Positive solutions for a class of sublinear elliptic systems
© Ma et al.; licensee Springer. 2014
Received: 17 October 2013
Accepted: 9 January 2014
Published: 30 January 2014
In this paper, we are concerned with the existence of positive solutions of the semilinear elliptic system
where is a parameter, is a continuous real function for each . Under some appropriate assumptions, we show that the above system has at least one positive solution in certain interval of λ. The proofs of our main results are based upon bifurcation theory.
where is a bifurcation parameter, is a continuous real function for each .
A solution of (1.1) is a pair . is called a positive solution of (1.1) if in Ω for each . In the following, also denotes the elements of .
The following definitions will be used in the statement of our main results.
Definition 1.1 
is said to be cooperative. Similarly, H is called a cooperative matrix.
Definition 1.2 
where B and D are square matrices, and is the transpose of Q. Otherwise, A is irreducible.
In the past few years, the existence of positive solutions to sublinear semilinear elliptic systems with two equations have been extensively studied, see for example, [3–6] and the references therein. The sublinear condition plays an important role. Very recently, Wu and Cui  considered the existence, uniqueness and stability of positive solutions to the sublinear elliptic system (1.1). By using bifurcation theory and the continuation method, they proved the following.
Theorem A Assume that
(H1) Each () is a smooth real function defined on satisfying .
(H2) , for all .
If at least one of () is positive and matrix is irreducible, then (1.1) has a unique positive solution for all ;
If , for each and matrix is irreducible, then for some , (1.1) has no positive solution when , and (1.1) has a unique positive solution for .
Moreover, (in the first case, we assume ) is a smooth curve so that is strictly increasing in λ, and as .
We are interested in the existence of positive solutions of (1.1) under weaker assumptions. More concretely, we consider the existence of positive solutions of (1.1) from the following two aspects: (a) To obtain the counterpart of Theorem A(ii) under the weaker assumptions than those of . In other words, we will not assume that () are smooth functions any more. (b) Furthermore, we will also consider the case that () may not exist. More precisely, the following two theorems, which are the main results of the present paper, shall be proved.
Theorem 1.1 Suppose that
(A3) , .
If the matrix is irreducible, then there exists such that (1.1) has no positive solution for and (1.1) has at least one positive solution for .
Theorem 1.2 Let (A1) and (A3) hold. Assume the following.
(A4) The matrix is irreducible, where .
Then for some , (1.1) has at least one positive solution for .
Remark 1.1 It follows from (A2) and (A2)′ that the matrices and are all cooperative.
Remark 1.2 We note that our assumptions in Theorems 1.1 and 1.2 are weaker than those of Theorem A, and, accordingly, our results are weaker than Theorem A. Since we just suppose that is continuous, we can only obtain the continua of positive solutions of (1.1) by applying bifurcation techniques, which are not necessarily curves of positive solutions, and thus the uniqueness and stability of positive solutions are not investigated. In , the authors obtained a smooth curve consisting of positive solutions of (1.1) by assuming stronger assumptions, under which the uniqueness and stability of positive solutions can be achieved.
The rest of the paper is arranged as follows. In Section 2, we recall some basic knowledges on the maximum principle of cooperative systems as well as the eigenvalues of cooperative matrices. Finally in Section 3, we prove our main results Theorems 1.1 and 1.2 by applying bifurcation theory.
The norm of will be defined as , where denotes the norm of . We use and for the standard Sobolev space. We use and to denote the null and the range space of a linear operator L, respectively.
is a real eigenvalue of , where is the spectrum of .
For , there exists a unique (up a constant multiple) eigenfunction , and in Ω.
For , the equation is uniquely solvable for any , and as long as .
(Maximum principle) For , assume that satisfies in Ω, on ∂ Ω, then in Ω.
If there exists satisfying in Ω, on ∂ Ω, and either on ∂ Ω or in Ω, then .
For the results and proofs, see Proposition 3.1 and Theorem 1.1 of Sweers . Moreover, from a standard compactness argument, there are countably many eigenvalues of , and as . We notice that () are not necessarily real-valued.
where . It is easy to verify that is the adjoint operator of , while both are considered as operators defined on subspaces of .
The following lemmas are crucial in the proof of our main results.
Lemma 2.2 
Let Y, Z, L and H be the same as in Lemma 2.1. Then the principal eigenvalue of is also a real eigenvalue of , , and for , there exists a unique eigenfunction of (up a constant multiple), and in Ω.
Lemma 2.3 [, Theorem 5.3.1]
Let matrix A be a nonnegative irreducible matrix. Then is a simple eigenvalue of A, associated to a positive eigenvector, where denotes the spectral radius of A. Moreover, .
Lemma 2.4 
is unbounded in ; or
such that in Ω and .
3 Proof of the main results
Proof of Theorem 1.1. We extend each to be a nonnegative continuous function, which is still denoted by , defined on ℝ in the following way: if , then .
is one dimensional. In addition, it is easy to see that and exist for .
We divide the rest of the proof into two steps.
Step 1. We show that is actually a bifurcation point.
Indeed, the proof of this is similar to the proof of Theorem A(ii), we state it here for the readers’ convenience.
Thus if and only if (3.6) holds, which implies that is one dimensional.
since (1.1) has only the trivial solution when .
Step 2: We claim that cannot blow up at some finite .
where . It is well known that is continuous and compact, and so is continuous and compact on .
Obviously, () in Ω and . In addition, we have . Or else, let , then by (3.14) we get () in Ω, which contradicts .
which contradicts .
Finally, by (3.10), the connectness of and above arguments, we can find some such that (1.1) has no positive solution for , and (1.1) has at least one positive solution for . □
To prove Theorem 1.2, we need the following lemmas as required.
which is a contradiction. Therefore, the closure of the set of nontrivial solutions of (3.25) is exactly Σ.
is the associated Nemytski operator. For , let , let denote the degree of on with respect to 0.
and so for k sufficiently large. Similarly, by (3.23) and (3.30), we can deduce that for k large enough. Consequently, for k sufficiently large we get , which contradicts . □
Corollary 3.2 For and , .
Hence for k sufficiently large, which contradicts . □
Corollary 3.4 For and , .
is unbounded in ; or
Consequently, (1.1) has at least one positive solution for . □
The authors are very grateful to the anonymous referees for their valuable suggestions. This work was supported by the NSFC (No. 11361054, No. 11201378), SRFDP(No. 20126203110004), Gansu provincial National Science Foundation of China (No. 1208RJZA258).
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