In this paper, we consider p-Laplacian 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:p-Laplacian system; singular weight; upper solution; lower solution; three solutions theorem
In this paper, we study one-dimensional p-Laplacian 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  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  have proved the theorem for scalar one-dimensional p-Laplacian problems with -Caratheodory condition which corresponds to case ; Henderson and Thompson  as well as Lü, O’Regan, and Agarwal  - for scalar second order ODEs and one-dimensional p-Laplacian with the derivative-dependent nonlinearity respectively; and De Coster and Nicaise  - for semilinear elliptic problems in nonsmooth domains. For noncooperative elliptic systems () with and Ω bounded, one may refer to Ali, Shivaji, and Ramaswamy . Specially, for subsuper solutions which are not completely ordered, this type of three solutions result was studied in .
The three solutions theorem for our system () or even for corresponding scalar p-Laplacian problems is not obviously extended from previous works mainly by the possibility . Caused by the delicacy of Leray-Schauder 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 p-Laplacian 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
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  and  for second order ODE systems, in [3,7,9,10,13,14,16] and  for semilinear elliptic systems on a bounded domain and in [5,15,17] and  for p-Laplacian systems on a bounded domain.
Among the reference works mentioned above, Hai and Shivaji  and Ali and Shivaji  (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 .
As a corollary, we obtain our second main result as follows.
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  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 p-Laplacian 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
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 , Lee and Lee  and Lü and O’Regan .
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;
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 concave-convex 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
Therefore, it follows from plugging (2.2) into (2.1) that
This contradicts (2.3) and the proof is done. □
We will show . Other facts can be proved by the same manner. Suppose . By Lemma 2.4 and the concave-convex argument, we may assume without loss of generality that there exists such that on . Then for given , by the fact , , there exists such that
Thus we have
Since ε is arbitrary, we have
On the other hand, we observe the inequality
Similarly, we have
Adding (2.10) and (2.11), we have
It is known that A is completely continuous . Define with norm . We note that
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
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
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
This completes the proof. □
We now prove three solutions theorem for (P).
Proof of Theorem 1.1
Let and be the first eigenvalues of for respectively and let and be corresponding eigenfunctions with . Since are positive and concave , we may choose such that and for
Similarly, we get
Similarly, we also get
and by the excision property, we have
This completes the proof.
Then it is known that is completely continuous  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 .
For later use, we introduce the following well-known result. See  for proof and details.
Thus we get
(Case I) Both f and g are bounded.
Thus we have
By similar argument as the above, with (3.3), we may show that
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 , we may choose and such that
The authors declare that they have no competing interests.
All authors have equally contributed in obtaining new results in this article and also read and approved the final manuscript.
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 Mid-career Researcher Program (No. 2010-0000377) and Basic Science Research Program (No. 2012005767) through NRF grant funded by the MEST.
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