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
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 in and satisfy , , . Then problem (P) has at least three solutions , and such 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.
For a precise description, let us give the list of assumptions that we consider. (k) = , where
Condition ( ) is sometimes called a combined sublinear effect at ∞ and simple examples satisfying ( ) ∽ ( ) can be given as follows:
where and , and also
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 .
We first transform ( ) into one-dimensional p-Laplacian 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 exists such that ( ) has no positive solution for , at least one positive solution at and at least two positive solutions for .
As a corollary, we obtain our second main result as follows.
Corollary 1.3Assume (k), ( ), ( ) and ( ). Then there exists such that ( ) has no positive radial solution for , at least one positive radial solution at and 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  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
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 , Lee and Lee  and Lü and O’Regan .
Theorem 2.2Let and be a lower solution and an upper solution of problem (P) respectively such that( ) = , for all ..Assume ( ). Also assume that there exist such that( ) = , , for all ..Then problem (P) has at least one solution such that
Remark 2.3 It is not hard to see that condition (H) implies the following condition;
For each , there exists such that
for and .
Lemma 2.4Assume (H) and ( ). Let be a nontrivial solution of (P). Then there exists such 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 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
Since , for sufficiently large n, we obtain
This contradicts (2.3) and the proof is done. □
Theorem 2.5Assume (H) and ( ). If is 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 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
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
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 right-hand 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 p-Laplacian 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 . 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 ( ). Let and be a strict lower solution and a strict upper solution of problem (P) respectively such that , and . Then problem (P) has at least one solution such that
Moreover, for large enough,
Proof Define given 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
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
from to t, we have
Similarly, we see that is bounded. Therefore, we get
Since every solution of ( ) is contained in Ω, the excision property implies that
Since on Ω, we finally get
This completes the proof. □
We now prove three solutions theorem for (P).
Proof of Theorem 1.1
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 , 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
Similarly, we also get
For , large enough, define
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.
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
and define by taking
where and are unique zeros of
respectively. And define by
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 .
Remark 3.1 If is a solution of ( ), then and .
For later use, we introduce the following well-known result. See  for proof and details.
Proposition 3.2LetXbe a Banach space, an order cone inX. Assume that and are bounded open subsets inXwith and . Let be a completely continuous operator such that either
(i) , , and ,
(ii) , , and , .
ThenAhas a fixed point in .
Lemma 3.3Assume , , ( ) and ( ). Let be a compact subset of . Then there exists a constant such that for all and all possible positive solutions of ( ), 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 solution for some , then ( ) has a solution such 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,
(Case II) as .
Using ( ), choose such that , and
Let . Then
Thus is an upper solution of ( ).
(Case III) g is bounded and as .
Choose such that , and and let
Consequently, by Theorem 2.2, ( ) has a solution satisfying
Lemma 3.5Assume , , ( ), ( ) and ( ). Then there exists such 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 exists such that for , ( ) has a positive solution with and .
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
If , then we have
If , then
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
Let , then and for ,
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