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.
MSC: 34B15, 34B18.
Keywords:sublinear elliptic systems; positive solutions; eigenvalues; bifurcation theory
The following definitions will be used in the statement of our main results.
is said to be cooperative. Similarly, H is called a cooperative matrix.
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 AAssume that
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 1.1Suppose that
Theorem 1.2Let (A1) and (A3) hold. Assume the following.
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.
which can be rewritten as
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.
Then it is clear that the results in Lemma 2.1 are also true for the eigenvalue problem
which is equivalent to
The following lemmas are crucial in the proof of our main results.
Lemma 2.3 [, Theorem 5.3.1]
LetVbe a real Banach space. Suppose that
Furthermore, assume that
3 Proof of the main results
Let us define
where . By (A2), all entries of J are positive. Therefore Lemma 2.3 yields the result that J has a positive principal eigenvalue , the corresponding eigenvector satisfying (). Moreover, it is not difficult to verify that
We divide the rest of the proof into two steps.
Indeed, the proof of this is similar to the proof of Theorem A(ii), we state it here for the readers’ convenience.
Let us consider the adjoint eigenvalue equation
Next, we verify that
Otherwise, we have
By using [, Theorem 1.7], we conclude that is a bifurcation point. Furthermore, by the Rabinowitz global bifurcation theorem , there exists a continuum of positive solutions of (1.1), which joins to infinity in . Clearly,
by Lebesgue control convergence theorem, we get
which together with (3.15) yields
On the other hand, we know from (A2) and (3.15) that
Hence we conclude from (3.17) and (3.18) that
To prove Theorem 1.2, we need the following lemmas as required.
where K is given as in the proof of Theorem 1.1. Hence is a nonnegative solution of (3.25). Moreover, from (3.24) it follows that is a solution of (1.1). On the other hand, (3.25) has no half-trivial solutions. Otherwise, U must have trivial and nontrivial components, and so there is a such that in Ω, and by the maximum principle of elliptic boundary value problems, we have
which is a contradiction. Therefore, the closure of the set of nontrivial solutions of (3.25) is exactly Σ.
where is given as in (3.13), and
Apparently, (3.27) is equivalent to
Proof of Theorem 1.2. For such that , let , and . For any , it is easy to see that the assumptions of Lemma 2.4 are all satisfied. Therefore there exists a continuum of solutions of (3.25) containing , and either
By Lemma 3.1, the case (ii) cannot occur, and hence is unbounded bifurcated from . Note that (3.25) has only trivial solutions when , and therefore . Moreover, from Lemma 3.1 it follows that for a closed interval , if , then in X is impossible. Thus must be bifurcated from . Finally, applying similar methods to the proof of Step 2 of Theorem 1.1, we can show that
The authors declare that they have no competing interests.
RM and RC completed the main study, carried out the results of this article and drafted the manuscript. YL checked the proofs and verified the calculation. All the authors read and approved the final manuscript.
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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