## Boundary Value Problems

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# On positive solutions to equations involving the one-dimensional p-Laplacian

Boundary Value Problems20132013:125

DOI: 10.1186/1687-2770-2013-125

Accepted: 29 April 2013

Published: 15 May 2013

## Abstract

We consider equations involving the one-dimensional p-Laplacian

with the Dirichlet boundary conditions. By using time map methods, we show how changes of the sign of lead to multiple positive solutions of the problem for sufficiently large λ.

MSC:34B10, 34B18.

### Keywords

positive solutions one-dimensional p-Laplacian uniqueness time map

## 1 Introduction

Let be continuous and change its sign. Let Ω be an open subset of with smooth boundary Ω. The semi-positone problems and their special cases
(1.1)
and
(1.2)

(and their finite difference analogues) have been extensively studied since early 1980s. Several different approaches such as variational methods, bifurcation theory, lower and upper solutions method and quadrature arguments have been successfully applied to show the existence of multiple solutions. See Brown and Budin [1], Peitgen et al. [2], Peitgen and Schmitt [3], Hess [4], Ambrosetti and Hess [5], Cosner and Schmitt [6], Dancer and Schmitt [7], Espinoza [8], Anuradha and Shivaji [9], Anuradha et al. [10], de Figueiredo [11], Lin and Pai [12], Clément and Sweers [13] and the references therein.

Very recently, Loc and Schmitt [14] considered the problem
(1.3)
where is the p-Laplace operator for . They assumed that the nonlinearity f is a continuous function on , , and there exist such that on and on for every . They proved that, for λ sufficiently large, if
(1.4)
then the problem (1.3) has at least positive bounded solutions which belong to the Sobolev space and are such that for each , where

In the special case that and , Brown and Budin [1] applied the quadrature arguments to get the following more detailed results.

Theorem A [[1], Theorem 3.8]

Assume that

(H1) ;

(H2) ;

(H3) There exists such that and for ;

(H4) If , there exist with such that and for , .

Then:
1. (a)

For all , there exists a solution of (1.2).

2. (b)
If , there exist at least two solutions of (1.2) such that

where
(1.5)
and
(1.6)
1. (c)
If is any solution of (1.2) such that , then

where .

Of course the natural question is whether or not the similar results still hold for the corresponding problem involving the one-dimensional p-Laplacian
(1.7)

We shall answer these questions in the affirmative if . More precisely, we get the following theorem.

Theorem 1.1 Let and let (H1), (H3), (H4) hold. Assume that

(H2′) either or and
(1.8)
Then:
1. (a)
For all , there exists a solution of (1.7), and is the least eigenvalue of BVP
(1.9)

2. (b)
If , there exist at least two solutions of (1.7) such that

3. (c)
If is any solution of (1.7) such that , then

where
(1.10)

We shall apply the time map method to show how changes of the sign of lead to multiple positive solutions of (1.7) for sufficiently large λ.

In the following, we extend f so that for all , then all the solutions of (1.7) are positive on .

## 2 Preliminaries

To prove our main results, we use the uniqueness results due to Reichel and Walter [15] on the initial value problem
(2.1)

where and .

Lemma 2.1 Let (H1) hold. If and , then the initial value problem (2.1) has a unique local solution. The extension remains unique as long as .

Proof It is an immediate consequence of Reichel and Walter [[15], Theorem 2]. □

Lemma 2.2 Let (H1) hold. Let , and let be such that
Then the initial value problem
(2.2)

has a unique local solution.

Proof (H1) implies that f is locally Lipschitzian. This together with the assumption and using [[15], (iii) and (v) in the case (β) of Theorem 4] yields that (2.2) has a unique solution in some neighborhood of a. □

Lemma 2.3 Let be continuous. Let u be a solution of the equation
(2.3)
with . Let be such that . Then
(2.4)
Proof Since g is independent of t, both and satisfy the initial value problem
(2.5)

By Lemmas 2.1 and 2.2, (2.5) has a unique solution defined on . Therefore, (2.4) is true. □

Lemma 2.4 Let be a positive solution of the problem
(2.6)
(2.7)
with and . Let be such that . Then
1. (a)

;

2. (b)

is the unique point on which u attains its maximum;

3. (c)

, .

Proof (a) Suppose on the contrary that , say , then
However, this is impossible since and in . Therefore .
1. (b)
Suppose on the contrary that there exists with and

We may assume that . The other case can be treated in a similar way.

If in the interval , then Lemma 2.3 yields that

This contradicts the boundary conditions . Therefore, in any subinterval of .

So, there exists , such that
Obviously,
Multiplying both sides of the equation in (2.6) by and integrating from t to , we get that
(2.8)
and subsequently,
This contradicts the facts that and . Therefore,
Similarly, we can prove that
1. (c)
Suppose on the contrary that there exists with . Then

This together with (2.8) implies that

This contradicts the facts that and . □

## 3 Proof of the main results

To prove Theorem 1.1, we need the following preliminary results.

Lemma 3.1 For any , there exists a unique such that
(3.1)
(3.2)

has a positive solution with . Moreover, is a continuous function on .

Proof By Lemma 2.4, is a positive solution of (3.1), (3.2) if and only if is a positive solution of
(3.3)
(3.4)
Suppose that is a solution of (3.3), (3.4) with . Then
and so
(3.5)
Putting , we obtain
(3.6)

Hence λ (if exists) is uniquely determined by ρ.

If , we define by (3.6) and by (3.5). It is straightforward to verify that u is twice differentiable, u satisfies (3.3), (3.4), in and . The continuity of is implied by (3.6) and this completes the proof. □

Let

Then .

Lemma 3.2 Let (H1) and (H2′) hold, and let . Then

where is the least eigenvalue of (1.9).

Proof We only deal with . The other one can be treated by the same method.

To this end, we divide the proof into two cases.

Case 1. We show that implies .

In this case, for any , there is a positive number R such that for . Thus, if , then
for . From (3.6), we have that for any ,
where
see Zhang [16]. Hence
Case 2. We show that for some implies that
(3.7)
where
(3.8)
In fact, (3.6) yields
(3.9)
for , where

We will show that the last integral in (3.9) converges to zeros as .

For , using l’Hospital’s rule, it follows that as ,
For ,
uniformly in v. Therefore, (3.9) implies
(3.10)

Therefore, (3.7) holds. □

From the definitions of and , we have that and for . Moreover, we have the following.

Lemma 3.3 Let . Then
1. (i)

;

2. (ii)

.

Proof (i) Suppose firstly that . Since S is open, and so there exists such that
Clearly k must be a local maximum for F and so . If , then
Let
Then if ,
Hence
As , is a nondecreasing sequence of measurable functions. Therefore, by the monotone convergence theorem and the assumption , it follows that

since .

Suppose next that . Then .

Since
Thus
1. (ii)
Let and . Since ,
(3.11)

Hence, if , then it follows from (3.11) that
Hence, if ,
where and denotes the characteristic function of . As is a nondecreasing sequence of measurable functions, by the monotone convergence theorem

□

Proof of Theorem 1.1 (a) follows from the continuity of and Lemma 3.2.
1. (b)

follows from the continuity of and Lemma 3.3.

2. (c)
is any solution of (3.1), (3.2) if and only if

Hence
Now, if , then
and so

□

## Declarations

### Acknowledgements

The authors are very grateful to the anonymous referees for their valuable suggestions. This work was supported by the NSFC (No. 11061030), NSFC (No. 11126296), SRFDP (No. 20126203110004) and Gansu Provincial National Science Foundation of China (No. 1208RJZA258).

## Authors’ Affiliations

(1)
Department of Mathematics, Northwest Normal University

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