0 Boundary Value Problems

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Multiplicity of positive solutions of superlinear semi-positone singular Neumann problems

Qiuyue Li1*, Fuzhong Cong1, Zhe Li2 and Jinkai Lv1

Correspondence: liqy609@163.com 1 Department of Foundation Courses, Aviation University of Airforce, Renmin Street 7855, Changchun, 130012, China Fulllist of author information is available at the end of the article

Abstract

Introduction: Neumann boundary value problems have been studied by many authors. We are mainly interested in the semi-positone case. This paper deals with the existence and multiplicity of positive solutions of a superlinear semi-positone singular Neumann boundary value problem.

Preliminaries: The proof of our main results relies on a nonlinear alternative of Leray-Schauder type, the method of upper and lower solutions and on a well-known fixed point theorem in cones.

Main results: We obtained the existence of at least two different positive solutions.

Keywords: positive solutions; superlinear; semi-positone; singular; Neumann problem

ft Spri

ringer

1 Introduction

We will be concerned with the existence and multiplicity of positive solutions of the superlinear singular Neumann boundary value problem in the semi-positone case

j-(p(x)u')' + q(x)u = g(x, u), x e I =[0,1], ....

[u'(0) = 0, u'(1) = 0.

Here the type of perturbations g(x, u) maybe singular near u = 0 andg (x, u) is superlinear near u = +cc. From the physical point of view, g(x, u) has an attractive singularity near u = 0 if

lim g(x, u) = uniformly in x

and the superlinearity of g(x, u) means that

lim g(x, u)/u = +cc uniformly in x.

By the semi-positone case of (1.1), we mean that g(x, u) may change sign and satisfies F(x, u) = g(x, u)+M > 0 where M > 0 is a constant.

© 2014 Li et al.; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the originalworkis properly credited.

It is well known that the existence of positive solutions of boundary value problems has been studied by many authors in [1-6] and references therein. They mainly considered the case of p(x) = 1 and q(x) = 0. In [7], the authors studied positive solutions of Neumann boundary problems of second order impulsive differential equations in the positone case, based on a nonlinear alternative principle of Leray-Schauder type and a well-known fixed point theorem in cones. This paper attempts to study the existence and multiplicity of positive solutions of second order superlinear singular Neumann boundary value problems in the semi-positone case. The techniques we employ here involve a nonlinear result of Leray-Schauder, the well-known fixed point theorem in cones and the method of upper and lower solutions. We prove that problem (1.1) has at least two different positive solutions. Moreover, we do not take the restrictions p(x) = 1 or q(x) = 0.

Throughout this paper, we assume that the perturbed partg(x, u) satisfies the following hypotheses:

(Hi) g(x, u) e C(I x R+,R+),p(x) e C1(I), q(x) e C(I),p(x) > 0, q(x) > 0.

(H2) There exists a constant M >0 such that F(x, u) = g(x, u)+ M > 0 for all x e I and

In Section 2, we perform a study of the sign of the Green's function of the corresponding linear problems

\-(p(x)u')' + q(x)u = h(x), x e I, \u! (0) = 0, u'(1) = 0.

In detail, we construct the Green's function G(x, y) and give a sufficient condition to ensure G(x,y) is positive. This fact is crucial for our arguments. We denote

A = min G(x,y), B = max G(x,y), a = A/B. (1.3)

(x,y)eIxI (x,y)eIxI

We also use m(x) to denote the unique solution of (1.2) with h(x) = 1, m(x) = /0 G(x,y) dy. In Section 3, we state and prove the main results of this paper.

2 Preliminaries

For the reader's convenience we introduce some results of Green's functions. Let Q = I x I, Q1 = {(x,y) e Q|0 < x < y < 1}, Q2 = {(x,y) e Q|0 < y < x < 1}. Considering the homogeneous boundary value problem

i-(p(x)u')' + q(x)u = 0, x e I,

\u! (0) = 0, u'(1) = 0, (.)

and let G(x,y) be the Green's function of problem (2.1). Then G(x,y) can be written as

u e (0, <x>).

G(x, y)

(x,y) e Q1, (x,y) e Q2,

where m and n are linearly independent, and m, n and m satisfy the following lemma.

Lemma 2.1 [7] Suppose that (H.) holds and problem (2.1) has only zero solution, then there exist two functions m(x) and n(x) satisfying:

(i) m(x) e C2(I, R) is increasing and m(x) >0, x e I;

(ii) n (x) e C2 (I, R) is decreasing and n (x)> 0, x e I;

(iii) Lm = -(p(x)m')' + q(x)m = 0, m(0) = 1, m'(0) = 0;

(iv) Ln = -(p(x)n')' + q(x)n = 0, n(1) = 1, n'(1) = 0;

(v) w = p(x)(m' (x)n(x) - m(x)n'(x)) is a positive constant.

Lemma 2.2 [7] The Green's function G(x, y) defined by (2.2) has the following properties:

(i) G(x, y) is continuous in Q;

(ii) G(x, y) is symmetrical on Q;

(iii) G(x, y) has continuous partial derivatives on Qi, Q2;

(iv) For each fixed y e I, G(x, y) satisfies LG(x, y) = 0for x = y, x e I. Moreover, G'x(0,y) = G'x(1,y) = 0for y e (0,1).

(v) For x = y, G'x has discontinuity point of the first kind, and

G'x(y + 0,y) - G'x(y-0,y) = -p!), y e (0,1).

Lemma 2.3 [8] Suppose that conditions in Lemma 2.1 hold and h : I ^ R is continuous. Then the problem

-(p(x)u')' + q(x)u = h(x), x e I, u (0) = 0, u'(l) = 0,

has a unique solution, which can be written as u(x)= I G(x,y)h(y) dy.

Next we state the theorem of fixed points in cones, which will be used in Section 3.

Theorem 2.1 [9] Let X be a Banach space andK (c X) be a cone. Assume that are

open subsets ofX with 0 e & 1 c &2, and let

T: K n (&2\&1) ^ K

be a continuous and compact operator such that either

(i) ||Tu|| > ||uM, u e Kn 9&1 and \\Tu\\ < ||uM, u e Kn d^; or

(ii) ||Tu| < ||u|, u e K n d&1 and ||Tu|| > ||u||, u e K n d&2. Then T has a fixed point in K n (&2\&1).

In applications below, we take X = C(I) with the supremum norm || • || and define K = \ u e X: u(x) > 0 and min u(x) > a ||u|| [. (.5)

One may readily verify that K is a cone in X. Now suppose that F: I x R ^ [0, to) is continuous and define an operator T: X ^ X by

(Tu)(x) = i G(x,y)F(y, u(y)) dy (2.6)

for u e X and x e [0,1].

Lemma 2.4 T is well defined and maps X into K. Moreover, T is continuous and completely continuous.

3 Main results

In this section we establish the existence and multiplicity of positive solutions to (1.1). Since we are mainly interested in the attractive-superlinear nonlinearities g(x, u) in the semi-positone case, we assume that the hypotheses of the following theorem are satisfied.

Theorem 3.1 Suppose that (H1) and (H2) hold. Furthermore, assume the following: (H3) There exist continuous, non-negative functions f (u) and g(u) such that

F(x, u) = g(x, u)+ M < f (u) + h(u) for all (x, u) e I x (0, to),

andf (u) > 0 is non-increasing and h(u)/f (u) is non-decreasing in u e (0, to). (H4) There exists r > such that-r-rr-r > IMI.

V ' a f (a r-M||®||){1+ hi)}

(H5) There exists a constant A > M, s >0 such that

F(x, u) > A, f (u) > A for all (x, u) e I x (0, s]. Then problem (1.1) has at least one positive solution v e C(I) with 0 < ||v + Mw|| < r.

Before we present the proof of Theorem 3.1, we state and prove some facts. First, it is easy to see that we can take c >0 and no > 1 such that

i A - M1 , x

c|NI < min s,iqri' (()

1 i s }

— < min< s, —, ca ||w||, ar -M||w||}. (.2)

n0 r M' " ", " "J ( )

Lemma 3.1 Suppose that (H1)-(H5) hold, then a(x) = (M + c)v(x) is a strict lower solution to the problem

I-(p(x)u')' + q(x)u = Fn(x, u -Mm(x)), x e I, n > n0, u (0) = 0, u'(1) = 0,

where Fn(x, u) = F(x,max{u, n}), (x, u) e I x R.

Proof It is easy to see that a'(0) = (M + c)w'(0) = 0 and a'(1) = (M + c)w'(1) = 0.

Since a(x) - Mw(x) = cw(x) > ca ||w| > > n, and using (3.1), we have e > a(x) -Mw(x) = cw(x) > n > 0. By assumption (H5), we have

Fn(x, a(x) -M«(x)) > A, Vn > n0.

This implies that a(x) is a strict lower solution to (3.3). □

Lemma 3.2 Suppose that (H1)-(H5) hold. Then the problem

-(p(x)u')' + q(x)u = fn(u - Mw(x))(1 + x e I, (3^)

u (0) = 0, u'(1) = 0, .

has at least one positive solution Pn(x) with ||Pn || < r.

Proof The existence is proved using the Leray-Schauder alternative principle together with a truncation technique. Since (H4) holds, we have

||w|f(ar - M||w||) (1 + h(r)/f (r)) < r.

Consider the family of problems

J-(p(x)u')' + q(x)u = Xfn(u -Mw(x))(1 + hfy), x e I, (35)

\u' (0) = 0, u'(1) = 0, .

where X e I andfn(u) =f (max{u, 1/n}), (x, u) e I x R.fn(u) is non-increasing. Problem (3.5) is equivalent to the following fixed point problem in C[0,1]

P = XTnfi, (3.6)

where Tn is defined by

Tn(P(x)) = f G(x,y)fn{p(y)-Mw(y))(1 + h(r)/f (r)) dy. (3.7)

We claim that any fixed point P of (3.6) for any X e [0,1] must satisfy ||P|| = r. Otherwise, assume that P is a solution of (3.6) for some X e [0,1] such that ||P || = r. Note that fn(x, u) > 0. By Lemma 3.1, for all x, P(x) -Mw(x) > ar - M||«|| > 1/n. Hence, for allx,

P(x)-Mw(x) > 1/n and P(x)-Mw(x) > ar -M||w||. (3.8)

Then we have, for all x,

P(x) = xfo G(x,y)fn(P(y)-Mw(y))^ 1+ jT^) dy < ^ 1 G(x,y)f (P(y) -Mw(y))( 1 + hT^ dy

< / G(x,y)f (ar - M|M|) (1 + h(r)/f (r)) dy

< |Mf (a r - MINI) (1 + h(r)/f(r)).

Therefore,

r = ||P1| < |Mf (ar - M|M|)(l + h(r)/f (r)) < r.

This is a contradiction and the claim is proved.

From this claim, the nonlinear alternative of Leray-Schauder guarantees that problem (3.6) (with X = l) has a fixed point, denoted by pn, in Br, i.e., problem (3.4) has a positive solution pn with ||pn|| < r. (Infact, itis easyto seethatpn(x) > l/n with ||^„|| = r.)

Lemma 3.3 Suppose that (Hl)-(H5) hold, then ¡3n(x) is an upper solution of problem (3.3).

Proof By Lemma 3.2 we know that jin(x) is a solution to equation (3.4). If (x) -Mrn(x) > ni, then

Fn(x, pn(x)-Ma>(x)) = F(x, pn(x)-Mw(x))

(3.10)

If fin(x) -Mrn(x) < n, then

(3.11)

Since (0) = (1) = 0, we have

-(p(x)fin(x))' + q(x)fin(x) > Fn(x,Pn(x) -Mw(x)), x e I,

P'n (o) = o, pn (1) = 0.

This implies that pn(x) is an upper solution of problem (3.3).

Lemma 3.4 Suppose that (H1)-(H5) hold, then pn(x) > a(x) (n > n0).

Proof Let z(x) = a(x)-pn(x), we will prove z(x) < 0.Ifthisisnottruefor n > n0,thereexists x0 e [0,1] such that z(x0) = maxz(x) > 0, z'(x0) = 0, z"(x0) < 0. Then (p(x0)z'(x0))' < 0.

WVA0) > cu n^ii > 1 > 1

is non-increasing, we have

Since a(x0) -M«(x0) = c«(x0) > cu ||«|| > ^ > ¿, a(x0) -Mrn(x0) < c||«|| < e, and/n(u)

/«(P(x0)-Mw(x0)) >/„(a(x0)-M«(x0)) =/(a(x0) -M«(x0))

> A (3.12)

^p(x0)z'(x0^ + q(x0)z(x0) = M + c -/„(P„(x0)-M«(x0))^1 + ^r)

/ h(r) \

< M + c -/„ (a(x0) - Mrn(x0)) + — J

< M + c -A^ 1+ ^<0. (3.13)

This is a contradiction and completes the proof of Lemma 3.4. □

Proof o/Theorem 3.1 To show (1.1) has a positive solution, we will show

(3.14)

{-(p(x)u')' + q(x)u = F(x, u(x) - Mrn(x)), x e I, u' (0) = u'(l) = 0

has a solution u e C(I), u(x) > Ma>(x), x e I.

If this is true, then v(x) = u(x) - Ma>(x) is a positive solution of (l.l) since

-(p(x)v')' + q(x)v = —p(x)u'(x) -p(x)Mw'(x))' + q(x)u(x) - Mq(x)a(x) = —p(x)u'(x))' + q(x)u(x) -M = F(x, u(x) - Mw(x)) - M = g(x, u(x) - M«(x)) = g(x, v(x)).

As a result, we will only concentrate our study on (3.14).

By Lemmas 3.1-3.4 and the upper and lower solutions method, we know that (3.3) has a solution un with (M + c)rn(x) = a(x) < un(x) < P„(x) < r. Thus we have un(x) - Ma>(x) > ca ||«||, u„(x) < P„(x) < r.

By the fact that u„ is a bounded and equi-continuous family on [0,1], the Arzela-Ascoli theorem guarantees that [u„}neN0 has a subsequence {u„k}keN, which converges uniformly on [0,1] to a function u e C[0,1]. Then u satisfies u(x) - Ma>(x) > ca ||«||, u(x) < r for all x. Moreover, u„k satisfies the integral equation

t (x)= /" G (x, y)F (y, unk (y) - Mw(yj) dy.

Letting k ^ to, we arrive at

u(x) = I G(x,y)F(y, u(y) - Mrn(y)) dy, Jo

where the uniform continuity of F(x, u(x)-Mm(x)) on [0,1] x [ca ||m||, r] is used. Therefore, u is a positive solution of (3.14).

Finally, it is not difficult to show that ||u|| < r. Assume otherwise: note that F(x, u) > 0. By Lemma 2.4, for all x, u(x) > 1/n and r > u(x) -Mm(x) > ar - M||m|| > 1/n. Hence, for all x,

u(x)-Mw(x) > 1/k and r > u(x)-Mrn(x) > ar - (3.15)

Then we have for all x,

u(x) = I G(x, y)F(y, u(y)-Ma>(y)) dy

, 11 G(x, yif <u(y)-M.<y))(I) dy

< I G(x,y)f <ar - M||w|)<1 + h(r)/f (r)) dy

< Mf <ar - M|M|)<1 + h(r)/f(r)). (.16) Therefore,

r = ||u||< Mf <ar -M|m|)<1 + h(r)/f(r)). This is a contradiction and completes the proof of Theorem 3.1. □

Corollary 3.1 Let us consider the following boundary value problem

{-(p(x)u')' + q(x)u = ¡x(u a + u + k(x)), x e I, u (0) = u'(1) = 0,

where a > 0, ji >0 and k: [0,1] ^ Ris continuous, ¡x > 0 is chosen such that

u(au - M|M|)a X < sup -—,

ue(MM.,to) !MH1 + 2Hua + ua+j}

here H = ||k||. Then problem (3.17) has a positive solution u e C[0,1]. Proof We will apply Theorem 3.1 with M = ¡¡H and

f (u) =fi(u) = ¡u-a, h(u) = uj + 2H), hi(u) = ¡uj. Clearly, (H1)-(H3) and (H5) are satisfied.

(3.17)

. u(au - M|M|)a /MM \

T(u) =-—, u e-,+to .

|M|{1 + 2Hua + ua+j} \ a )

Since T() = 0, T(to) = 0, then there exists r e (, to)such that

T (r) = su u(a u - MIM)a

ue(MM,to) I«I{l + 2Hua + ua+j}•

This implies that there exists r e (, to) such that ^ < ||mrfiW(rHra}, s° H is satisfied.

Since j > l. Thus all the conditions of Theorem 3.1 are satisfied, so the existence is guaranteed. □

Next we will find another positive solution to problem (1.1) by using Theorem 2.1.

Theorem 3.2 Suppose that conditions (H1)-(H5) hold. In addition, it is assumed that the following two conditions are satisfied:

(H6) F (x, u) = g(x, u)+ M > f1(u) + h1(u) for some continuous non-negative functions f(u) and h1(u) with the properties that f1(u) > 0 is non-increasing and h1(u)/f1(u) is non-decreasing.

(H7) There exists R > r such that hR(aR-M||(||h < ||(1.

af1(R){1+ A1M-M||((||) }

Then, besides the solution u constructed in Theorem 3.1, problem (1.1) has another positive solution v e C[0,1] with r < || v + M(l < R.

Proof To show (1.1) has a positive solution, we will show (3.14) has a solution u e C[0,1] with u(x) > M((x) for x e [0,1] and r < ||u|| < R. Let X = C[0,1] and K be a cone in X defined by (2.5). Let

¿r = {u e U: nun < r}, ¿R = {u e X: ||u! < R}

and define the operator T: K n (¿2R \ ¿r) ^ K by

(Tu)(x) = i G(x,y)F(y, u(y) -Mrn(y)) dy, 0 < x < 1, (3.19)

where G(x,y) is as in (2.2).

For each u e K n (¿2R \ ¿r)r < ||Un < R, we have 0 < ar - M|(|| < u(x) -Ma>(x) < R. Since F: [0,1] x [ar - M|(||,R] ^ [0, to) is continuous, it follows from Lemma 2.4 that the operator T: K n (¿2R \ ¿r) ^ K is well defined, is continuous and completely continuous. First we show

||Tu|| < ||un for u e K n dQr. (.20)

In fact, if ü e K n dQr, then \\ü\\ = r and ü(x) > ar > M\\«\\ for x e I. So we have (Tü)(x) = i G(x,y)F(y, ü(y)-Mo(y)) dy

I! G(x, y)f (ü ö™))1 + dy

G(x, y)f (a r - M\\o\\){ 1+ Kryj dy = o(x)f (ar -M\\o\\){ 1+ hr)]

< \\o\f (a r - M\\«\\){ 1+ ^J

< r = \\u\\.

This implies ||ru|| < ||ui.e., (3.20) holds. Next we show

HTuH > ||u|| for ue Kn dQR. (.21)

To see this, let ue K n 3&R, then 11 u | = R and u (x) > a R > M||«|| for x e I .Asa result, it follows from (H6) and (H7) that, for x e I,

(Tu)(x) = f G(x,y)F(y, u(y)-Mw(y)) dy J0

> J; G(x, y)f1(u «-**)){ 1 + |||fM||} dy

> f1 G(x,y)/1(R)(1+ ^R - ^ ) dy

-j0 (,q /1(ar - M|M|)Jy

h (aR - M|M|) /1(a R - M||«||)

h1(aR -M||«|

= «(x]fi(R)j1 +

> a\\o\fiCR) 1 +

fi(aR -M\\o\\ > R = \\ü\.

Now (3.20), (3.21) and Theorem 2.1 guarantee that T has a fixed point ü e K n (¿2R \ ) with r < \\ü\\ < R. Clearly, this ü is a positive solution of (3.14). This completes the proof of Theorem 3.2. □

Let us consider again example (3.17) in Corollary 3.1 for the superlinear case, i.e., a >0, ß >1 and k: [0,1] ^ R is continuous, ¡x > 0 is chosen suchthat (3.18) holds, here H = \\k\\. Then problem (3.17) has a positive solution ü e C[0,1]. Clearly, (Hi)-(H6) are satisfied.

Since ß >1, then (H7) is satisfied for R large enough because when R ^ro,

R Ra+1

af1(R){1 + ^(^f:!) r ax(1 + (aR -M\\«\\)a+ß)

Thus all the conditions of Theorem 3.2 are satisfied, so the existence is guaranteed.

Corollary 3.2 Assume that a > 0, j >1 and k : I ^ R is continuous, ¡i >0 is chosen such that (3.18) holds. Take H = ||k||. Then problem (3.17) has at least two different positive solutions.

Competing interests

The authors declare that they have no competing interests.

Authors' contributions

All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.

Author details

1 Department of Foundation Courses, Aviation University of Airforce, Renmin Street 7855, Changchun, 130012, China.

2Department of Aviation Mechanical Engineering, Aviation University of Airforce, Nanhu Road 2222, Changchun, 130012,

China.

Acknowledgements

The authors express their thanks to the referee for his valuable suggestions. The work was supported by the National

Natural Science Foundation of China (No: 11171350).

Received: 5 February 2014 Accepted: 12 September 2014 Published online: 08 October 2014

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doi:10.1186/s13661-014-0217-0

Cite this article as: Li et al.: Multiplicity of positive solutions of superlinear semi-positone singular Neumann problems. Boundary Value Problems 2014 2014:217.

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