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Positive solutions for Sturm-Liouville BVPs on time scales via sub-supersolution and variational methods
Boundary Value Problems volume 2013, Article number: 123 (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.
In , we considered the problem
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.
with the norm
We know the immersion is compact. Analogous to the proof in the real numbers situation, one can deduce the following result on time scales.
Lemma 2.1 If , , , then
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 , .
For convenience, we denote
and for , we set
In order to discuss the existence of a positive solution of (1.2), we consider the following problem:
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 , .
Proof Let u be a solution of (2.1). In view of Lemma 2.2, we know and , Δ-a.e. in . Multiplying (2.1) by , integrating over and employing the integration by parts formula for an absolutely continuous function on , we find that
Therefore, for .
Next, we show that if is nondecreasing in , and for , then , .
In fact, if the conclusion is false, we can suppose that there exists such that and one of the following two cases holds:
for and ,
for and there exists such that for .
For the case (i), if , then , there exists such that for . According to the nonnegativity of u on and , it is easy to see that is impossible. Thus we have
If , then we know , and . If , then , . Hence, in this case, we always have
However, since u is a solution of (1.2), (2.2) and , we know that
which contradicts (2.4).
For the case (ii), it can be divided into two cases to consider.
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 ,
in order to prove the existence of a positive solution of (1.2), it suffices to consider the existence of a solution of (2.1). Now we establish the corresponding variational formulations for (2.1). We set
then E is a Banach space with the norm , and we can find that can be taken as an equivalent norm on E. Define the functional I on E as
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.
Lemma 2.5 The following statements are valid.
, and for every ,
J is weakly continuous in E and is compact.
The solutions of (2.1) match up to the critical points of I in E.
For the eigenvalue problem
we have the following lemma.
Lemma 2.6 [, Lemma 3.1]
The eigenvalues of (2.6) may be arranged as , and there exists a countable orthonormal basis of E consisting of eigenfunction associated eigenvalues of (2.6) and
Remark 2.7 By (2.7) and Lemma 2.2, we know the eigenfunction corresponding to the eigenvalue satisfies for . Furthermore, by the Krein-Rutman theorem [, Theorem 7.C], we know with
The problem (1.2) has the Green function
where , ψ, φ are solutions of
respectively, and satisfy , , , , , , , .
Lemma 2.9 The function defined by
belongs to , where is given in Remark 2.7.
Proof Clearly, is well defined in .
If , by and , we have . Hence .
If , then . By Remark 2.7, we know . If , then . If , then there exists such that , for , then by L’Hôspital rule [, Theorem 1.119] and Lemma 2.8, we know
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]
Suppose satisfies the P.S. condition. Suppose and
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.
Theorem 3.1 Assume that (H1) holds and there are constants and such that
where is fixed, , represents the unique positive solution of
Then the problem (1.2) has at least one positive solution.
Proof For fixed , let v be the unique positive solution of
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 . □
Remark 3.2 Furthermore, by Lemma 2.8, we know
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.
In fact, if is not a local minimizer of I in E, then for every there is such that , and . By the Lagrange multiplier rule, we know there exists a constant such that
Note that is a solution of (1.2), so
Thus, from (3.2), we have
Therefore, is a solution of
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.
Let and be an increasing positive sequence with as . For , define
where . Then
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 .
Suppose that is unbounded. Since
we have as . Denote , then . So, without loss of generality, we can assume that in E, in . Note that
by the definition of and passing to limit in (3.4), one can derive that .
In view of (H2) and the definition of , for small enough, there exist independent of n, such that
Since , then
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 .
Proof If u is a nonnegative solution of (1.2), then by the definition of , we have
Condition (H2) implies there exist , such that
Hence, from (3.7) and (3.8), we derive that
So, using (3.7), we know
Thus, by (H2), we know is uniformly bounded. Note that by Lemma 2.9, for ,
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 . □
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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).
The authors declare that they have no competing interests.
All the authors typed, read, and approved the final manuscript.