Abstract
In this paper, we consider pLaplacian systems with singular weights. Exploiting Amann type three solutions theorem for a singular system, we prove the existence, nonexistence, and multiplicity of positive solutions when nonlinear terms have a combined sublinear effect at ∞.
MSC: 35J55, 34B18.
Keywords:
pLaplacian system; singular weight; upper solution; lower solution; three solutions theorem1 Introduction
In this paper, we study onedimensional pLaplacian system with singular weights of the form where , λ is a nonnegative parameter, , is a nonnegative measurable function on , on any open subinterval in and with . In particular, may be singular at the boundary or may not be in . It is easy to see that if , then all solutions of () are in . On the other hand, if , then this regularity of solutions is not true in general; for example, even for scalar case, if we take , and , , then , and the solution u for corresponding scalar problem of () is given by which is not in .
For more precise description, let us introduce the following two classes of weights;
We note that h given in the above example satisfies but . The main interest of this paper is to establish Amann type three solutions theorem [4] when with possibility of . The theorem generally describes that two pairs of lower and upper solutions with an ordering condition imply the existence of three solutions. Recently, Ben Naoum and De Coster [6] have proved the theorem for scalar onedimensional pLaplacian problems with Caratheodory condition which corresponds to case ; Henderson and Thompson [18] as well as Lü, O’Regan, and Agarwal [23]  for scalar second order ODEs and onedimensional pLaplacian with the derivativedependent nonlinearity respectively; and De Coster and Nicaise [11]  for semilinear elliptic problems in nonsmooth domains. For noncooperative elliptic systems () with and Ω bounded, one may refer to Ali, Shivaji, and Ramaswamy [3]. Specially, for subsuper solutions which are not completely ordered, this type of three solutions result was studied in [26].
The three solutions theorem for our system () or even for corresponding scalar pLaplacian problems is not obviously extended from previous works mainly by the possibility . Caused by the delicacy of LeraySchauder degree computation, the crucial step for the proof is to guarantee regularity of solutions, but with condition , regularity is not known yet. Due to the singularity of weights on the boundary, the regularity heavily depends on the shape of nonlinear terms f and g. Therefore, the first step is to investigate certain conditions on f and g to guarantee regularity of solutions. Another difficulty is to show that a corresponding integral operator is bounded on the set of functions between upper and lower solutions in . To overcome this difficulty, we give some restrictions on upper and lower solutions such that their boundary values are zero. As far as the authors know, our three solutions theorem (Theorem 1.1 in Section 2) is new and first for singular pLaplacian systems with weights of class.
To cover a larger class of differential system, we consider the systems of the form where are continuous. We give more conditions on F and G as follows: () = For each , and are nondecreasing in u.; (H) = There exist and such that
and
for all and .; () = and , for all .. We now state our first main result related to three solutions theorem as follows. See for more details in Section 2.
Theorem 1.1Assume (H), () and (). Let, be a lower solution and an upper solution and, be a strict lower solution and a strict upper solution of problem (P) respectively. Also, assume that all of them are contained inand satisfy, , . Then problem (P) has at least three solutions, andsuch that, , and, .
As an application of Theorem 1.1, we study the existence, nonexistence, and multiplicity of positive radial solutions for the following quasilinear system on an exterior domain: where , , , , , and with .
In recent years, the existence of positive solutions for such systems has been widely studied, for example, in [1] and [27] for second order ODE systems, in [3,7,9,10,13,14,16] and [8] for semilinear elliptic systems on a bounded domain and in [5,15,17] and [2] for pLaplacian systems on a bounded domain.
For a precise description, let us give the list of assumptions that we consider. (k) = , where
; () = and ,; () = for all ,; () = f and g are nondecreasing..
Condition () is sometimes called a combined sublinear effect at ∞ and simple examples satisfying () ∽ () can be given as follows:
Among the reference works mentioned above, Hai and Shivaji [17] and Ali and Shivaji [2] (with more general nonlinearities) considered problem () with case and Ω bounded. For monotone functions f and g with and satisfying condition (), they proved that there exists such that the problem has at least one positive solution for .
We first transform () into onedimensional pLaplacian systems () with change of variables , , and where is given by
It is not hard to see that if in () satisfies (k), then in () satisfies , for . Mainly by making use of Theorem 1.1, we prove the following existence result for problem ()
Theorem 1.2Assume, , (), () and (). Then there existssuch that () has no positive solution for, at least one positive solution atand at least two positive solutions for.
As a corollary, we obtain our second main result as follows.
Corollary 1.3Assume (k), (), () and (). Then there existssuch that () has no positive radial solution for, at least one positive radial solution atand at least two positive radial solutions for.
We finally notice that the first eigenfunctions of make an important role to construct upper solutions in the proofs of Theorem 1.2 and Theorem 1.1. This is possible due to a recent work of Kajikiya, Lee, and Sim [19] which exploits the existence of discrete eigenvalues and the properties of corresponding eigenfunctions for problem (E) with .
This paper is organized as follows. In Section 2, we state a regularity result and a three solutions theorem for singular pLaplacian systems. In addition, we introduce definitions of (strict) upper and lower solutions, a related theorem and a fixed point theorem for later use. In Section 3, we prove Theorem 1.2.
2 Three solutions theorem
In this section, we give definitions of upper and lower solutions and prove three solutions theorem for the following singular system where are continuous.
We call a solution of (P) if , and satisfies (P).
Definition 2.1 We say that is a lower solution of problem (P) if , and
We also say that is an upper solution of problem (P) if , and it satisfies the reverse of the above inequalities. We say that and are strict lower solution and strict upper solution of problem (P), respectively, if and are lower solution and upper solution of problem (P), respectively and satisfying , , , for .
We note that the inequality on can be understood componentwise. Let . Then the fundamental theorem on upper and lower solutions for problem (P) is given as follows. The proof can be done by obvious combination from Lee [20], Lee and Lee [21] and Lü and O’Regan [22].
Theorem 2.2Letandbe a lower solution and an upper solution of problem (P) respectively such that() = , for all..Assume (). Also assume that there existsuch that() = , , for all..Then problem (P) has at least one solutionsuch that
Remark 2.3 It is not hard to see that condition (H) implies the following condition;
For each , there exists such that
Lemma 2.4Assume (H) and (). Letbe a nontrivial solution of (P). Then there existssuch that bothuandvhave no interior zeros in.
Proof Let be a nontrivial solution of (P). Suppose, on the contrary, that there exist sequences , of interior zeros of u and v respectively with . We note that both sequences should exist simultaneously. Indeed, if one of the sequences say, , does not exist, then assuming without loss of generality, on for some , we get for by (). From the monotonicity of , we know that v is concave on the interval. Thus v should have at most one interior zero in , a contradiction. With this concaveconvex argument, we know that , on and if and are local extremal points of u and v on and respectively, thus both and are in . We consider the case that , and in . All other cases can be explained by the same argument. If , then by using Remark 2.3, we have
and similarly,
Therefore, it follows from plugging (2.2) into (2.1) that
Since , for sufficiently large n, we obtain
This contradicts (2.3) and the proof is done. □
Theorem 2.5Assume (H) and (). Ifis a solution of (P), then.
Proof Let be a nontrivial solution of (P). Then so that it is enough to show
We will show . Other facts can be proved by the same manner. Suppose . By Lemma 2.4 and the concaveconvex argument, we may assume without loss of generality that there exists such that on . Then for given , by the fact , , there exists such that
Let . Then integrating (P) over and using Remark 2.3, we have
where we use the fact that is decreasing since v is concave. From and (2.4), we know . This implies that conditions and are equivalent. From (2.4), we have
Thus we have
Since ε is arbitrary, we have
Using the fact , with same argument, we have
On the other hand, we observe the inequality
where
Since , we may choose such that
Integrating (P) over with and using Remark 2.3, we get
here we use the fact that is increasing in . Using (2.7), we have
Integrating (2.9) over with respect to s and using (2.8), we have
Similarly, we have
Adding (2.10) and (2.11), we have
on . From (2.5) and (2.6), we see that the righthand side of (2.12) goes to zero as . This is a contradiction and the proof is complete. □
Now, we consider the three solutions theorem for singular pLaplacian system (P). For , if
then the zero of , denoted by is uniquely determined by ν. Define by taking
It is known that A is completely continuous [24]. Define with norm . We note that
If F and G satisfy condition (H), then for , from Remark 2.3 and (2.13), we get
This implies and by similar computation, we also get . This fact enables us to define the integral operator for problem (P) and the regularity of solutions (Theorem 2.5) is crucial in this argument. Now, define an operator T by
then we see that and completely continuous.
Lemma 2.6Assume (H), () and (). Letandbe a strict lower solution and a strict upper solution of problem (P) respectively such that, and. Then problem (P) has at least one solutionsuch that
and also define
Let us consider the following modified problem We first show that there exists a constant such that if is a solution of (), then . In fact, every solution of () satisfies on . From (H), () and the fact that , , we get
Similarly, we see that is bounded. Therefore, , for some . Thus it is enough to show that
Assume, on the contrary, that there exists such that
Then choosing with , we get the following contradiction:
Now, assume . Since on and , there exists such that and we get the same contradiction from the above calculation by using 0 instead of . For case, we also get the same contradiction. Consequently, we get . The other cases can be proved by the same manner. Taking , we see that every solution of () is contained in Ω. We now compute . For this purpose, let us consider the operator defined by
Then it is obvious that is completely continuous. We show that there exists such that and . Indeed, since , there is such that . By integrating
Similarly, we see that is bounded. Therefore, we get
Since every solution of () is contained in Ω, the excision property implies that
This completes the proof. □
We now prove three solutions theorem for (P).
Proof of Theorem 1.1
Define
and let us consider Then noting that every solution of () satisfies , we may choose , by (H) such that
Let and be the first eigenvalues of for respectively and let and be corresponding eigenfunctions with . Since are positive and concave [19], we may choose such that and for
We show that and are a strict upper solution and a strict lower solution of () respectively. Indeed,
Similarly, we get
Moreover,
Similarly, we also get
Then by Theorem 2.2, there exist two solutions and of (P) satisfying and . Therefore, by Lemma 2.6, we get
and by the excision property, we have
This completes the proof.
3 Application
In this section, we prove the existence, nonexistence, and multiplicity of positive solutions for () by using three solutions theorem in Section 2. Let us define a cone
Then it is known that is completely continuous [25] and in is equivalent to the fact that is a positive solution of (). We know from Theorem 2.5 that under assumptions and (), any solution of problem () is in .
Remark 3.1 If is a solution of (), then and .
For later use, we introduce the following wellknown result. See [12] for proof and details.
Proposition 3.2LetXbe a Banach space, an order cone inX. Assume thatandare bounded open subsets inXwithand. Letbe a completely continuous operator such that either
or
Lemma 3.3Assume, , () and (). Letbe a compact subset of. Then there exists a constantsuch that for alland all possible positive solutionsof (), one has.
Proof If it is not true, then there exist and solutions of () such that . We note that
This implies both and . Moreover, by the above estimation,
Thus we get
as and this contradiction completes the proof. □
Lemma 3.4Assume, , () and (). If () has a lower solutionfor some, then () has a solutionsuch that.
Proof It suffices to show the existence of an upper solution of () satisfying . Let and be positive solutions of
(Case I) Both f and g are bounded.
Since () are positive concave functions and , we may choose such that and . We now show that is an upper solution of (). In fact,
Similarly,
Using (), choose such that , and
And
Thus is an upper solution of ().
(Case III) g is bounded and as .
Choose such that , and and let
Then
And
Consequently, by Theorem 2.2, () has a solution satisfying
□
Lemma 3.5Assume, , (), () and (). Then there existssuch that if () has a positive solution, then.
Proof Let be a positive solution of (). Without loss of generality, we may assume . From (), we know that
From (3.1) and (), we can choose such that
where , . Using (3.2) and (), we have
Thus we have
□
Lemma 3.6Assume, , (), () and (). Then for each, there existssuch that for, () has a positive solutionwithand.
Proof We know that if satisfies and , then is a solution of (). Since are completely continuous, is also completely continuous. Given , choose
where . Let . If , then for , . From the definition of , we know that is the maximum value of on . If , then from the choice of , we have
By the concavity of , we get for ,
By similar argument as the above, with (3.3), we may show that
Let , . For , from (), we may choose such that and
By Proposition 3.2, () has a positive solution such that and . We know that is a lower solution of () for and by Lemma 3.4, the proof is complete. □
We now prove one of the main results for this paper.
Proof of Theorem 1.2
From Lemma 3.6 and Lemma 3.5, we know that the set is not empty and . By Lemma 3.3 and complete continuity of T, there exist sequences and such that and in with a solution of (). We claim that is a nontrivial solution of (). Suppose that it is not true, then there exists a sequence of solutions for () such that and . As in the proof of Lemma 3.3, we get
But from (), we have a contradiction to the fact that the right side of the above inequality converges to zero as . Thus is a nontrivial solution of (). According to Lemma 3.4 and the definition of , we know that () has at least one positive solution at and no positive solution for . To prove the existence of the second positive solution of () for , we will use Theorem 1.1. Let . Then we have a lower solution of () and a strict lower solution of () in satisfying . For upper solutions, let and be the first eigenvalues of for respectively and let and be corresponding eigenfunctions with . Since and are in and positive [19], we may choose and such that
Also by the fact , there exists such that
Let . Then and it is a strict upper solution of () in . Indeed,
and
Finally, from Lemma 3.6, there exists such that () has a positive solution satisfying and . By using the concavity of solutions, it is easily verified that
Therefore, is an upper solution of () in . Now by Theorem 1.1, () has at least two positive solutions and such that and and .
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors have equally contributed in obtaining new results in this article and also read and approved the final manuscript.
Acknowledgements
The authors express their thanks to Professors Ryuji Kajikiya, Yuki Naito and Inbo Sim for valuable discussions related to regularity of solutions and also thank to the referees for their careful reading and valuable remarks and suggestions. The first author was supported by Pusan National University Research Grant, 2011. The second author was supported by Midcareer Researcher Program (No. 20100000377) and Basic Science Research Program (No. 2012005767) through NRF grant funded by the MEST.
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