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
Let Ω be a bounded smooth domain in ( ). In this paper, we study the existence of positive solutions of the semilinear elliptic system
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.
Let ( ) be smooth real functions defined on . Define the Jacobian of the vector field as
If ( ) for all , then the semilinear elliptic system
is said to be cooperative. Similarly, H is called a cooperative matrix.
An matrix A is reducible if for some permutation matrix Q,
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 AAssume that
(H1) Each ( ) is a smooth real function defined on satisfying .
(H2) , for all .
(i) If at least one of ( ) is positive and matrix is irreducible, then (1.1) has a unique positive solution for all ;
(ii) 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 inλ, 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 1.1Suppose that
(A1) ( ) are continuous real functions satisfying
(A2) There exist constants such 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 1.2Let (A1) and (A3) hold. Assume the following.
(A2)′ For each , there exist such that
(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.
We shall essentially work in Banach space , here
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.
Let be a solution of (1.1). Suppose that ( ) are smooth real functions. Then we can deduce the eigenvalue problem
which can be rewritten as
Let and with . Suppose thatL, Hare given as in (2.3), andHis cooperative and irreducible. Then we have the following:
(a) is a real eigenvalue of , where is the spectrum of .
(b) For , there exists a unique (up a constant multiple) eigenfunction , and in Ω.
(c) For , the equation is uniquely solvable for any , and as long as .
(d) (Maximum principle) For , assume that satisfies in Ω, on∂Ω, then in Ω.
(e) 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.
In this section, we also need to consider the adjoint operator of . Let the transpose matrix of H be
Then it is clear that the results in Lemma 2.1 are also true for the eigenvalue problem
which is equivalent to
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.
LetY, Z, LandHbe 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 matrixAbe a nonnegative irreducible matrix. Then is a simple eigenvalue ofA, associated to a positive eigenvector, where denotes the spectral radius ofA. Moreover, .
LetVbe a real Banach space. Suppose that
is completely continuous and for all . Let ( ) such that is the isolated solution of the equation
Furthermore, assume that
where is an isolated neighborhood of trivial solutions. Let
Then there exists a continuum (i.e., a closed connected set) ofcontaining , and either
(i) is unbounded in ; or
Finally, let be the principal eigen-pair of the linear eigenvalue problem
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 .
Let us define
where . Then it follows from (A1) that is continuous, and is always a solution of (1.1). Moreover, (A2) implies that F is differentiable at , and
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
where . This implies that is a positive eigenvector of the operator . Similarly, has the same principal eigenvalue and the corresponding eigenvector is , where ( ) is a positive constant. Obviously,
Hence when , is not invertible and is a potential bifurcation point. More precisely, the null space
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.
Suppose . Then there exists such that
Let us consider the adjoint eigenvalue equation
where , . Multiplying the system (3.4) by , multiplying the system (3.5) by , integrating on Ω and subtracting, then we obtain
Thus if and only if (3.6) holds, which implies that is one dimensional.
Next, we verify that
Otherwise, we have
multiplying the system (3.9) by and using (3.8), we can get a contradiction that
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,
since (1.1) has only the trivial solution when .
Step 2: We claim that cannot blow up at some finite .
Otherwise, a sequence can be taken such that
where . Let be the Green operator of −Δ subject to Dirichlet boundary conditions, i.e., if and only if
By the elliptic regularity, satisfies
Here also denotes the Nemytski operator generated by itself. Clearly, (3.12) is equivalent to
where . It is well known that is continuous and compact, and so is continuous and compact on .
Let ( ). Then in Ω and . Dividing both sides of (3.13) with , we have
For each , from (A2) and (A3) it follows that is bounded in . Moreover, we have
is bounded in X. This together with the compactness of implies that has a subsequence, denoted by itself, satisfying, in X,
Obviously, ( ) in Ω and . In addition, we have . Or else, let , then by (3.14) we get ( ) in Ω, which contradicts .
Then for each ,
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
Now, let in (3.14), using (3.19) and the fact that we can obtain
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.
By Remark 1.1 and Lemma 2.3, the matrices and have the principal eigenvalues and , respectively, and the corresponding positive eigenvectors are and . Moreover, it is easy to obtain
where is given as in (2.6). By Lemma 2.2, the matrices and have principal eigenvalues and , respectively, the associated positive eigenvectors are and . We can easily verify that
Let be the closure of the set of positive solutions to (1.1). We extend each to be a function defined on ℝ by
then on ℝ. Let be a solution of
Then by (3.24), for each ,
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.
In the following, we shall apply the Leray-Schauder degree theory, mainly to the mapping ,
where is given as in (3.13), and
is the associated Nemytski operator. For , let , let denote the degree of on with respect to 0.
Lemma 3.1Let be a compact interval with . Then there exists a such that
Proof Suppose on the contrary that there exist sequences and so that
Apparently, (3.27) is equivalent to
and ( ) in Ω, and therefore , . Furthermore, it follows from (A2)′ that, for sufficiently small, . This together with (3.28) implies that, for k large enough,
Multiplying (3.29) by , multiplying (3.22) by , integrating on Ω and adding, using (3.30) and the fact , , we know that, for k large enough,
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 3.2For and , .
Proof Lemma 3.1, applied to the interval , guarantees the existence of such that, for ,
Hence for any ,
Lemma 3.3Suppose that . Then there exists such that
Proof Suppose on the contrary that there exist and a sequence with and such that
Clearly, ( ) in Ω. Multiplying (3.22) by , multiplying (3.32) by , integrating over Ω and adding, then by (3.30) we know that, for k large enough,
Hence for k sufficiently large, which contradicts . □
Corollary 3.4For and , .
Proof Let , where is the constant given as in Lemma 3.3. Since is bounded in , there exists a constant such that , . By Lemma 3.3, we get
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
(i) is unbounded in ; or
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
Consequently, (1.1) has at least one positive solution for . □
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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