Positive solutions for a class of sublinear elliptic systems

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]

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

Definition 1.2 [2]

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

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

(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 $\lambda <\hat{\lambda }$ 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.

(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.

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.

Lemma 2.1 [1, 4]

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.

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]

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 $s<0$, then $g_{ij}(s)\equiv g_{ij}(0)$.

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.

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.

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 ${\lambda }^{\ast }\in (0,\mathrm{\infty })$.

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 $(0,\mathrm{\infty })\times X$.

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