Positive solutions for Sturm-Liouville BVPs on time scales via sub-supersolution and variational methods
© Zhang et al.; licensee Springer 2013
Received: 20 January 2013
Accepted: 26 April 2013
Published: 13 May 2013
This paper is concerned with the existence of one and two positive solutions for the following Sturm-Liouville boundary value problem on time scales
Under a locally nonnegative assumption on the nonlinearity f and some other suitable hypotheses, positive solutions are sought by considering the corresponding truncated problem, constructing the variational framework and combining the sub-supersolution method with the mountain pass lemma.
The theory of dynamic equations on time scales has become a new important mathematical branch [1, 2] since it was initiated by Hilger in 1988 . Since then, boundary value problems (BVPs) for dynamic equations on time scales have received considerable attention, various fixed point theorems, sub-supersolution method and Leray-Schauder degree theory have been applied to get many interesting results about the existence of solutions; see [1, 2, 4–12] and the references therein. Variational method is also an important method for dealing with the existence of BVPs. Recently, some authors have used the theory to study the existence of solutions of some BVPs on time scales [4, 5, 13, 14].
with the Dirichlet boundary condition. They gave some sufficient conditions for the existence of single and multiple positive solutions by using the variational method and critical point theory.
and obtained the existence of many solutions depending on the value of the parameter λ which lie in some different intervals under some suitable hypotheses. The main approaches are also the variational method and some known critical point theorems and a three critical point theorem established in . Erbe et al.  also established some existence criteria of positive solutions by a fixed point theorem in a cone with the globally nonnegative hypothesis of f.
Here is a time scale and , , , , , , , and . The purpose of this paper is to discuss the existence and multiplicity of positive solutions to the problem (1.2) under the local non-negativity assumption of f and some other hypotheses. The main tools are the truncated method, the variational method, the sub-supersolution method and the mountain pass lemma. First, inspired by the method in , we convert the existence of a positive solution of (1.2) to the existence of a solution of an associated problem of (1.2). In contrast with the paper , the appearance of term , our problem is more complicated and the proof is also different from  (see Lemma 2.3 for details). Next, we construct a supersolution of (1.2) and give the existence of one positive solution. Finally, under our weaker assumption on f (see (H1) and (H2) for details), since we cannot verify that the corresponding functional for the associated problem satisfies the P.S. condition, we consider the corresponding truncated problem. To prove the existence of the second positive solution by the mountain pass lemma, we also give an estimate of a nonnegative solution of (1.2) and prove the solution of a truncated problem is also a solution of (1.2) for n large enough (see Theorem 3.3 for details). To the best of our knowledge, the results are new both in the continuous and in the discrete case.
The paper is organized as follows. In Section 2, we present some basic properties of some related Sobolev space on time scales, construct the variational framework, give some properties of this framework and some necessary lemmas. In Section 3, we firstly get the existence of a single positive solution of (1.2) by using the sub-supersolution method; then applying the truncated method, analytic technique and mountain pass lemma, we establish the existence of two positive solutions.
2 Preliminaries and variational formulation
In this section, we list the definition and properties of the Sobolev space on time scales , give some lemmas which we need for the proof of the main result and construct a variational framework.
We know the immersion is compact. Analogous to the proof in the real numbers situation, one can deduce the following result on time scales.
Using Lemma 2.1, by a similar proof of , one can derive the following.
Lemma 2.2 If , then and , Δ-a.e. in , where , .
First, we give an important lemma.
Lemma 2.3 If for , u is a solution of (2.1), then u is nonnegative in . Furthermore, if for , and is nondecreasing in , then , .
Therefore, for .
Next, we show that if is nondecreasing in , and for , then , .
for and ,
for and there exists such that for .
which contradicts (2.4).
If , then one can deduce that , . From this and , we have a contradiction.
If , then we always have . If , then , . Similar to (i), we get (2.2) and (2.3). But this is impossible from (2.4) and (2.5).
If and , then , , . So, we get a contradiction to (2.4) and (2.5).
If and , we have , . Hence, we get (2.2) and (2.3). But this contradicts (2.4) and (2.5).
If and , then . By assumption, u is a solution of (1.2), so we have , which contradicts and for . □
Remark 2.4 From the proof of Lemma 2.3, we can easily find that if , then the monotonicity assumption of can be omitted.
By Lemma 2.3, under the hypothesis
(H1) is nondecreasing in , and for ,
Note that the appearance of the term in the functional I guarantees that I is , see next lemma. By the definition of Fréchet derivative and the fact that the immersion is compact, we have the following results.
- (i), and for every ,
- (ii)We define
The solutions of (2.1) match up to the critical points of I in E.
we have the following lemma.
Lemma 2.6 [, Lemma 3.1]
respectively, and satisfy , , , , , , , .
belongs to , where is given in Remark 2.7.
Proof Clearly, is well defined in .
If , by and , we have . Hence .
Hence, is bounded for s close to 0.
Similarly, we can derive that is bounded for s close to . □
In order to derive the main result, we list the following well-known mountain pass lemma.
Lemma 2.10 [, Theorem 6.1]
there exist , such that for with ;
there is such that and .
Then is a critical value.
3 Main results
In this section, we establish some existence criteria of a positive solution of (1.1) by employing the sub-supersolution method and critical point theory.
First, using a method analogous to that in , we construct a supersolution to employ the sub-supersolution method.
Then the problem (1.2) has at least one positive solution.
Then, from the definition of , we have . Then, by the assumptions, it is easy to see that v is a supersolution of (1.2). In addition, condition (H1) guarantees that the constant function 0 is a strict subsolution of (1.2). Therefore, the sub-supersolution method implies (1.2) has a positive solution . □
Theorem 3.3 Under the hypothesis of Theorem 3.1 and suppose the condition
(H2) uniformly for
holds, then the problem (1.2) has at least two positive solutions.
In order to prove this theorem, we first present some necessary lemmas.
Lemma 3.4 Let v, be given in the proof of Theorem 3.1, then is a local minimizer of I in E.
By the assumptions and Lemma 2.8, we easily find , . Hence is a local minimizer of I in W.
Next, by a similar argument to that in , we assert that is also a local minimizer of I in E.
It is easy to show that as in . But this contradicts the fact that is a local minimizer of I in W. □
Next, under hypothesis (H2), in order to show the existence of the second positive solution of (1.2) by employing the mountain pass lemma, we need to show that I satisfies the P.S. condition. However, by (H2), we cannot justify this; therefore, we consider the truncation function and the truncation functional defined as follows.
Lemma 3.5 Assume that (H1) and (H2) hold, then there exists such that the functional satisfies the P.S. condition in E for .
Proof For given n, let be the P.S. sequence of , that is, is bounded and as . If is bounded, one can deduce that I satisfies the P.S. condition by a similar proof to Proposition B.35 in .
by the definition of and passing to limit in (3.4), one can derive that .
Passing to the limit in (3.6), we know . Hence , which contradicts . Therefore is bounded. So, satisfies the P.S. condition for . □
By Lemmas 3.4 and 3.5, we deduce that for n large enough, has a nontrivial critical point by using the mountain pass lemma and Theorem 1 in . In order to obtain a solution of (1.2), we need to get an estimate of . Therefore, we first give an estimate of a nonnegative solution of (1.2) employing a method similar to that in .
Lemma 3.6 Suppose (H2) holds, then there is such that for any nonnegative solution u of (1.2), we have .
Hence, there exists such that . □
Remark 3.7 Note that we only need (H2) to derive (3.8). Hence, (3.5) implies Lemma 3.6 is also valid for the truncation problem.
Proof of Theorem 3.3 Since the positive solution derived from Theorem 3.1 is a local minimizer of I and , we can choose large enough such that for every . Hence, is also a local minimizer of for . Then, from the definition of and Lemma 3.5, we know the mountain pass lemma and Theorem 1 in  imply that has the second critical point . Furthermore, by Remark 3.7, we know is also a critical point of I. Thus the problem (1.2) has the second positive solution . □
Dedicated to Professor Hari M Srivastava.
The research of HR Sun has been supported by the program for New Century Excellent Talents in University (NECT-12-0246) and FRFCU (lzujbky-2013-k02).
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