r.sub.0]) < [u.sub.2] ([r.sub.0]). Note that [u.sub.1](0) = [u.sub.2](0) and [u.sub.1](b) = [u.sub.2](b), we might as well take

[b.sub.1] = inf r; 0 [less than or equal to] r < [r.sub.0], [u.sub.1](s) < [u.sub.2](s), s [member of] (r,[r.sub.0])

and

[b.sub.2] = sup r; [r.sub.0] < r [less than or equal to] b, [u.sub.1](s) < [u.sub.2](s), s [member of] ([r.sub.0], r).

Then for any r [member of] ([b.sub.1], [b.sub.2]), we have [u.sub.1](r) < [u.sub.2](r) and [u.sub.1]([b.sub.1]) = [u.sub.2]([b.sub.1]), [u.sub.1] ([b.sub.2]) = [u.sub.2]([b.sub.2]). Obviously, [u.sub.1](r) - [u.sub.2](r) must have a minimal value point c [member of] ([b.sub.1], [b.sub.2]), such that [u'.sub.1] (c) = [u.sub.2](c), [u".sub.1(c)] > [u".sub.2]' (c). A simple calculation for the first equation of the problem (3.1) shows that the solution u of the problem (3.1) satisfies

(n - 1)[r.sup.n-2][(u').sup.p-1] + (p - 1)[r.sup.n-1])[(u').sup.p-2]u" + [r.sup.n-1]h(r)f (r,u,u') = 0.

Therefore, for the two solutions [u.sub.1](r) and [u.sub.2](r), we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

By virtue of (H1) and (H2), we see that f(c, [u.sub.1] (c), [u.sub.1] (c)) > f(c, [u.sub.2](c), [u.sub.2](c)) and h(c) > 0, hence we conclude that [u".sub.1](c) < [u".sub.2](c), which is a contradiction. The proof is complete.

Lemma 3.2. Assume [u.sub.1](r) and [u.sub.2](r) are two solutions of the problem (3.1) with boundary value conditions

[u.sub.1] (0) = [[alpha].sub.1] > 0, [u'.sub.1] (1) = [[beta].sub.1] > 0;

[u.sub.2] (0) = [[alpha].sub.2] > 0, [u'.sub.2] (1) = [[beta].sub.2] > 0,

respectively. If there exists a point b [member of] (0,1) such that [u.sub.1](b) < [u.sub.2](b), [u'.sub.1] (b) < ['u.sub.2] (b), then for any r [member of] (b, 1], we have [u.sub.1](r) < [u.sub.2] (r) and [u'sub.1](r) < [u1.sub.2] (r).

Proof. Suppose to the contrary, that is, there exists a point [r.sub.0] [member of] (b, 1] such that [u'.sub.1]([r.sub.0]) > [u'.sub.2] ([r.sub.0]), then we take

[r.sup.*] = inf {r; b < r [less than or equal to] [r.sub.0], [u'.sub.1](s) > [u'.sub.2](s), s [member of] (b, r]}.

Obviously, [r.sup.*] is exist, hence we have

[u.sub.1](r) < [u.sub.2](r), r [member of] [b, [r.sup.*]],

[u'.sub.1](r) < [u'.sub.2](r), r [member of] [b, [r.sup.*]],

and

[u'.sub.1] ([r.sup.*]) = [u'.sub.2] ([r.sup.*]).

Then on the point [r.sup.*], by utilizing a similar method of the proof of Lemma 3.1, we obtain [u".sub.1] ([r.sup.*]) < [u".sub.2] ([r.sup.*]), which implies that [r.sup.*] is not a stationary point of [u'.sub.1](r) - [u'.sub.2](r). Clearly, it is a contradiction since [u'.sub.1]([r.sup.*]) = [u'.sub.2]([r.sup.*]). Thus we have

[u'.sub.1](r) < [u'.sub.2](r), r [member of] [b,1].

Since [u.sub.1](b) < [u.sub.2](b), we further have

[u.sub.1] (r) < [u.sub.2](r), r [member of] [b, 1].

The proof is complete.

Lemma 3.3. For any fixed [alpha] [greater than or equal to] 0 and [beta] > 0, the problem (3.1) admits at most one solution.

Proof. We give the proof by contradiction. Without loss of generality, assume that [u.sub.1] (r) and [u.sub.2](r) are two solutions of the problem (3.1), and there exists one point b [member of](0,1], such that [u.sub.1](b) < [u.sub.2](b). By a simple analysis, we see that there must exist another point [r.sub.0] [member of] (0, b] such that [u.sub.1] ([r.sub.0]) < [u.sub.2] ([r.sub.0]) and [u'.sub.1] ([r.sub.0]) < [u'.sub.2] ([r.sub.0]). Recalling Lemma 3.2, we can conclude that

[u.sub.1](r) < [u.sub.2](r) and [u1.sub.1](r) < [u1.sub.2](r), for any r [member of] [[r.sub.0], 1).

Then from (H1) we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The contradiction implies that the lemma is proved.

We use the following fixed point theorem, which can be found in [16], to obtain the solution of the problem (3.1).

Lemma 3.4. Assume U is a relatively open subset of a convex set K in a Banach space X. Let G: [bar.U] [right arrow] K bea compact map, p [member of] U, and N[lambda] (u) = N(u, [lambda]):[bar.U] x [0,1] [right arrow] K a family of compact maps (i.e., N ([bar.U] x [0,1]) is contained in a compact subset of K and N: [bar.U] x [0,1] [right arrow] K is continuous) with [N.sub.1] = G and [N.sub.0] = p, the constant map to p. Then either

(i) G has a fixed point in [bar.U]; or

(ii) There is a point u [member of] [partial derivative]U and [lambda][member of] (0,1) such that u = [N.sub.[lambda]]u.

Lemma 3.5. If p > n, then for any fixed [alpha] [greater than or equal to] 0 and [beta] > 0, the problem (3.1) has a solution u(r) [member of] C([0,1]) [intersection] [C.sup.1] ((0,1]).

Proof. We first consider the following problem for any fixed parameter [lambda] [member of] (0,1),

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3.2)

Obviously, since Remark 3.1 also holds for the problem (3.2), we can see that u [member of] X for any solution u of the problem (3.2). Therefore, solving the problem (3.2) is equivalent to finding a nonnegative solution u(r) [member of] X with u(r) satisfying

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3.3)

Since p > n, it follows that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

is integrable on (0,1), then the right side of (3.3) is reasonable in [0,1]. Define the operator [N.sub.[lambda]] : K [right arrow] K by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3.4)

where K = {u [member of] X; u(0) = [alpha], u' (1) = [beta]} .

In what follows, we will show that the operator [N.sub.1] has a fixed point in X. The proof will be given in several steps.

Step 1: We shall show that there is a constant M*, independent of [lambda], such that [parallel]u[parallel] [less than or equal to] M* for any solution u(r) of the problem (3.2) for each [lambda] [member of] (0,1).[parallel]u[parallel] [less than or equal to] M* for any solution u(r) of the problem (3.2) for each [lambda] [member of] (0,1).

Let u(r) be a solution of the problem (3.2). According to the equation (3.3), we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where

[M.sub.1] > [[integral].sup.1.sub.0] [r.sup.n-1] h (r) f (r,0,0) dr + [[beta].sup.p-1]

is a constant. Thus there is a constant M*, independent of A, such that [parallel]u[parallel] < M* for any solution u(r) of the problem (3.2) for each [lambda] [member of] (0, 1).

Step 2: It is easy to see that [N.sub.[lambda]] is continuous for any fixed [lambda]. We will show that [N.sub.[lambda]] is even completely continuous for fixed [lambda] by Arzela-Ascoli theorem.

Let [OMEGA] be a bounded subset of K, i.e., [parallel]u[parallel] < C for all u [member of] [OMEGA]. Here C > 0 is a constant. Firstly, following the proof in Step 1, we can see that [N.sub.[lambda]][OMEGA] is closed and bounded, and there exist the following two inequalities

[absolute value of ([N.sub.[lambda]]u)(r)] [less than or equal to] [alpha] + [[integral].sup.1.sub.0] [[phi].sub.q] (1/[s.sup.n-1][M.sub.2]) ds (3.5)

and

[absolute value of ([N.sub.[lambda]](u)] [less than or equal to][[phi].sub.q](M2), (3.6)

where

[M.sub.2] = sup{[absolute value of f(r,u,0)];0[less than or equal to]r [less than or equal to] 1, -C [less than or equal to] u [less than or equal to]C}[[integral.sup.1.sub.0][t.sup.n-1]h(t)dt[[beta].sup.p-1]

We next show the equicontinuity of [N.sub.[lambda]][OMEGA] on [0,1]. For u [member of][OMEGA] and [r.sub.1], [r.sub.2][member of] [0,1], we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3.7)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3.8)

The equicontinuity of [N.sub.[lambda]][OMEGA] on [0,1] now follows from the inequality (3.7) and the equation (3.8). Therefore, the Arzela-Ascoli theorem implies that [N.sub.[lambda]] is completely continuous.

Step 3: We shall show that N([bar.U] x [0,1]) is contained in a compact subset of K, where

U = {u [member of] K; [parallel]u[parallel] [less than or equal to] M* + 1}, ([N.sub.0]u)(r)=[alpha] + [beta][[integral].sup.r.sub.0][[phi].sub.q]([1/[s.sup.n-1]])ds.

Let N([u.sub.n], [[lambda].sub.n]) be any sequence in N([bar.U] x [0,1]). Then it is easy to see that N([u.sub.n], [[lambda].sub.n]) is uniformly bounded and equicontinuous on [0, 1] since the inequality (3.5)-(3.7) and the equation (3.8) does not depend on the fixed [lambda], thus the Arzela-Ascoli theorem again yields the result. By Lemma 3.4, we deduce that [N.sub.1] has a fixed point, i.e., the problem (3.1) has a solution u(r) [member of] C([0,1]) [intersection] [C.sup.1] ((0,1]).

Summing up, we complete the proof of Lemma 3.5.

Lemma 3.3 and Lemma 3.5 implies that the problem (3.1) has an unique solution. According to this, in what follows, by using the shooting method, precisely speaking, by selecting [beta] > 0 suitably in the problem (3.1) such that the nonlocal boundary value condition (1.5) holds, we will show that the problem (1.4)-(1.6) admits a unique solution. Before going further, we need the following lemmas.

Lemma 3.6. If [u.sub.1] (r), [u.sub.2] (r) are two solutions of the problem (3.1) with [u'.sub.1](1) = [[beta].sub.1], [u'.sub.2](1) = [[beta].sub.2], and [[beta].sub.1] > [[beta].sub.2] > 0, then

[u.sub.1] (r) [greater than or equal to] [u.sub.2](r), r [member of] (0,1].

Proof. If the lemma were not true, there must exist one point [r.sub.0] [member of] (0, 1) such that [u.sub.1] ([r.sub.0]) < [u.sub.2]([r.sub.0]). Since [u.sub.1] (0) = [u.sub.2](0), it is easy to see that there exists another point [r.sub.1] [member of] (0, [r.sub.0]), such that [u.sub.1] ([r.sub.1]) < [u.sub.2]([r.sub.1]), [u'.sub.1] ([r.sub.1]) < [u'.sub.2]([r.sub.1]) by considering the continuity of the solution. According to Lemma 3.2, we conclude that [u'.sub.1] (1) < [u'.sub.2](l), namely, [[beta].sub.1] < [[beta].sub.2], which is a contradiction. The proof is complete. ?

Lemma 3.7. If [u.sub.1] (r), [u.sub.2] (r) are two solutions of the problem (3.1) with [u.sub.1](1) = [[beta].sub.1], [u.sub.2](1) = [[beta].sub.2], and [[beta].sub.1] > [[beta].sub.2] > 0, then

[u'.sub.1] (r) [greater than or equal to] [u'.sub.2](r), r [member of] (0,1).

Proof. If the lemma were not true, there exists a point [r.sub.0] [member of] (0,1) such that [u'.sub.1]([r.sub.0]) < [u'.sub.2]([r.sub.0]). According to Lemma 3.6, we see that [u.sub.1]([r.sub.0]) [greater than or equal to][u.sub.2]([r.sub.0]). We will show that [u'.sub.1]([r.sub.0]) > [u'.sub.2]([r.sub.0]). Otherwise, we have [u.sub.1]([r.sub.0]) = [u.sub.2]([r.sub.0]). Since that [u'.sub.1] ([r.sub.0]) < [u'.sub.2]([r.sub.0]) and [u.sub.1] (0) = [u.sub.2](0), therefore, there must exist one point b [member of] (0, [r.sub.0]) such that [u.sub.1](b) > [u.sub.2](b) and [u'.sub.1] (b) > [u'.sub.2](b) by considering the continuity of the solution. According to Lemma 3.2, we have [u.sub.1]([r.sub.0]) > [u.sub.2]([r.sub.0]) which is a contradiction. Since [u.sub.1](0) = [u.sub.2](0), [u.sub.2](r) - [u.sub.1](r) must have a minimal value point c [member of] (0, [r.sub.0]), hence we have [u.sub.1](c) > [u.sub.2](c), [u'.sub.1](c) = [u'.sub.2](c). By utilizing a similar method in Lemma 3.1, we see that [u".sub.1](c) > [u".sub.2](c), clearly, which contradicts the fact that c is a minimal value point of [u.sub.2](r) - [u.sub.1](r) . Therefore the proof is complete.

Combining Lemma 3.6 and Lemma 3.7, we obtain the following conclusion.

Corollary 3.1. Let [u.sub.1](r), [u.sub.2](r) be two solutions of the problem (3.1) with [u'.sub.1](1) = [beta], [u'.sub.2](1) = [[beta].sub.2], and [[beta].sub.1] > [[beta].sub.2] > 0. Then there exists a point b [member of] [0,1) such that

[u.sub.1](r) = [u.sub.2](r), 0 [less than or equal to] r [less than or equal to] b, (3.9)

[u.sub.1](r) > [u.sub.2](r), b < r [less than or equal to] 1, (3.10)

[u.sub.1](r) > [u'.sub.2](r), b < r [less than or equal to] 1. (3.11)

Proof. Let

b = sup {r; 0 [less than or equal to] r < 1, [u.sub.1] (r) = [u.sub.2](r)}.

By virtue of [u.sub.1](0) = [u.sub.2](0) and Lemma 3.1, we have 0 [less than or equal to] b < 1. Furthermore, together with Lemma 3.6, we also have

[u.sub.1] (r) = [u.sub.2](r), 0 [less than or equal to] r [less than or equal to] b, [u.sub.1] (r) > [u.sub.2](r), b < r [less than or equal to] 1.

So, it suffices to consider the inequality (3.11). If it were not true, then according to Lemma 3.7, there exists a point [r.sub.0] [member of] (b, 1) such that [u'.sub.1]([r.sub.0]) = [u'.sub.2] ([r.sub.0]). By utilizing a similar method to the proof of Lemma 3.1, we see that [u".sub.1] ([r.sub.0]) > [u".sub.2] ([r.sub.0]), which implies that the point [r.sub.0] is a minimal value point of [u.sub.2](r) - [u.sub.1](r) . Hence, there must exist a point [r.sub.1][member of](b,[r.sub.0]) such that [u'.sub.1]([r.sub.1]) < [u'.sub.2]([r.sub.1]), which contradicts Lemma 3.7. The proof is complete.

Combining with the above lemmas, we are now in a position to get the existence and uniqueness of solution of the problem (1.4)-(1.6) by the shooting methods. Define

k([beta]) = [[phi].sub.p]([u'.sub.[beta]](1))[[integral].sup.1.sub.0][s.sup.n-1][[phi].sub.p]([u'.sub.[beta]](s)) dg(s), [beta] > 0,

where [u.sub.[beta]] is the solution of the problem (3.1) with boundary value condition [u'.sub.[beta]](1) = [beta]. Then the desired unique solution of the problem (1.4)-(1.6) will be obtained by selecting an unique constant [beta]* > 0, such that k([beta]*) = 0. First of all, we need to establish the strict monotonicity and continuity of the function k([beta]) with respect to [beta].

Lemma 3.8. k([beta]) is continuous and strictly monotonous with respect to [beta] > 0.

Proof. Let [[beta].sub.1] > [[beta].sub.2] > 0. According to Corollary 3.1 and the first equation of the problem (3.1), we derive

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] are the solutions of the problem (3.1) with [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. By the assumption (H1), it is easy to see that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Thus

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

namely,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

By the above inequality, we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Furthermore, we can see the continuity of k([beta]) when [beta] > 0 from the following inequality

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Hence, the proof of Lemma 3.8 is complete.

Now, by virtue of the above established lemmas about the approximate problem (3.1), we are going to prove the main results in this paper.

Proof of Theorem 2.1. A simple calculation for the first equation of the problem (3.1) implies that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Noticing the assumption (H1)-(H3) and Fubini's Theorem, we see that, when [beta] > 1,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

On the other hand, when [beta] [less than or equal to] 1, let [delta] be a positive constant, which is small enough, and g([delta]) > 0. From the proof of Lemma 3.5, we know that if u is a solution of the problem (3.1), then there exists a constant M* > 0 such that [parallel]u[parallel] < M*, that is,

u(r) [less than or equal to] M* and u'(r) [less than or equal to] M* /[delta], r [member of] [[delta],1].

Applying the strict monotonicity of f and the positivity of f and h, we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3.13)

Combining the inequality (3.12) with the inequality (3.13), it follows

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Then recalling the strict monotonicity and continuity of k([beta]), there is one and only one [beta]* > 0, such that k([beta]*) = 0, and [u.sub.[beta]*] is the unique positive solution of nonlocal boundary value problem (1.4)-(1.6).

Finally, we consider the property of the solution of the problem (1.4)-(1.6). Suppose that there exists a point [r.sub.0] [member of] (0,1) such that y([r.sub.0]) is the local maximum value of y(r). Without loss of generality, we assume that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], then we see that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Following the proof of Lemma 3.1, however, we can obtain [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], which is a contradiction.

Summing up, we complete the Proof of Theorem 2.1.

Proof of Theorem 2.2. Suppose that the nonlocal boundary value problem (1.4)-(1.6) admits a solution u [member of] C([0,1]) [intersection] [C.sup.1((0,1])]. Then by a simple calculation to the equation (1.4), we see that u'(r) satisfies

u'(r) = [r.sup.-(n-1)/(p-1)][[phi].sub.q]([integral].sup.1.sub.r][s.sup.n-1]h(s)f(s, u, u')ds + [[phi].sub.p] (u' (1))), r [member of] (0,1].

Obviously, u'(r) is not integrable since p [less than or equal to] n, which contradicts the definition of u. Hence, we complete the proof of Theorem 2.2. ?

Received by the editors in June 2012 - In revised form in January 2013.

Communicated by P. Godin.

Acknowledgement The authors would like to express their deep thanks to the referees for their valuable suggestions for the revision of the manuscript. The authors also would like to thank Professor Jingxue Yin and Professor Chunpeng Wang, under whose guidance this paper was completed.

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* This work is partially supported by NSFC, partially supported by Ph.D. Specialities of Educational Department of China, partially supported by the Fundamental Research Funds for the Central Universities and the Research Funds of Renmin University of China (2010030171).

Hailong Ye

Yuanyuan Ke ([dagger])

School of Mathematical Sciences, South China Normal University, Guangzhou 510631, P.R. China

School of Information, Renmin University of China, Beijing 100872, P.R. China

Email: ke_yy@163.com.

([dagger]) Corresponding author.

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Author: | Ye, Hailong; Ke, Yuanyuan |
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Publication: | Bulletin of the Belgian Mathematical Society - Simon Stevin |

Article Type: | Report |

Geographic Code: | 9CHIN |

Date: | Dec 1, 2013 |

Words: | 5733 |

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