- Open Access
A result on three solutions theorem and its application to p-Laplacian systems with singular weights
© Lee and Lee; licensee Springer 2012
- Received: 16 February 2012
- Accepted: 18 May 2012
- Published: 22 June 2012
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 ∞.
- p-Laplacian system
- singular weight
- upper solution
- lower solution
- three solutions theorem
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.
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 1.1 Assume (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, .
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
() = and ,;
() = for all ,;
() = f and g are nondecreasing..
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 .
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 1.2 Assume, , (), () 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 1.3 Assume (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.
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.
where are continuous.
We call a solution of (P) if , and satisfies (P).
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 2.2 Letandbe a lower solution and an upper solution of problem (P) respectively such that
() = , for all.
Remark 2.3 It is not hard to see that condition (H) implies the following condition;
for and .
Lemma 2.4 Assume (H) and (). Letbe a nontrivial solution of (P). Then there existssuch that both u and v have no interior zeros in.
This contradicts (2.3) and the proof is done. □
Theorem 2.5 Assume (H) and (). Ifis a solution of (P), then.
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. □
then we see that and completely continuous.
This completes the proof. □
We now prove three solutions theorem for (P).
Proof of Theorem 1.1
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 .
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.
, , and ,
, , and , .
Then A has a fixed point in.
Lemma 3.3 Assume, , () and (). Letbe a compact subset of. Then there exists a constantsuch that for alland all possible positive solutionsof (), one has.
as and this contradiction completes the proof. □
Lemma 3.4 Assume, , () and (). If () has a lower solutionfor some, then () has a solutionsuch that.
(Case I) Both f and g are bounded.
(Case II) as .
Thus is an upper solution of ().
(Case III) g is bounded and as .
Lemma 3.5 Assume, , (), () and (). Then there existssuch that if () has a positive solution, then.
Lemma 3.6 Assume, , (), () and (). Then for each, there existssuch that for, () has a positive solutionwithand.
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
Therefore, is an upper solution of () in . Now by Theorem 1.1, () has at least two positive solutions and such that and and .
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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