- Open Access
On positive solutions to equations involving the one-dimensional p-Laplacian
© Ma et al.; licensee Springer 2013
- Received: 15 October 2012
- Accepted: 29 April 2013
- Published: 15 May 2013
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.
- positive solutions
- one-dimensional p-Laplacian
- time map
(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 , Peitgen et al. , Peitgen and Schmitt , Hess , Ambrosetti and Hess , Cosner and Schmitt , Dancer and Schmitt , Espinoza , Anuradha and Shivaji , Anuradha et al. , de Figueiredo , Lin and Pai , Clément and Sweers  and the references therein.
In the special case that and , Brown and Budin  applied the quadrature arguments to get the following more detailed results.
Theorem A [, Theorem 3.8]
(H3) There exists such that and for ;
(H4) If , there exist with such that and for , .
For all , there exists a solution of (1.2).
- (b)If , there exist at least two solutions of (1.2) such that
- (c)If is any solution of (1.2) such that , then
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
- (a)For all , there exists a solution of (1.7), and is the least eigenvalue of BVP(1.9)
- (b)If , there exist at least two solutions of (1.7) such that
- (c)If is any solution of (1.7) such that , then
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 .
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 [, Theorem 2]. □
has a unique local solution.
Proof (H1) implies that f is locally Lipschitzian. This together with the assumption and using [, (iii) and (v) in the case (β) of Theorem 4] yields that (2.2) has a unique solution in some neighborhood of a. □
By Lemmas 2.1 and 2.2, (2.5) has a unique solution defined on . Therefore, (2.4) is true. □
is the unique point on which u attains its maximum;
- (b)Suppose on the contrary that there exists with and
We may assume that . The other case can be treated in a similar way.
This contradicts the boundary conditions . Therefore, in any subinterval of .
- (c)Suppose on the contrary that there exists with . Then
This contradicts the facts that and . □
To prove Theorem 1.1, we need the following preliminary results.
has a positive solution with . Moreover, is a continuous function on .
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. □
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 .
We will show that the last integral in (3.9) converges to zeros as .
Therefore, (3.7) holds. □
From the definitions of and , we have that and for . Moreover, we have the following.
Suppose next that . Then .
- (ii)Let and . Since ,(3.11)
follows from the continuity of and Lemma 3.3.
- (c)is any solution of (3.1), (3.2) if and only if
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).
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