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
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
(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.
Very recently, Loc and Schmitt  considered the problem
where is the p-Laplace operator for . They assumed that the nonlinearity f is a continuous function on ℝ, , and there exist such that on and on for every . They proved that, for λ sufficiently large, if
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  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 , .
(a) 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
Of course the natural question is whether or not the similar results still hold for the corresponding problem involving the one-dimensional p-Laplacian
We shall answer these questions in the affirmative if . More precisely, we get the following theorem.
Theorem 1.1Let and let (H1), (H3), (H4) hold. Assume that
(H2′) either or and
(a) For all , there exists a solution of (1.7), and is the least eigenvalue of BVP
(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 .
To prove our main results, we use the uniqueness results due to Reichel and Walter  on the initial value problem
where and .
Lemma 2.1Let (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]. □
Lemma 2.2Let (H1) hold. Let , and let be such that
Then the initial value problem
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. □
Lemma 2.3Let be continuous. Letube a solution of the equation
with . Let be such that . Then
Proof Since g is independent of t, both and satisfy the initial value problem
By Lemmas 2.1 and 2.2, (2.5) has a unique solution defined on . Therefore, (2.4) is true. □
Lemma 2.4Let be a positive solution of the problem
with and . Let be such that . Then
(b) is the unique point on whichuattains its maximum;
(c) , .
Proof (a) Suppose on the contrary that , say , then
However, this is impossible since and in . Therefore .
(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
Multiplying both sides of the equation in (2.6) by and integrating from t to , we get that
This contradicts the facts that and . Therefore,
Similarly, we can prove that
(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 3.1For any , there exists a unique such that
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
Suppose that is a solution of (3.3), (3.4) with . Then
Putting , we obtain
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. □
Lemma 3.2Let (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 ,
see Zhang . Hence
Case 2. We show that for some implies that
In fact, (3.6) yields
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 ,
uniformly in v. Therefore, (3.9) implies
Therefore, (3.7) holds. □
From the definitions of and , we have that and for . Moreover, we have the following.
Lemma 3.3Let . Then
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
Then if ,
As , is a nondecreasing sequence of measurable functions. Therefore, by the monotone convergence theorem and the assumption , it follows that
Suppose next that . Then .
(ii) Let and . Since ,
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.
(b) follows from the continuity of and Lemma 3.3.
(c) is any solution of (3.1), (3.2) if and only if
Now, if , then
The authors declare that they have no competing interests.
RM completed the main study, carried out the results of this article. YL drafted the manuscript. AOMA checked the proofs and verified the calculation. All the authors read and approved the final manuscript.
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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