In this paper, we make use of a new stability result and bifurcation theory to study the existence and uniqueness of positive solutions to semilinear elliptic systems with some general sublinear conditions. Moreover, we obtain the precise global bifurcation diagrams of the system in a single monotone solution curve.
MSC: 35J55, 35B32.
Keywords:semilinear elliptic systems; positive solution; stability; existence; uniqueness
We consider positive solutions of a semilinear elliptic system with n equations ( )
where is a positive parameter, Ω is a bounded smooth domain in , and ( ) are smooth real-valued functions defined on satisfying the following.
Cooperativeness Define the Jacobian of the vector field as
Then ( ) for .
The purpose of this paper is to study the existence, uniqueness and stability of positive solutions of such cooperative system (1.1) under certain conditions of .
The existence, uniqueness and stability of positive solutions to sublinear semilinear elliptic systems with two equations have been recently studied in [1-4]. The sublinear condition plays an important role. In this paper, we continue the effort in  to prove the stability of a positive solution to (1.1) under some reasonable sublinear conditions, and the stability implies the uniqueness of the positive solution. We also prove corresponding existence results using bifurcation theory and the continuation method. This is motivated by the existence study of exact multiplicity (and uniqueness) of positive solutions to the scalar semilinear elliptic equation:
starting from Korman et al.[5,6]. In Ouyang and Shi [7,8], their result classified the different global exact multiplicity of (1.3) for more general nonlinearity f. There are more results on the existence and uniqueness of solution to the semilinear cyclic elliptic system
The notation of sublinearity and superlinearity of the nonlinear vector field or the ones in higher dimension was considered in Sirakov . Our definition of sublinear nonlinearity is quite different, and ours is similar to the one in Ouyang and Shi  for the scalar case. Dalmasso  obtained the existence and uniqueness result for a more special sublinear system, and it was extended by Shi and Shivaji . The uniqueness of a positive solution for large λ was proved in Hai [11,12], Hai and Shivaji . If Ω is a finite ball or the whole space, then the positive solutions of systems are radically symmetric and decreasing in radical direction by the result of Troy . Hence the system can be converted into a system of ODEs. Several authors have taken that approach for the existence of the solutions, see Serrin and Zou [15,16], and much success has been achieved for Lane-Emden systems. Using the scaling invariant in the Lane-Emden system, the uniqueness of the radial positive solution was shown in Dalmasso , Korman and Shi . Cui et al.[18,19] considered cyclic systems with three equations, and the uniqueness and existence of positive solutions were obtained. For the Lane-Emden systems with n equations, Maniwa  obtained the uniqueness and existence of positive solutions to systems under the sublinear conditions.
We organize the rest of this paper in the following way. In Section 2, we recall the maximum principle and prove the main stability result. In Section 3, we use the stability result and bifurcation theory to prove the existence and uniqueness of a positive solution. We also obtain the precise global bifurcation diagrams of the system (the bifurcation diagram is a single monotone solution curve in all cases) and give some examples. In Section 4, we consider the similar question for merely Hölder continuous nonlinearities, and we use monotone methods for existence. We use and for the standard Sobolev space, for the space of continuous functions defined on , and . We use and to denote the null space and the range space of a linear operator L.
2 Stability and linearized equations
In this section, we study the stability result about a positive solution. Let be a solution of (1.1). We shall denote the partial derivative of with respect to by or . The stability of U is determined by the eigenvalue equation
which can be written as
An matrix A is reducible if for some permutation matrix Q,
where B and C are square matrices and is the transpose of Q. Otherwise, A is irreducible.
Throughout this paper, H is assumed to be irreducible, since if not the case, the linearized system (2.1) can be reduced to two subsystems with one being not coupled with the other. If we assume that H is cooperative and irreducible, then the maximum principles hold for such systems. Before stating our results, we recall some known results as required.
Lemma 2.2Let , and let for . Suppose thatL, Hare given as in (2.3), and is irreducible and satisfies ( ) for . Then we have the following.
1. is a real eigenvalue of , where is the spectrum of .
2. For , there exists a unique (up a constant multiple) eigenfunction , and in Ω.
3. For , the equation is uniquely solvable for any , and as long as .
4. (Maximum principle) For , suppose that satisfies in Ω, on∂Ω, then in Ω.
5. If there exists satisfying in Ω, on∂Ω, and either on∂Ω or in Ω, then .
For the result and proofs, see Sweers , Proposition 3.1 and Theorem 1.1. Moreover, from a standard compactness argument, there are countably many eigenvalues of , and as . We notice that is not necessarily real-valued. We call a solution Ustable if ; and otherwise, it is unstable ( ).
For our purpose, in this section, we also need to consider the adjoint operator of . Let the transpose matrix of the Jacobian be
Then, evidently, the results in Lemma 2.2 also hold for the eigenvalue problem
where . It is easy to verify that is the adjoint operator of , while both are considered as operators defined on subspaces of . By using the well-known functional analytic techniques (see [21,22]), one can show the following.
Lemma 2.3LetX, Y, LandHbe the same as in Lemma 2.2. 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.
Cui et al. obtained the stability result of a positive solution for the system with two equations. We give the following stability result about a positive solution of (1.1).
Theorem 2.4Suppose that is a positive solution of (1.1), is cooperative and is irreducible, thenUis stable if satisfies one of the following conditions: for any ,
Proof Let be a positive solution of (1.1), and let be the corresponding principal eigen-pair of (2.6) such that in Ω for any .
We assume that satisfies ( ). Multiplying the system (1.1) by , the system (2.6) by U, integrating over Ω and subtracting, we obtain
Hence if ( ) is satisfied.
Similar to the proof above, let be the corresponding principal eigen-pair of (2.1) such that in Ω for any . We assume that satisfies ( ). Multiplying the system (1.1) by u, the system (2.1) by U, integrating over Ω and subtracting, we can get
Hence if ( ) is satisfied. □
On the other hand, the same proof also implies the following instability result under the opposite condition of ( ) and ( ).
Theorem 2.5Suppose that is a positive solution of (1.1), is cooperative and is irreducible, thenUis unstable if satisfies one of the following conditions: for any ,
1. In , for positive solutions of the scalar equation
the function is called a sublinear function if , and it is superlinear if . It was proved in Proposition 3.14 of  that a positive solution u is stable if h is sublinear, and u is unstable if h is superlinear. Now our conclusions, Theorems 2.4 and 2.5, are generalizations of the corresponding results. The condition ( ) for is the generalization of sublinearity (or superlinearity) to n-variable vector fields.
2. The conditions ( ) and ( ) can be written in a vector form and , where , , and H is the original Jacobian matrix of the vector field , and is the transpose matrix of the Jacobian H. The condition ( ) is clearly more natural as the conditions for are separate. Hence the sublinearity can be defined for a single n-variable function. The condition ( ) is defined for the whole vector field .
3. If a solution U is stable, then it is necessarily a non-degenerate solution. That is, any eigenvalue of (2.1) has a positive real part. But when a solution is proved to be unstable, it can be a degenerate one with zero or pure imaginary eigenvalues.
3 Existence and uniqueness
In this section, we consider the uniqueness and existence of positive solutions for the following problem:
Suppose that each ( ) is a smooth real function defined on satisfying
( ) ;
( ) , , for all .
The Perron-Frobenius theorem plays a critical role in our main result.
Lemma 3.1 (Perron-Frobenius theorem: strong form [, 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, .
Here let be the principal eigen-pair of
such that in Ω and . We have the following existence and uniqueness result for this sublinear problem.
Theorem 3.2Assume that each of satisfies ( ), ( ) and
( ) .
1. If at least one of ( ) is positive and matrix is irreducible, then (3.1) has a unique positive solution for all ;
2. If , for each and matrix is irreducible, then for some , (3.1) has no positive solution when , and (3.1) 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 .
Proof Our proof follows that of Theorem 6.1 in . Firstly, we extend to be defined on R and they are continuously differentiable on R. From ( ), implies that , so satisfies ( ). Hence from Theorem 2.4, any positive solution of (3.1) is stable.
Let us define
where and . Here are at least , then is continuously differentiable, where and . For weak solutions , one can also consider and where is properly chosen.
It is easy to see that is a solution of (3.1). We apply the implicit function theorem at . The Fréchet derivative of F is given by
Then is an isomorphism from X to Y, and the implicit function theorem implies that has a unique solution for for some small , and is the unique solution of
where e is the unique positive solution of
If for any , then is positive for . If there exists i such that , then for . For , and is positive, hence as well.
Next we assume that and ( ). The linearized operator is
Since , all entries of matrix J are positive. Therefore, by using Lemma 3.1, there exist a positive principal eigenvalue and the corresponding eigenvector of J for some such that is a positive eigenvector of , where . Similarly, has the same principal eigenvalue , and the corresponding eigenvector , where ( ) is a positive constant.
Hence when , is not invertible and is a potential bifurcation point. More precisely, the null space is one-dimensional.
Suppose that , then there exist such that
Consider the adjoint eigenvalue equation
Multiplying the system (3.8) by , the system (3.9) by , integrating over Ω and subtracting, we get
Hence if and only if (3.9) holds, which implies that the codimension of is one.
Next we verify that . Indeed,
But this contradicts with
By using a bifurcation from a simple eigenvalue theorem of Crandall-Rabinowitz , we conclude that is a bifurcation point. The nontrivial solutions of near are in the form of for , where ( ). From the stability of positive solutions, each positive solution is non-degenerate.
Next we claim that (1.1) has no positive solution when . Let is a positive solution of (1.1) and satisfy
Multiplying the system (3.1) by , the system (3.13) by , integrating over Ω and subtracting, by ( ) and , we obtain
Hence (3.1) has no positive solution when . And the solution can also be parameterized as for . With the implicit function theorem, we can extend this curve to the largest .
Let . We show that is strictly increasing in λ for . In fact, satisfies the equation
Hence from the maximum principle (Lemma 2.2 part 3) and the fact that from the stability of positive solutions. We claim that . Suppose not, then and . Then one can show that the curve Γ can be extended to from some standard elliptic estimates, then from the implicit function theorem, Γ can be extended beyond , which is a contradiction; if and , a contradiction can be derived with the solution curve which cannot blow up at finite (see similar arguments for the scalar equation in ). Hence we must have .
Finally, we claim the uniqueness. If there is another positive solution for some , then the arguments above show that this solution also belongs to a solution curve defined for , and the solutions on the curve are increasing in λ, but the nonexistence of positive solutions for and the local bifurcation at exclude the possibility of another solution curve. Hence the positive solution is unique for all . This completes the proof. □
We consider the following cyclic system:
where are smooth real functions defined on satisfying ( ), ( ) and ( ). Then Theorem 3.2 implies the existence and uniqueness of a positive solution of problem (3.16), and the bifurcation diagram is a single monotone solution curve. The n-dimensional cyclic positone and semipositone system was considered in  and . They got the existence and multiplicity of positive solutions result for some combined sublinear condition by the method of sub-super solutions.
Consider the general Lane-Emden system
where is a positive parameter, , ( ) are positive constants, ( ) are nonnegative constants satisfying and denotes a bounded domain of class , . For the case , (3.17) has been studied by many authors. Especially, Dalmasso  proved the uniqueness and existence of positive solutions to (3.17) for the case , , .
In this section, we show the uniqueness and existence of positive solutions for (3.17) by using super-subsolution methods and the stability of positive solutions by using Theorem 2.4.
Let be the unique positive solution of
We construct a super-solution . There exists a suitable positive constant M such that
where . We construct a sub-solution in the form of , where ε will be specified later. Recall that is the positive principal eigenfunction with . Now, for the ith equation of (3.17), we have
where . Hence is a subsolution of (3.17), if we choose ε smaller, and it satisfies . Hence if we choose ε smaller so that , then is a subsolution of (4.1). Therefore, there exists a positive solution of (3.17) between the supersolution and the subsolution.
Next, we show the solution is stable. Letting , by simple calculation, we get
So, satisfies ( ), hence from Theorem 2.4, any positive solution of (3.17) is stable.
4 Application: Hölder continuous case
In this section, we consider that for .
where is a positive parameter, .
We use the monotone method to prove the existence of a solution. For the supersolution, we choose . Then
Similarly, we have . Then it is clear that is a supersolution of (4.1). We construct a sub-solution in the form of , where ε will be specified later. Recall that is the positive principal eigenfunction with . Now, for the equation of u or the equation of v, we have
Hence if we choose ε smaller and it satisfies , so that , then is a subsolution of (4.1). Therefore, there exists a positive solution of (4.1) between the supersolution and the subsolution when . We remark that the stability defined in Section 2 can still be established for a nonlinear function to become ∞ near ∂Ω by using Remark 3.1 in . Thus the positive solution of (4.1) is stable.
Remark 4.2 Since (4.1) is a cooperative model from ecology with logistic growth rate and sublinear interaction term, we can get the stable result. When the interaction terms are uv (Lotka-Volterra type) and as proposed here, and they do not satisfy the conditions of Theorem 3.2, thus, the solution may not be unique.
The authors declare that they have no competing interests.
BW and RC carried out the proof of the main part of this article, RC corrected the manuscript and participated in its design and coordination. All authors have read and approved the final manuscript.
Partially supported by the National Natural Science Foundation of China (No. 11071051, 11271100 and 11101110), the Aerospace Supported Fund, China, under Contract (Grant 2011-HT-HGD-06), Science Research Foundation of the Education Department of Heilongjiang Province (Grant No. 12521153), Science Foundation of Heilongjiang Province (Grant A201009) and Harbin Normal University advanced research Foundation (11xyg-02).