Positive solutions for a class of sublinear elliptic systems

In this paper, we are concerned with the existence of positive solutions of the semilinear elliptic system

where \lambda >0 is a parameter, g_{ij}:[0,\mathrm{\infty })\to [0,\mathrm{\infty }) is a continuous real function for each i,j=1,2,\dots ,n. 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.

Let Ω be a bounded smooth domain in {\mathbb{R}}^N (N\ge 1). In this paper, we study the existence of positive solutions of the semilinear elliptic system

where \lambda >0 is a bifurcation parameter, g_{ij}:[0,\mathrm{\infty })\to [0,\mathrm{\infty }) is a continuous real function for each i,j=1,2,\dots ,n.

A solution of (1.1) is a pair (\lambda ,U):=(\lambda ,(u_1,u_2,\dots ,u_n))\in (0,\mathrm{\infty })\times {[C(\overline{\mathrm{\Omega }})]}^n. (\lambda ,U) is called a positive solution of (1.1) if u_i>0 in Ω for each i=1,2,\dots ,n. In the following, (u_1,u_2,\dots ,u_n) also denotes the elements of {\mathbb{R}}_{+}^n=\{(u_1,u_2,\dots ,u_n)\in {\mathbb{R}}^n:u_i\ge 0,i=1,2,\dots ,n\}.

The following definitions will be used in the statement of our main results.

Definition 1.1 [1]

Let f_i (i=1,2,\dots ,n) be smooth real functions defined on {\mathbb{R}}_{+}^n. Define the Jacobian of the vector field (f_1,f_2,\dots ,f_n) as

If \frac{\partial f_i}{\partial u_j}\ge 0 (i\ne j) for all (u_1,u_2,\dots ,u_n)\in {\mathbb{R}}_{+}^n, then the semilinear elliptic system

is said to be cooperative. Similarly, H is called a cooperative matrix.

Definition 1.2 [2]

An n\times n matrix A is reducible if for some permutation matrix Q,

where B and D are square matrices, and Q^T 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 [1] 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 g_{ij} (i,j=1,2,\dots ,n) is a smooth real function defined on {\mathbb{R}}_{+} satisfying g_{ij}(0)\ge 0.

(H2) g_{ij}^{\prime }(s)\ge 0, {(g_{ij}(s)/s)}^{\prime }\le 0 for all s\ge 0.

(H3) {lim}_{s\to \mathrm{\infty }}\frac{g_{ij}(s)}{s}=0.

1. (i)

   If at least one of g_{ij}(0) (i=1,2,\dots ,n) is positive and matrix G={(g_{ij}(0))}_{n\times n} is irreducible, then (1.1) has a unique positive solution U(\lambda )=(u_1(\lambda ),u_2(\lambda ),\dots ,u_n(\lambda )) for all \lambda >0;

2. (ii)

   If g_{ij}(0)=0, g_{ij}^{\prime }(0)>0 for each i,j=1,2,\dots ,n and matrix G^{\prime }={(g_{ij}^{\prime }(0))}_{n\times n} is irreducible, then for some {\lambda }_{\ast }>0, (1.1) has no positive solution when \lambda \le {\lambda }_{\ast }, and (1.1) has a unique positive solution U(\lambda )=(u_1(\lambda ),u_2(\lambda ),\dots ,u_n(\lambda )) for \lambda >{\lambda }_{\ast }.

Moreover, \{(\lambda ,u_1(\lambda ),u_2(\lambda ),\dots ,u_n(\lambda )):\lambda >{\lambda }_{\ast }\} (in the first case, we assume {\lambda }_{\ast }=0) is a smooth curve so that u_i(\lambda ) is strictly increasing in λ, and u_i(\lambda )\to 0 as \lambda \to {\lambda }_{\ast }^{+}.

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 [1]. In other words, we will not assume that g_{ij} (i,j=1,2,\dots ,n) are smooth functions any more. (b) Furthermore, we will also consider the case that {lim}_{s\to 0}\frac{g_{ij}(s)}{s} (i,j=1,2,\dots ,n) 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

(A1) g_{ij}:[0,\mathrm{\infty })\to [0,\mathrm{\infty }) (i,j=1,2,\dots ,n) are continuous real functions satisfying

(A2) There exist constants g_{ij}^0\in (0,\mathrm{\infty }) such that

(A3) {lim}_{s\to \mathrm{\infty }}\frac{g_{ij}(s)}{s}=0, \mathrm{\forall }i,j=1,2,\dots ,n.

If the matrix {(g_{ij}^0)}_{n\times n} is irreducible, then there exists \hat{\lambda }>0 such that (1.1) has no positive solution for and (1.1) has at least one positive solution for \lambda \ge \hat{\lambda }.

Theorem 1.2 Let (A1) and (A3) hold. Assume the following.

(A2)′ For each i,j\in \{1,2,\dots ,n\}, there exist {\underline{g}}_{ij},{\overline{g}}_{ij}\in (0,\mathrm{\infty }) such that

(A4) The matrix J_{\alpha }:={((1-\alpha ){\underline{g}}_{ij}+\alpha {\overline{g}}_{ij})}_{n\times n} is irreducible, where \alpha \in [0,1].

Then for some \tilde{\lambda }>0, (1.1) has at least one positive solution for \lambda >\tilde{\lambda }.

Remark 1.1 It follows from (A2) and (A2)′ that the matrices {(g_{ij}^0)}_{n\times n} and J_{\alpha },\alpha \in [0,1] 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 g_{ij} 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 [1], 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.

Remark 1.3 For related results, established via bifurcation techniques, for other kind of problems, we refer the readers to [7–9] and the references therein.

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 X={[C_0(\overline{\mathrm{\Omega }})]}^n, here

The norm of U\in X will be defined as {\parallel U\parallel }_X={\sum }_{l=1}^n\parallel u_l\parallel, where \parallel \cdot \parallel denotes the norm of C(\overline{\mathrm{\Omega }}). We use W^{2,p}(\mathrm{\Omega }) and W_{\mathrm{loc}}^{2,p}(\mathrm{\Omega }) for the standard Sobolev space. We use N(L) and R(L) to denote the null and the range space of a linear operator L, respectively.

Let (\lambda ,U)=(\lambda ,(u_1,u_2,\dots ,u_n)) be a solution of (1.1). Suppose that g_{ij}:[0,\mathrm{\infty })\to [0,\mathrm{\infty }) (i,j=1,2,\dots ,n) are smooth real functions. Then we can deduce the eigenvalue problem

which can be rewritten as

where

Lemma 2.1 [1, 4]

Let Y={[W_{\mathrm{loc}}^{2,p}(\mathrm{\Omega })\cap C_0(\overline{\mathrm{\Omega }})]}^n and Z={[L^p(\mathrm{\Omega })]}^n with p>N. Suppose that L, H are given as in (2.3), and H is cooperative and irreducible. Then we have the following:

1. (a)

   {\mu }_1=inf\{Re(\mu ):\mu \in spt(L-H)\} is a real eigenvalue of L-H, where spt(L-H) is the spectrum of L-H.

2. (b)

   For \mu ={\mu }_1, there exists a unique (up a constant multiple) eigenfunction {\mathbf{u}}_1\in Y, and {\mathbf{u}}_1>0 in Ω.

3. (c)

   For , the equation L\mathbf{u}=H\mathbf{u}+\mu \mathbf{u}+\mathbf{f} is uniquely solvable for any \mathbf{f}\in Z, and \mathbf{u}>0 as long as \mathbf{f}\ge 0.

4. (d)

   (Maximum principle) For , assume that \mathbf{u}\in {[W_{\mathrm{loc}}^{2,p}(\mathrm{\Omega })\cap C(\overline{\mathrm{\Omega }})]}^n satisfies L\mathbf{u}\ge H\mathbf{u}+\mu \mathbf{u} in Ω, \mathbf{u}\ge 0 on ∂ Ω, then \mathbf{u}\ge 0 in Ω.

5. (e)

   If there exists \mathbf{u}\in {[W_{\mathrm{loc}}^{2,p}(\mathrm{\Omega })\cap C(\overline{\mathrm{\Omega }})]}^n satisfying L\mathbf{u}\ge H\mathbf{u} in Ω, \mathbf{u}\ge 0 on ∂ Ω, and either \mathbf{u}\not\equiv 0 on ∂ Ω or L\mathbf{u}\not\equiv H\mathbf{u} in Ω, then {\mu }_1>0.

For the results and proofs, see Proposition 3.1 and Theorem 1.1 of Sweers [10]. Moreover, from a standard compactness argument, there are countably many eigenvalues \{{\mu }_i\} of L-H, and |{\mu }_i-{\mu }_1|\to \mathrm{\infty } as i\to \mathrm{\infty }. We notice that {\mu }_i (i\ge 2) are not necessarily real-valued.

In this section, we also need to consider the adjoint operator of L-H. 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 {\mathbf{u}}^{\ast }={({\xi }_1^{\ast },{\xi }_2^{\ast },\dots ,{\xi }_n^{\ast })}^T. It is easy to verify that L-H^T is the adjoint operator of L-H, while both are considered as operators defined on subspaces of {[L^2(\mathrm{\Omega })]}^n.

The following lemmas are crucial in the proof of our main results.

Lemma 2.2 [1]

Let Y, Z, L and H be the same as in Lemma  2.1. Then the principal eigenvalue {\mu }_1 of L-H is also a real eigenvalue of L-H^T, {\mu }_1=inf\{\mu \in spt(L-H^T)\}, and for \mu ={\mu }_1, there exists a unique eigenfunction {\mathbf{u}}_1^{\ast }\in {[W_{\mathrm{loc}}^{2,2}(\mathrm{\Omega })\cap C_0(\overline{\mathrm{\Omega }})]}^n of L-H^T (up a constant multiple), and {\mathbf{u}}_1^{\ast }>0 in Ω.

Lemma 2.3 [[11], Theorem 5.3.1]

Let n\times n matrix A be a nonnegative irreducible matrix. Then \rho (A) is a simple eigenvalue of A, associated to a positive eigenvector, where \rho (A) denotes the spectral radius of A. Moreover, \rho (A)>0.

Lemma 2.4 [12]

Let V be a real Banach space. Suppose that

is completely continuous and F(\lambda ,0)=0 for all \lambda \in \mathbb{R}. Let a,b\in \mathbb{R} () such that u=0 is the isolated solution of the equation

Furthermore, assume that

where B_r(0) is an isolated neighborhood of trivial solutions. Let

Then there exists a continuum (i.e., a closed connected set) \mathcal{C} of \mathcal{S} containing [a,b]\times \{0\}, and either

1. (i)

   \mathcal{C} is unbounded in \mathbb{R}\times V; or

2. (ii)

   \mathcal{C}\cap [(\mathbb{R}\mathrm{\setminus }[a,b])\times \{0\}]\ne \mathrm{\varnothing }.

Finally, let ({\lambda }_1,{\phi }_1) be the principal eigen-pair of the linear eigenvalue problem

such that {\phi }_1>0 in Ω and \parallel {\phi }_1\parallel =1.

Proof of Theorem 1.1. We extend each g_{ij} to be a nonnegative continuous function, which is still denoted by g_{ij}, defined on ℝ in the following way: if , then g_{ij}(s)\equiv g_{ij}(0).

Let us define

where \lambda \in \mathbb{R}. Then it follows from (A1) that F:\mathbb{R}\times X\to X is continuous, and (\lambda ,U)=(\lambda ,(0,0,\dots ,0)) is always a solution of (1.1). Moreover, (A2) implies that F is differentiable at (\lambda ,U)=(\lambda ,(0,0,\dots ,0)), and

where J={(g_{ij}^0)}_{n\times n}. By (A2), all entries of J are positive. Therefore Lemma 2.3 yields the result that J has a positive principal eigenvalue {\chi }_J, the corresponding eigenvector {(k_1,k_2,\dots ,k_n)}^T satisfying k_i>0 (i=1,2,\dots ,n). Moreover, it is not difficult to verify that

where {\lambda }_{\ast }=\frac{{\lambda }_1}{{\chi }_J}. This implies that {(k_1{\phi }_1,k_2{\phi }_1,\dots ,k_n{\phi }_1)}^T is a positive eigenvector of the operator F_U({\lambda }_{\ast },(0,0,\dots ,0)). Similarly, J^T has the same principal eigenvalue {\chi }_J and the corresponding eigenvector is {(k_1^{\ast },\dots ,k_n^{\ast })}^T, where k_i^{\ast } (i=1,2,\dots ,n) is a positive constant. Obviously,

Hence when \lambda ={\lambda }_{\ast }=\frac{{\lambda }_1}{{\chi }_J}, F_U({\lambda }_{\ast },(0,0,\dots ,0)) is not invertible and \lambda ={\lambda }_{\ast } is a potential bifurcation point. More precisely, the null space

is one dimensional. In addition, it is easy to see that F_{\lambda }(\lambda ,U) and F_{\lambda U}(\lambda ,(0,0,\dots ,0)) exist for (\lambda ,U)\in \mathbb{R}\times X.

We divide the rest of the proof into two steps.

Step 1. We show that ({\lambda }_{\ast },(0,0,\dots ,0)) 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 {(h_1,h_2,\dots ,h_n)}^T\in R(F_U({\lambda }_{\ast },(0,0,\dots ,0))). Then there exists ({\psi }_1,{\psi }_2,\dots ,{\psi }_n)\in X such that

Let us consider the adjoint eigenvalue equation

where w_i^{\ast }=k_i^{\ast }{\phi }_1, i=1,2,\dots ,n. Multiplying the system (3.4) by {(w_1^{\ast },w_2^{\ast },\dots ,w_n^{\ast })}^T, multiplying the system (3.5) by {({\psi }_1,{\psi }_2,\dots ,{\psi }_n)}^T, integrating on Ω and subtracting, then we obtain

Thus {(h_1,h_2,\dots ,h_n)}^T\in R(F_U({\lambda }_{\ast },(0,0,\dots ,0))) if and only if (3.6) holds, which implies that R(F_U({\lambda }_{\ast },(0,0,\dots ,0))) is one dimensional.

Next, we verify that

Otherwise, we have

Since

multiplying the system (3.9) by {(k_1^{\ast }{\phi }_1,k_2^{\ast }{\phi }_1,\dots ,k_n^{\ast }{\phi }_1)}^T and using (3.8), we can get a contradiction that

By using [[13], Theorem 1.7], we conclude that ({\lambda }_{\ast },(0,0,\dots ,0)) is a bifurcation point. Furthermore, by the Rabinowitz global bifurcation theorem [14], there exists a continuum {\mathcal{C}}_1^{+} of positive solutions of (1.1), which joins ({\lambda }_{\ast },(0,0,\dots ,0)) to infinity in \mathbb{R}\times X. Clearly,

since (1.1) has only the trivial solution (0,(0,0,\dots ,0)) when \lambda =0.

Step 2: We claim that {\mathcal{C}}_1^{+} cannot blow up at some finite {\lambda }^{\ast }\in (0,\mathrm{\infty }).

Otherwise, a sequence \{({\lambda }^k,U^k)\}\subset {\mathcal{C}}_1^{+} can be taken such that

where U^k=(u_1^k,u_2^k,\dots ,u_n^k). Let K:C_0(\overline{\mathrm{\Omega }})\to C_0(\overline{\mathrm{\Omega }}) be the Green operator of −Δ subject to Dirichlet boundary conditions, i.e., u=Kv if and only if

By the elliptic regularity, ({\lambda }^k,U^k) satisfies

Here g_{ij} also denotes the Nemytski operator generated by itself. Clearly, (3.12) is equivalent to

where \mathcal{K}=diag(K,K,\dots ,K). It is well known that K:C_0(\overline{\mathrm{\Omega }})\to C_0(\overline{\mathrm{\Omega }}) is continuous and compact, and so \mathcal{K} is continuous and compact on (0,\mathrm{\infty })\times X.

Let w_i^k=\frac{u_i^k}{{\parallel U^k\parallel }_X} (i=1,2,\dots ,n). Then w_i^k>0 in Ω and {\parallel (w_1^k,w_2^k,\dots ,w_n^k)\parallel }_X=1. Dividing both sides of (3.13) with {\parallel U^k\parallel }_X, we have

For each i,j=1,2,\dots ,n, from (A2) and (A3) it follows that \frac{g_{ij}(s)}{s} is bounded in [0,\mathrm{\infty }). Moreover, we have

Therefore,

is bounded in X. This together with the compactness of \mathcal{K} implies that \{(w_1^k,w_2^k,\dots ,w_n^k)\} has a subsequence, denoted by itself, satisfying, in X,

Obviously, {\tilde{w}}_j\ge 0 (j=1,2,\dots ,n) in Ω and {\parallel ({\tilde{w}}_1,{\tilde{w}}_2,\dots ,{\tilde{w}}_n)\parallel }_X=1. In addition, we have {\lambda }^{\ast }>0. Or else, let k\to \mathrm{\infty }, then by (3.14) we get {\tilde{w}}_j\equiv 0 (j=1,2,\dots ,n) in Ω, which contradicts {\parallel ({\tilde{w}}_1,{\tilde{w}}_2,\dots ,{\tilde{w}}_n)\parallel }_X=1.

We define

Then for each j=1,2,\dots ,n,

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 k\to \mathrm{\infty } in (3.14), using (3.19) and the fact that {\lambda }^{\ast }>0 we can obtain

which contradicts {\parallel ({\tilde{w}}_1,{\tilde{w}}_2,\dots ,{\tilde{w}}_n)\parallel }_X=1.

Finally, by (3.10), the connectness of {\mathcal{C}}_1^{+} and above arguments, we can find some \hat{\lambda }>0 such that (1.1) has no positive solution for , and (1.1) has at least one positive solution for \lambda \ge \hat{\lambda }. □

To prove Theorem 1.2, we need the following lemmas as required.

By Remark 1.1 and Lemma 2.3, the matrices J_0={({\underline{g}}_{ij})}_{n\times n} and J_1={({\overline{g}}_{ij})}_{n\times n} have the principal eigenvalues {\chi }_0:=\rho (J_0)>0 and {\chi }_1:=\rho (J_1)>0, respectively, and the corresponding positive eigenvectors are {(k_1^0,k_2^0,\dots ,k_n^0)}^T and {(k_1^1,k_2^1,\dots ,k_n^1)}^T. Moreover, it is easy to obtain

and

where {\lambda }_1 is given as in (2.6). By Lemma 2.2, the matrices{(J_0)}^T and {(J_1)}^T have principal eigenvalues {\chi }_0 and {\chi }_1, respectively, the associated positive eigenvectors are {(k_1^{0\ast },k_2^{0\ast },\dots ,k_n^{0\ast })}^T and {(k_1^{1\ast },k_2^{1\ast },\dots ,k_n^{1\ast })}^T. We can easily verify that

and