2006 International Conference in Honor of Jacqueline Fleckinger. Electronic Journal of Differential Equations, Conference 16, 2007, pp. 59–80. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu (login: ftp)

EXISTENCE OF SOLUTIONS FOR ELLIPTIC SYSTEMS INVOLVING OPERATORS IN DIVERGENCE FORM

LAURE CARDOULIS

Dedicated to Jacqueline Fleckinger on the occasion of an international conference in her honor

Abstract. In this paper, we obtain some results about the existence of solutions to the system

− ∑_{k,j=1}^{N} \frac{∂}{∂x_{k}} (ρ_{kj,i} \frac{∂u_{i}}{∂x_{j}} ) + q_{i}u_{i} = μ_{i}m_{i}u_{i} + g_{i}(x, u_{1}, …, u_{n}),

for i = 1, …, n defined in ℝ^{N}.

[section]

1. Introduction

1.1. The problem settings. We study the elliptic system

− ∑_{k,j=1}^{N} \frac{∂}{∂x_{k}} (ρ_{kj,i} \frac{∂u_{i}}{∂x_{j}} ) + q_{i}u_{i} = μ_{i}m_{i}u_{i} + g_{i}(x, u_{1}, …, u_{n}) in ℝ^{N},  (1.1)

for i = 1, …, n. We consider the following hypotheses for each i = 1, …, n:

(H1) q_{i} ∈ L_{loc}^{2}(ℝ^{N}) ∩ Lloc^{\frac{p}{2}}(ℝ^{N}), p > N, such that lim_{|x|→+∞} q_{i}(x) = +∞ and

q_{i} ≥ const > 0. (H2) For all j, k = 1, …, N, ρ_{kj,i} = ρ_{jk,i} and there exists positive constants

α_{i}, β_{i} such that for all ξ = (ξ_{1}, …, ξ_{N}) ∈ ℝ^{N},

α_{i}|ξ|^{2} ≤ ∑_{k,j=1}^{N} ρ_{kj,i}ξ_{j}ξ_{k} ≤ β_{i}|ξ|^{2}.

(H3) m_{i} ∈ L^{∞}(ℝ^{N}), m_{i} ≥ const > 0. We will specify later the form and the hypotheses on each g_{i} and we denote by μ_{i} real parameters for i = 1, …, n. The variational space is denoted by V_{q_{1}}(ℝ^{N}) × ⋯ × V_{q_{n}}(ℝ^{N}) where for i = 1, …, n, V_{q_{i}}(ℝ^{N}) is the completion of D(ℝ^{N}), the set of C^{∞} functions with compact support, under the norm

2000 Mathematics Subject Classification. 35J10.

Key words and phrases. Elliptic systems; sub and super solutions; bifurcation method.

\begin{matrix}c \\ ◯ \end{matrix}2007 Texas State University - San Marcos.

Published May 15, 2007.

∥u∥_{ρ_{i},q_{i}} = ( ∫_{ℝ^{N}} [ ∑_{k,j=1}^{N} ρ_{kj,i} \frac{∂u}{∂x_{j}} \frac{∂u}{∂x_{k}} + q_{i}u^{2}])^{1/2}.  (1.2)

Due to hypothesis (H2), V_{q_{i}}(ℝ^{N}) is also the completion of D(ℝ^{N}) under the norm

∥u∥_{q_{i}} = ( ∫_{ℝ^{N}} |∇u|^{2} + q_{i}u^{2})^{1/2}.  (1.3)

We recall that the embedding of each V_{q_{i}}(ℝ^{N}) into L^{2}(ℝ^{N}) is compact. We denote by

∥u∥_{m_{i}} = ( ∫_{ℝ^{N}} m_{i}u^{2})^{1/2}  (1.4)

for all u ∈ L^{2}(ℝ^{N}). According to the hypothesis (H3), ∥ · ∥_{m_{i}} is a norm in L^{2}(ℝ^{N}), equivalent to the usual norm so the embedding of V_{q_{i}}(ℝ^{N}) into (L^{2}(ℝ^{N}), ∥ · ∥_{m_{i}}) is still compact. We denote by M_{i} the operator of multiplication by m_{i} in L^{2}(ℝ^{N}) and by L_{ρ_{i}} the operator defined by

L_{ρ_{i}}u := − ∑_{k,j=1}^{N} \frac{∂}{∂x_{k}} (ρ_{kj,i} \frac{∂u}{∂x_{j}}).  (1.5)

The operator (L_{ρ_{i}} + q_{i})^{−1}M_{i} : (L^{2}(ℝ^{N}), ∥ · ∥_{m_{i}}) → (L^{2}(ℝ^{N}), ∥ · ∥_{m_{i}}) is positive self-adjoint and compact. So its spectrum is discrete and consists of a positive sequence tending to 0. We denote by λ_{i} the inverse of the first eigenvalue and by ϕ_{i} the corresponding eigenfunction which satisfy

(L_{ρ_{i}} + q_{i})ϕ_{i} = λ_{i}m_{i}ϕ_{i} in ℝ^{N},  (1.6)

λ_{i} > 0 and ∥ϕ_{i}∥_{m_{i}} = 1. (We recall that λ_{i} is simple and ϕ_{i} > 0 (see for example [1, 2, 4, 9, 21, 24] and Proposition 1.1 below.) By the Courant-Fischer formulas,

λ_{i} = inf \{\frac{∫_{ℝ^{N}} [∑_{k,j=1}^{N} ρ_{kj,i} \frac{∂ϕ}{∂x_{j}} \frac{∂ϕ}{∂x_{k}} + q_{i}ϕ^{2}]}{∫_{ℝ^{N}} m_{i}ϕ^{2}}, ϕ ∈ D(ℝ^{N})\}.  (1.7)

The aim of this paper is to study the existence of solutions for the system (1.1). This extends earlier results obtained for the Laplacian operator in a bounded domain (see [17, 18]), for an operator of divergence form in a bounded domain (when each ρ_{kj,i} is independent of j, see [14]), for equations or systems involving Schrödinger operators −Δ+q_{i} in ℝ^{N} (see [3, 11, 12, 13, 15]). The methods to get the existence of solutions are the Lax-Milgram Theorem for linear systems, applications of the sub and super solutions method, or the bifurcation method. Note that an important tool to obtain positive solutions is the Maximum Principle. We present in this paper a classification of different results for the existence of solutions for the system (1.1). Since these results are refinements of results obtained for Schrödinger operators whose potentials tend to infinity at infinity, we will only express the results for operators of divergence form studied here with some simple sketches of the proofs.

1.2. Review of results for the scalar case (i = 1). We consider here the following equation, in a variational sense,

− ∑_{k,j=1}^{N} \frac{∂}{∂x_{k}} (c_{k,j} \frac{∂u}{∂x_{j}}) + qu = λmu + g in ℝ^{N}.  (1.8)

We assume the following: The potential q satisfies (H1), the coefficients c_{k,j} satisfy (H2), the weight m satisfies (H3), the constant λ is a real parameter and finally g ∈ L^{2}(ℝ^{N}). We let L_{c} be the operator defined by L_{c}u := − ∑_{k,j=1}^{N} \frac{∂}{∂x_{k}} (c_{k,j} \frac{∂u}{∂x_{j}}) and M the multiplication operator given by the function m. Since the operator (L_{c} + q)^{−1}M : (L^{2}(ℝ^{N}), ∥ · ∥_{m}) → (L^{2}(ℝ^{N}), ∥ · ∥_{m}) is positive self-adjoint and compact, then its spectrum is discrete and consists of a positive sequence tending to 0. We denote by λ_{c,q} the inverse of the first (largest) eigenvalue of the operator (L_{c} + q)^{−1}M.

As in [1, 2, 4, 9, 20], we have the following result.

Proposition 1.1. The eigenvalue λ_{c,q} is simple and there exists an associated eigenfunction ϕ_{c,q} which is a strictly positive and continuous function in ℝ^{N}.

We recall that the above result is well known for the case of bounded domains or the case where L_{c} + q = −Δ + q is a Schrödinger operator in ℝ^{ℕ}.

Proof. First, we conclude from [25, Theorem 7.1] that every solution ψ ∈ L^{2}(ℝ^{N}) of the equation (L_{c} + q − λ_{c,q}m)ψ = 0 in L^{2}(ℝ^{N}) is a continuous function in ℝ^{N}.

Next, we will show that λ_{c,q} is simple and it has an eigenfunction which is strictly positive. To do this, we follow [1]. Note that λ_{c,q} is of finite multiplicity. If in contrast λ_{c,q} is not simple, then there exists an eigenfunction, say, ψ, which changes sign. It follows that ψ^{+} := max (ψ, 0) is also an eigenfunction associated with λ_{c,q}. Let Ω_{+} := {x ∈ ℝ^{N}, ψ(x) > 0}; we denote by λ_{+} the inverse of the first eigenvalue of (L_{c} + q)^{−1}M defined on Ω_{+} with the Dirichlet boundary condition. Then λ_{c,q} = λ_{+}. Since Ω_{+} ≠ ℝ^{N}, ψ and ψ^{+} are linearly independent. Hence, by considering all domains Ω_{∗} such that Ω_{+} ⊂ Ω_{∗} ⊂ ℝ^{N}, we can construct an infinite number of linearly independent eigenfunctions associated with λ_{c,q}, contradicting the fact that λ_{c,q} is of finite multiplicity. Therefore, we have shown that λ_{c,q} is simple and every eigenfunction of it does not change sign.

To complete the proof, we consider a non-negative eigenfunction ϕ of λ_{c,q} and need to showing that ϕ > 0. Assuming that there exists y ∈ ℝ^{N} such that ϕ(y) = 0. Let R > 0 and r > 0 be such that B(y, r) ⊂ B(0, R). Using the Harnack Inequality for the operator L_{c} + q − λ_{c,q}m (see [20, Theorem 8.20], [25, Theorem 8.1]), we have sup_{B(y,4r)} ϕ ≤ C inf_{B(y,r)} ϕ = 0, and we deduce that ϕ = 0 in B(y, r) (for any r > 0). Therefore ϕ ≡ 0, which is impossible, since ϕ is an eigenfunction. □

We have the following weak Maximum Principle for (1.8).

Theorem 1.1 (The weak Maximum Principle). Assume that λ < λ_{c,q}, g ≥ 0 and u is a solution of the equation (1.8). Then u ≥ 0.

Proof. The idea is to multiply the equation (1.8) by u^{−} = Max(0, −u), then to use the characterisation of λ_{c,q} (see (1.7)) in order to obtain (λ_{c,q} − λ) ∫_{ℝ^{N}} m(u^{−})^{2} ≤ 0. Therefore, under the conditions λ_{c,q} − λ > 0 and m > 0 we obtain that u^{−} = 0. □

By using the Lax-Milgram Theorem we obtain the existence of a solution for the equation (1.8).

Theorem 1.2. Assume λ < λ_{c,q}. Then there exists a unique solution u ∈ V_{q}(ℝ^{N}) for the equation (1.8). Moreover, by the Maximum Principle, if g ≥ 0, then this solution u satisfies u ≥ 0.

Proof. Let l : (V_{q}(ℝ^{N}))^{2} → ℝ be the quadratic form given by

l(u, v) := ∫_{ℝ^{N}} [ ∑_{k,j=1}^{N} c_{k,j} \frac{∂u}{∂x_{j}} \frac{∂v}{∂x_{k}} + quv − λmuv] ∀(u, v) ∈ (V_{q}(ℝ^{N}))^{2}.

Let α be a real number such that λ + α > 0. Consider the norm

∥u∥_{c,q+αm} := ( ∫_{ℝ^{N}} [ ∑_{k,j=1}^{N} c_{kj} \frac{∂u}{∂x_{j}} \frac{∂u}{∂x_{k}} + (q + αm)u^{2}])^{1/2} u ∈ V_{q}(ℝ^{N}).

It follows from the characterisation (1.7) of λ_{c,q} that l(u, u) ≥ \frac{λ_{c,q}−λ}{λ_{c,q}+α}∥u∥_{c,q+αm}^{2}. So the continuous bilinear form l is coercive. By the Lax-Milgram theorem, we get the existence of a unique weak solution u for the equation (1.8). Moreover, if g ≥ 0, by Theorem 1.1, we have u ≥ 0. □

1.3. Properties of M-matrices. We say that a matrix is positive if all its entries are nonnegative and we say that a matrix is positive definite if this matrix is symmetric and if all its principal minors are strictly positive. We recall some results about the M-matrices (see [5, Theorem 2.3, p.134]). Let I be the identity matrix. A matrix M = sI − B is called a non singular M-matrix, if B is a positive matrix and s is a real number such that s > ρ(B), where ρ(B) denotes the spectral radius of B.

Proposition 1.2. If M is a matrix with nonpositive off-diagonal entries, then the following five conditions are equivalent.

(P0) M is a non singular M-matrix.

(P1) All the principal minors of M are strictly positive.

(P2) M is semi-positive, i.e., there exists X >> 0 such that MX >> 0. Here

X >> 0 means that the entries of X are strictly positive. (P3) M has a positive inverse. (P4) There exists a diagonal matrix D, D > 0 such that ^{t}MD + DM is positive

definite.

2. Results for linear systems

In this section, we consider the system (1.1) in the form

− ∑_{k,j=1}^{N} \frac{∂}{∂x_{k}} (ρ_{kj,i} \frac{∂u_{i}}{∂x_{j}}) + q_{i}u_{i} = μ_{i}m_{i}u_{i} + ∑_{j=1;j≠i}^{n} a_{ij}u_{j} + f_{i} in ℝ^{N},  (2.1)

i = 1, …, n. We consider the hypotheses:

(H4) For all i, j = 1, …, n, a_{ij} ∈ L^{∞}(ℝ^{N}).

(H5) For all i = 1, …, n, f_{i} ∈ L^{2}(ℝ^{N}).

2.1. Case of a cooperative system: a_{ij} ≥ 0 (∀i ≠ j). We obtain here results for the Maximum Principle and the existence of solutions for a linear cooperative system. We use classical methods as in [1, 3, 12, 14, 17, 18]. For a cooperative system we suppose also the following hypothesis

(H6) For all i, j, i ≠ j ⇒ a_{ij} ≥ 0. For each i ≠ j, since each weight m_{i} is bounded below by a positive constant, we deduce the existence of positive constants K_{ij} such that a_{ij} ≤ K_{ij}\sqrt{m_{i}}\sqrt{m_{j}}. Note that in the particular case where m_{i} = 1 for each i, we can take K_{ij} = ∥a_{ij}∥_{L^{∞}(ℝ^{N})}. We denote by L = (l_{ij}) the n × n-matrix given as follows

l_{ii} := λ_{i} − μ_{i} and l_{ij} := −K_{ij} (i ≠ j).  (2.2)

For such a system, we have the following maximum principle.

Theorem 2.1. Assume (H1)–(H6) are satisfied. If the matrix L is a non singular M-matrix, then the cooperative system (2.1) satisfies the Maximum Principle.

Proof. Assume that for all i = 1, …, n, f_{i} ≥ 0. Let u = (u_{1}, …, u_{n}) be a solution of the system (2.1) and define u_{i}^{−} = max(0, −u_{i}). Multiplying by u_{i}^{−} and integrating over ℝ^{N}, we obtain

∫_{ℝ^{N}} [ ∑_{k,j=1}^{N} ρ_{kj,i} \frac{∂u_{i}}{∂x_{j}} \frac{∂u_{i}^{−}}{∂x_{k}} +q_{i}u_{i}u_{i}^{−}] = μ_{i} ∫_{ℝ^{N}} m_{i}u_{i}u_{i}^{−}+ ∑_{j=1;j≠i}^{n} ∫_{ℝ^{N}} a_{ij}u_{j}u_{i}^{−}+∫_{ℝ^{N}} f_{i}u_{i}^{−}.

Due to the characterisation of λ_{i} (see (1.7)), we obtain

λ_{i} ∫_{ℝ^{N}} m_{i}(u_{i}^{−})^{2} ≤ μ_{i} ∫_{ℝ^{N}} m_{i}(u_{i}^{−})^{2} + ∑_{j=1;j≠i}^{n} K_{ij}(∫_{ℝ^{N}} m_{i}(u_{i}^{−})^{2})^{1/2}(∫_{ℝ^{N}} m_{j}(u_{j}^{−})^{2})^{1/2}.

Thus, if we denote by ^{t}X = (x_{1}, …, x_{n}) where x_{i} = (∫_{ℝ^{N}} m_{i}(u_{i}^{−})^{2})^{1/2}, we have LX ≤ 0. Since L is a non singular M-matrix, we can deduce that X ≤ 0, so X = 0 and therefore u_{i} ≥ 0 for each i. □

Existence and uniqueness of a solution is stated as follows.

Theorem 2.2. Assume (H1)–(H6) are satisfied. If the matrix L (given by (2.2)) is a non singular M-matrix, then the cooperative system (2.1) has a unique solution u = (u_{1}, …, u_{n}) ∈ V_{q_{1}}(ℝ^{N}) × ⋯ × V_{q_{n}}(ℝ^{N}). Moreover, due to the Maximum Principle, if f_{i} ≥ 0 for all i, then this solution u satisfies u_{i} ≥ 0 for i = 1, …, n.

Proof. We suppose that L is a non singular M-matrix. Using (P4) (see Proposition 1.2), we introduce D a diagonal positive matrix such that ^{t}LD + DL is positive definite. We denote by d_{1}, …, d_{n} the diagonal entries of the diagonal matrix D. As for one equation, the method is based on the Lax-Milgram theorem.

Let α be a positive number such that for all 1 ≤ i ≤ n, μ_{i} + α > 0. Let l : (V_{q_{1}}(ℝ^{N}) × ⋯ × V_{q_{n}}(ℝ^{N}))^{2} → ℝ be defined by

l(u, v) = ∑_{i=1}^{n} d_{i} ∫_{ℝ^{N}} [ ∑_{k,j=1}^{N} ρ_{kj,i} \frac{∂u_{i}}{∂x_{j}} \frac{∂v_{i}}{∂x_{k}}

+ (q_{i} + αm_{i})u_{i}v_{i} − (μ_{i} + α)m_{i}u_{i}v_{i} − ∑_{j=1;j≠i}^{n} a_{ij}u_{j}v_{i}],

if u = (u_{1}, …, u_{n}) ∈ V_{q_{1}}(ℝ^{N}) × ⋯ × V_{q_{n}}(ℝ^{N}), and v = (v_{1}, …, v_{n}) ∈ V_{q_{1}}(ℝ^{N}) × ⋯ × V_{q_{n}}(ℝ^{N}). We denote by

∥u_{i}∥_{ρ_{i},q_{i}+αm_{i}} = ( ∫_{ℝ^{N}} [ ∑_{k,j=1}^{N} ρ_{kj,i} \frac{∂u_{i}}{∂x_{j}} \frac{∂u_{i}}{∂x_{k}} + (q_{i} + αm_{i})u_{i}^{2}])^{1/2}.  (2.3)

By the characterisation of λ_{i} (see (1.7)) and the Cauchy-Schwartz inequality, we get:

l(u, u) ≥ ∑_{i=1}^{n} d_{i} \frac{λ_{i} − μ_{i}}{λ_{i} + α} ∥u_{i}∥_{ρ_{i},q_{i}+αm_{i}}^{2}

− ∑_{i,j=1;i≠j}^{n} \frac{d_{i}K_{ij}}{\sqrt{λ_{i} + α}\sqrt{λ_{j} + α}}∥u_{i}∥_{ρ_{i},q_{i}+αm_{i}}∥u_{j}∥_{ρ_{j},q_{j}+αm_{j}}.

Setting ^{t}X = (x_{1}, …, x_{n}) with x_{i} = \frac{∥u_{i}∥_{ρ_{i},q_{i}+αm_{i}}}{\sqrt{λ_{i}+α}}, we get:

l(u, u) ≥^{t} XDLX = \frac{1}{2} ^{t}X[^{t}LD + DL]X.

Since ^{t}LD + DL is positive definite, we deduce that l is coercive. Therefore, by the Lax-Milgram Theorem, we get the existence and the uniqueness of a weak solution for the system (2.1). □

2.2. Case of a non necessarily cooperative system. We give a very similar result for the existence and uniqueness of a solution for the system (2.1) as the one obtained for a cooperative system. We do not give the proof which is exactly the same as for Theorem 2.2. We note that for all i, j, i ≠ j there exists positive constants K′_{ij} such that |a_{ij}| ≤ K′_{ij}\sqrt{m_{i}}\sqrt{m_{j}}. We denote by L′ = (l′_{ij}) the n × nmatrix given by:

l′_{ii} := λ_{i} − μ_{i} and l′_{ij} := −K′_{ij} (i ≠ j).  (2.4)

Existence and uniqueness of a solution is stated as follows.

Theorem 2.3. Assume (H1)–(H5) are satisfied. If the matrix L′ (given by (2.4)) is a non singular M-matrix, then the system (2.1) has a unique solution in V_{q_{1}}(ℝ^{N})× ⋯ × V_{q_{n}}(ℝ^{N}).

3. Results for semilinear systems

In this section, we consider system (1.1) in the form:

− ∑_{k,j=1}^{N} \frac{∂}{∂x_{k}} (ρ_{kj,i} \frac{∂u_{i}}{∂x_{j}}) + q_{i}u_{i} = μ_{i}m_{i}u_{i} + ∑_{j=1;j≠i}^{n} a_{ij}u_{j} + f_{i}(x, u_{1}, …, u_{n}) in ℝ^{N},

(3.1)

i = 1, …, n.

We consider also the following hypotheses which hold for each i = 1, …, n: (H7) f_{i} is Lipschitz respect to u_{i} uniformly in x. (H8) There exists θ_{i} ∈ L^{2}(ℝ^{N}) such that 0 ≤ f_{i}(x, u_{1}, …, u_{n}) ≤ θ_{i} for all

u_{1} ≥ 0, …, u_{n} ≥ 0. (H9) There exists θ_{i} ∈ L^{2}(ℝ^{N}) such that |f_{i}(x, u_{1}, …, u_{n})| ≤ θ_{i} for all u_{1}, …, u_{n}.

We obtain two results for the existence of a solution for the system (3.1), which is either cooperative or non cooperative.

3.1. Case of a cooperative system: a_{ij} ≥ 0 (∀i ≠ j). As in [3, 14, 18], we use a sub and super solutions method with a Schauder Fixed Point Theorem to obtain the existence of a positive solution if system (3.1) is cooperative.

Theorem 3.1. Assume (H1)–(H4), (H6)–(H8) are satisfied. If the matrix L (given by (2.2) is a non singular M-matrix, then the system (3.1) has at least one positive solution in V_{q_{1}}(ℝ^{N}) × ⋯ × V_{q_{n}}(ℝ^{N}).

Proof. We consider the system

− ∑_{k,j=1}^{N} \frac{∂}{∂x_{k}} (ρ_{kj,i} \frac{∂u_{i}}{∂x_{j}}) + q_{i}u_{i} = μ_{i}m_{i}u_{i} + ∑_{j=1;j≠i}^{n} a_{ij}u_{j} + θ_{i} in ℝ^{N},  (3.2)

i = 1, …, n. Applying Theorem 2.2 we deduce that the system (3.2) has a unique solution (which is positive by the Maximum Principle) u^{0} = (u_{1}^{0}, …, u_{n}^{0}). Moreover, by (H8), u^{0} is a super solution of the system (3.1). Note also that u_{0} = (0, …, 0) is a sub solution of the system (3.1). We denote by σ = [u_{0}, u^{0}].

To show the existence of positive solutions to system (3.1), we choose a positive real number α to be such that μ_{i} +α > 0 for all i. Let T : (L^{2}(ℝ^{N}))^{n} → (L^{2}(ℝ^{N}))^{n} be defined as follows: If u = (u_{1}, …, u_{n}), then T (u) := v = (v_{1}, …, v_{n}), v satisfies the equations

− ∑_{k,j=1}^{N} \frac{∂}{∂x_{k}} (ρ_{kj,i} \frac{∂v_{i}}{∂x_{j}})+(q_{i}+αm_{i})v_{i} = (μ_{i}+α)m_{i}u_{i}+ ∑_{j=1;j≠i}^{n} a_{ij}u_{j}+f_{i}(x, u_{1}, …, u_{n})

(3.3)

in ℝ^{N} for each i = 1, …, n. Note that, by the scalar case, T is well defined and T (σ) ⊂ σ. As in [3], using the compact embedding of each V_{q_{i}}(ℝ^{N}) into L^{2}(ℝ^{N}), we can prove that T is continuous and that T (σ) is compact. By the Schauder Fixed Point Theorem, we deduce the existence of u = (u_{1}, …, u_{n}) ∈ σ such that T (u) = u. Equivalently, u is a positive solution of the system (3.1). □

3.2. Case of a non cooperative semilinear system. For a non cooperative system, we obtain the following result.

Theorem 3.2. Assume (H1)–(H4), (H7), (H9) are satisfied. If the matrix L′ (given by (2.4)) is a non singular M-matrix, then the system (3.1) has at least one solution in V_{q_{1}}(ℝ^{N}) × ⋯ × V_{q_{n}}(ℝ^{N}).

Proof. We proceed exactly as for Theorem 3.1 by considering a sub and a super solution of the system (3.1) and using the Schauder Fixed Point Theorem. To do so, we consider the following system

− ∑_{k,j=1}^{N} \frac{∂}{∂x_{k}} (ρ_{kj,i} \frac{∂u_{i}}{∂x_{j}}) + q_{i}u_{i} = μ_{i}m_{i}u_{i} + ∑_{j=1;j≠i}^{n} |a_{ij}|u_{j} + θ_{i} in ℝ^{N},  (3.4)

for i = 1, …, n. Applying Theorem 2.2, we deduce that the system (3.4) has a unique solution (which is positive by the Maximum Principle) u^{0} = (u_{1}^{0}, …, u_{n}^{0}). Moreover, by (H8), u^{0} is a super solution of the system (3.1). Note also that −u^{0} is a sub solution of the system (3.1). We denote by σ = [−u^{0}, u^{0}].

Let α be a positive real number such that μ_{i} + α > 0 for all i. Let T : (L^{2}(ℝ^{N}))^{n} → (L^{2}(ℝ^{N}))^{n} be defined as follows: If u = (u_{1}, …, u_{n}), then T (u) := v = (v_{1}, …, v_{n}), where v is determined by the equations:

− ∑_{k,j=1}^{N} \frac{∂}{∂x_{k}} (ρ_{kj,i} \frac{∂v_{i}}{∂x_{j}})+(q_{i}+αm_{i})v_{i} = (μ_{i}+α)m_{i}u_{i}+ ∑_{j=1;j≠i}^{n} a_{ij}u_{j}+f_{i}(x, u_{1}, …, u_{n})

in ℝ^{N} for i = 1, …, n. Note that, by the scalar case, T is well defined and such that T (σ) ⊂ σ. As in Theorem 3.1, we can prove that T is continuous and that T (σ) is compact. By the Schauder Fixed Point Theorem, we deduce the existence of u = (u_{1}, …, u_{n}) ∈ σ such that T (u) = u. Clearly, u is also a solution of the system (3.1). □

If we relax the hypothesis about each weight m_{i} i.e. if we do not suppose that each weight m_{i} is bounded, then for the case where the system (3.1) is non necessarily cooperative we cannot apply the method developed for the Theorem 3.2. Indeed, the operator T would no longer be continuous from (L^{2}(ℝ^{N}))^{n} to (L^{2}(ℝ^{N}))^{n} ; moreover, even if we define T from (L^{2^{∗}}(ℝ^{N}))^{n} to (L^{2^{∗}}(ℝ^{N}))^{n}, we would lose the compact embedding of each V_{q_{i}}(ℝ^{N}) into L^{2^{∗}}(ℝ^{N}). So we use, as in [11, 12], an approximation method due to Boccardo, Fleckinger and de Thélin (see [6]). For the following result, we assume that

N ≥ 3, 0 ≤ m_{i} ∈ L^{\frac{N}{2}} (ℝ^{N}), m_{i} ≠ 0.

We recall that λ(m_{i}) has been defined by (1.7). We denote by L′′ = (l′′_{ij}) the n × n-matrix given by:

l′′_{ii} := λ(m_{i}) − μ_{i} and l′′_{ij} := −K′_{ij} (i ≠ j),

where K′_{ij} is a positive constant such that |a_{ij}| ≤ K′_{ij}\sqrt{m_{i}}\sqrt{m_{j}}.

Theorem 3.3. Assume (H1), (H2), (H4), (H7), (H9) are satisfied. Assume that N ≥ 3, 0 ≤ m_{i} ∈ L^{\frac{N}{2}} (ℝ^{N}), m_{i} ≠ 0. Assume also that for all i = 1, …, n, m_{i} ∈ L_{loc}^{∞}(ℝ^{N}) ∩ L^{1}(ℝ^{N}). If the matrix L′′ is a non singular M-matrix, then the system (3.1) has at least one solution in V_{q_{1}}(ℝ^{N}) × ⋯ × V_{q_{n}}(ℝ^{N}).

Proof. Let ϵ ∈]0, 1[ and B_{ϵ} = B(0, 1/ϵ) = {x ∈ ℝ^{N}, |x| < 1/ϵ}. Let 1_{B_{ϵ}} be the indicator function of B_{ϵ} and let α be a positive real such that for i, μ_{i} + α > 0. Define Λ = 1_{B_{ϵ}}.

Let T_{ϵ} : (L^{2}(ℝ^{N}))^{n} → (L^{2}(ℝ^{N}))^{n} be defined as follows: If u = (u_{1}, …, u_{n}) ∈ (L^{2}(ℝ^{N}))^{n}, then T_{ϵ}(u) := v = (v_{1}, …, v_{n}) where v is determined by the equations:

L_{ρ_{i}}v_{i}+(q_{i}+αm_{i})v_{i} = (μ_{i}+α)\frac{m_{i}u_{i}}{1 + ϵm_{i}|u_{i}|}Λ+ ∑_{j=1,j≠i}^{n} a_{ij} \frac{u_{j}}{1 + ϵ|u_{j}|}Λ+f_{i}(x, u_{1}, …, u_{n})

(3.5)

in ℝ^{N}, for i = 1, …, n. By the scalar case, T_{ϵ} is well defined and if we denote by h_{i} = max_{j,j≠i}((μ_{i} + α), |a_{ij}|))\frac{1}{ϵ}Λ ∈ L^{2}(ℝ^{N}), the equation (L_{ρ_{i}} + q_{i} + αm_{i})ξ_{ϵ,i} = nh_{i} + θ_{i} in ℝ^{N} admits a positive solution ξ_{ϵ,i} in V_{q_{i}}(ℝ^{N}). So ξ_{ϵ} = (ξ_{ϵ,1}, …, ξ_{ϵ,n}) is a super solution for (3.5) and ζ_{ϵ} = (ζ_{ϵ,1}, …, ζ_{ϵ,n}) = −ξ_{ϵ} is a sub solution for (3.5). Let σ_{ϵ} := [ζ_{ϵ}, ξ_{ϵ}]. Then T_{ϵ}(σ_{ϵ}) ⊂ σ_{ϵ}.

We prove easily that T_{ϵ} is a continuous function (by the hypotheses (H4), (H7) and using the function l : ℝ → ℝ defined by l(x) = \frac{x}{1+|x|}, which is Lipschitz and which satisfies: for all x, y ∈ ℝ, |l(x) − l(y)| ≤ |x − y|).

We prove also easily that T_{ϵ}(σ_{ϵ}) is compact (due to the hypotheses (H4) and (H9)). So by the Schauder Fixed Point Theorem, we can deduce the existence of u_{ϵ} = (u_{1,ϵ}, …, u_{n,ϵ}) ∈ σ_{ϵ} such that T_{ϵ}(u_{ϵ}) = u_{ϵ}.

Note that for i, (ϵu_{i,ϵ})_{ϵ} is a bounded sequence in V_{q_{i}}(ℝ^{N}). Indeed, by (3.5) and (H9) we get

∥ϵu_{i,ϵ}∥_{ρ_{i},q_{i}+αm_{i}}^{2} ≤ \frac{μ_{i} + α}{λ(m_{i}) + α}∥ϵu_{i,ϵ}∥_{ρ_{i},q_{i}+αm_{i}}^{2}

+ ∑_{j=1,j≠i}^{n} K′_{ij}(∫_{ℝ^{N}} m_{j})^{1/2}∥ϵu_{i,ϵ}∥_{m_{i}} + ∥θ_{i}∥_{L^{2}(ℝ^{N})}∥ϵu_{i,ϵ}∥_{L^{2}(ℝ^{N})},

and so we can deduce the existence of a positive constant K, (independent of ϵ) such that ∥ϵu_{i,ϵ}∥_{ρ_{i},q_{i}+αm_{i}} ≤ K.

Since the embedding of V_{q_{i}}(ℝ^{N}) into L^{2}(ℝ^{N}) is compact, we deduce that (for a subsequence) there exists u_{i}^{∗} such that ϵu_{i,ϵ} → u_{i}^{∗} as ϵ → 0 strongly in L^{2}(ℝ^{N}) and weakly in V_{q_{i}}(ℝ^{N}). Using the Dominated Convergence Theorem, due to (H9), we can pass to the limit (see (3.5)) and we can conclude that u_{i}^{∗} is a weak solution of

(L_{ρ_{i}} + q_{i} + αm_{i})u_{i}^{∗} = (μ_{i} + α)\frac{m_{i}u_{i}^{∗}}{1 + m_{i}|u_{i}^{∗}|} + ∑_{j=1,j≠i}^{n} a_{ij} \frac{u_{j}^{∗}}{1 + |u_{j}^{∗}|} in ℝ^{N}. (3.6)

Moreover, if we set ^{t}X = (x_{1}, …, x_{n}) with x_{i} = (∫_{ℝ^{N}} m_{i}u_{i}^{∗2})^{1/2}, using (3.6) we obtain L′′X ≤ 0. Since L′′ is a non singular M-matrix, we deduce that X = 0 i.e. for i, u_{i}^{∗} = 0.

We prove now by contradiction that for i, (u_{i,ϵ})_{ϵ} is bounded in V_{q_{i}}(ℝ^{N}), We suppose that (for a subsequence) there exists i_{0} such that ∥u_{i_{0},ϵ}∥_{ρ_{i_{0}} ,q_{i_{0}} +αm_{i_{0}}} → +∞ as ϵ → 0. Let

t_{ϵ} = max_{i} ∥u_{i,ϵ}∥_{ρ_{i},q_{i}+αm_{i}} and z_{i,ϵ} = \frac{1}{t_{ϵ}} u_{i,ϵ}.

Since (z_{i,ϵ})_{ϵ} is a bounded sequence in V_{q_{i}}(ℝ^{N}), there exists z_{i} such that z_{i,ϵ} → z_{i} as ϵ → 0 strongly in L^{2}(ℝ^{N}) and weakly in V_{q_{i}}(ℝ^{N}). We can pass to the limit (see (3.5)) and conclude that z_{i} is a weak solution of

(L_{ρ_{i}} + q_{i} + αm_{i})z_{i} = (μ_{i} + α)m_{i}z_{i} + ∑_{j=1,j≠i}^{n} a_{ij}z_{j} in ℝ^{N}.

Then, we can prove that for all i, z_{i} = 0. However, there exists a sequence (ϵ_{n}) such that there exists i_{1}, ∥z_{i_{1},ϵ_{n}}∥_{ρ_{i_{1}} ,q_{i_{1}} +αm_{i_{1}}} = 1. But z_{i_{1},ϵ_{n}} → z_{i_{1}} = 0 as n → +∞. So we get a contradiction.

Finally, there exists u_{i}^{0} such that u_{i,ϵ} → u_{i}^{0} strongly in L^{2}(ℝ^{N}) and weakly in V_{q_{i}}(ℝ^{N}). We can pass to the limit (see (3.5)) and to obtain

(L_{ρ_{i}} + q_{i} + αm_{i})u_{i}^{0} = (μ_{i} + α)m_{i}u_{i}^{0} + ∑_{j=1,j≠i}^{n} a_{ij}u_{j}^{0} + f_{i}(u_{1}^{0}, …, u_{n}^{0}) in ℝ^{N}

for all i. This completes the proof. □

3.3. A bifurcation result. In this section, we obtain a result for the existence of solutions for the system (1.1) by considering bifurcating solutions from the zero solution. We proceed here as in [10, 13, 15]. In this section, we suppose that the hypotheses (H1)–(H3) as well as (H10) are satisfied, where (H10) reads as follows:

(i) for i = 1, …, n, g_{i} : ℝ^{N} ×ℝ^{n} → ℝ, defined by g_{i}(x, y_{1}, …, y_{n}) with x ∈ ℝ^{N} and (y_{1}, …, y_{n}) ∈ ℝ^{n}, satisfies For all x ∈ ℝ^{N}, g_{i}(x, 0, …, 0) = 0. (ii) For i = 1, …, n, g_{i} is Frechet differentiable with respect to each variable y_{j} and each derivative \frac{∂g_{i}}{∂y_{j}} (x, .) is continuous and bounded, uniformly in x.

(iii) For all i, j = 1, …, n, \frac{∂g_{i}}{∂y_{j}} (x, 0, …, 0) = 0.

Note that g_{i} is Lipschitz in (y_{1}, …, y_{n}) uniformly in x. We denote by V = Π_{i=1}^{n}V_{q_{i}}(ℝ^{N}) and by ⟨., .⟩_{V} the inner product in V (i.e. for v = (v_{1}, …, v_{n}) ∈ V and all w = (w_{1}, …, w_{n}) ∈ V, ⟨v, w⟩_{V} = ∑_{i=1}^{n}⟨v_{i}, w_{i}⟩_{ρ_{i},q_{i}} (see (1.2))). We define the operator T : ℝ^{n} × V → V, T = (T^{1}, …, T^{n}) by:

⟨T^{i}(μ, u), v_{i}⟩_{ρ_{i},q_{i}} = ∫_{ℝ^{N}} [ ∑_{k,j=1}^{N} ρ_{kj,i} \frac{∂u_{i}}{∂x_{j}} \frac{∂v_{i}}{∂x_{k}} + q_{i}u_{i}v_{i} − μ_{i}m_{i}u_{i}v_{i} − g_{i}(x, u)v_{i}]. (3.7)

for i = 1, …, n, T^{i} : ℝ^{n} × V → V_{q_{i}}(ℝ^{N}), if μ = (μ_{1}, …, μ_{n}) ∈ ℝ^{n}, u = (u_{1}, …, u_{n}) ∈ V, for all v_{i} ∈ V_{q_{i}}(ℝ^{N}). We proceed in this section exactly as in [15] and so we give only the steps on each proofs.

Proposition 3.1. The operator T is well defined. Furthermore, for each i, the operator T^{i} is continuous, Frechet differentiable with continuous derivatives given by: For all ϕ ∈ V_{q_{i}}(ℝ^{N}) and for all ψ ∈ V_{q_{i}}(ℝ^{N}),

if j ≠ i, T_{μ_{j}}^{i} = 0, ⟨T_{u_{j}}^{i}(μ, u)ϕ, ψ⟩_{ρ_{i},q_{i}} = − ∫_{ℝ^{N}} \frac{∂g_{i}}{∂y_{j}} (x, u)ϕψ,

if j = i, ⟨T_{μ_{i}}^{i}(μ, u), ϕ⟩_{ρ_{i},q_{i}} = − ∫_{ℝ^{N}} m_{i}u_{i}ϕ and

⟨T_{u_{i}}^{i}(μ, u)ϕ, ψ⟩_{ρ_{i},q_{i}} = ∫_{ℝ^{N}} [ ∑_{k,j=1}^{N} ρ_{kj,i} \frac{∂ϕ}{∂x_{j}} \frac{∂ψ}{∂x_{k}} + q_{i}ϕψ − μ_{i}m_{i}ϕψ − \frac{∂g_{i}}{∂y_{i}} (x, u)ϕψ],

if j ≠ i, T_{μ_{j} u_{i}}^{i} = 0 = T_{μ_{i} u_{j}}^{i},
if j = i, ⟨T_{μ_{i} u_{i}}^{i}(μ, u)ϕ, ψ⟩_{ρ_{i},q_{i}} = − ∫_{ℝ^{N}} m_{i}ϕψ and T_{μ_{i} u_{i}}^{k} = 0 if k ≠ i.

Proposition 3.2. The operator T_{u}(λ, 0) is a continuous self-adjoint operator with λ = (λ_{1}, …, λ_{n}). The kernel of T_{u}(λ, 0), denoted by N(T_{u}(λ, 0)) is generated by {Φ_{1}, …, Φ_{n}} where for i = 1, …, n, Φ_{i} = (0, …, 0, ϕ_{i}, 0, …, 0).

Moreover, if we denote by R(T_{u}(λ, 0)) the range of the operator T_{u}(λ, 0), we have:

(1) codim(R(T_{u}(λ, 0))) = n
(2) for i = 1, …, n, T_{μ_{i} u}(λ, 0)Φ_{i} ∉ R(T_{u}(λ, 0))
(3) dim(Span{T_{μ_{i} u}(λ, 0)Φ_{i}, 1 ≤ i ≤ n}) = n.

Proof. Since T_{u}(λ, 0) = (T_{u}^{1}(λ, 0), …, T_{u}^{n}(λ, 0)), using (H10)(iii), we get that for i = 1, …, n, T_{u}^{i}(λ, 0) is a continuous self-adjoint operator. Therefore, T_{u}(λ, 0) is a continuous self-adjoint operator too.

We have: for all v = (v_{1}, …, v_{n}) ∈ V,

v ∈ N(T_{u}(λ, 0))

if and only if for all w ∈ V, ⟨T_{u}(λ, 0)v, w⟩_{V} = 0

if and only if for i = 1, …, n, v_{i} ∈ Span{ϕ_{i}}

if and only if v ∈ Span{Φ_{1}, …, Φ_{n}} where Φ_{i} = (0, …, 0, ϕ_{i}, 0, …, 0).

Therefore codim R(T_{u}(λ, 0)) = n.

Now we prove that T_{μ_{i}u}(λ, 0)Φ_{i} ∉ R(T_{u}(λ, 0)) for all i. Note that we have identified T_{μ_{i}u}(λ, 0).(1, Φ_{i}) with T_{μ_{i}u}(λ, 0)Φ_{i}. We have: ⟨T_{μ_{i}u}(λ, 0)Φ_{i}, Φ_{i}⟩_{V} = − ∫_{ℝ^{N}} m_{i}ϕ_{i}^{2} ≠ 0. Therefore, T_{μ_{i}u}(λ, 0)Φ_{i} is not orthogonal to Φ_{i}. Now since N(T_{u}(λ, 0)) = Span{Φ_{1}, …, Φ_{n}} and R(T_{u}(λ, 0)) = N(T_{u}(λ, 0))^{⊥_{V}}, we deduce that T_{μ_{i}u}(λ, 0)Φ_{i} ∉ R(T_{u}(λ, 0)). Moreover, let (α_{1}, …, α_{n}) ∈ ℝ^{n} be such that: ∑_{j=1}^{n} α_{j}T_{μ_{j}u}(λ, 0)Φ_{j} = 0. Fix i, 1 ≤ i ≤ n. So that ⟨∑_{j=1}^{n} α_{j}T_{μ_{j}u}(λ, 0)Φ_{j}, Φ_{i}⟩_{V} = 0 and

∑_{j=1}^{n} α_{j}⟨T_{μ_{j}u}^{i}(λ, 0)Φ_{j}, ϕ_{i}⟩_{ρ_{i},q_{i}} = 0.

This implies that −α_{i} ∫_{ℝ^{N}} m_{i}ϕ_{i}^{2} = 0 and thus α_{i} = 0. Consequently, we have that dim(Span{T_{μ_{i}u}(λ, 0)Φ_{i}, 1 ≤ i ≤ n}) = n, completing the proof. □

Although we cannot apply directly the results obtained in [16], as in [15] we follow the proof of [16, Theorem 1.7] to obtain the following result.

Theorem 3.4. Assume that the hypotheses (H1), (H2), (H10) are satisfied. Then there exist a constant ϵ_{0} > 0, a neighbourhood U of (λ, 0) (with λ = (λ_{1}, …, λ_{n}) and 0 = (0, …, 0) ∈ V) and a continuous function H : (−ϵ_{0}, ϵ_{0}) → U such that T (H(ϵ)) = 0 for all ϵ ∈ (−ϵ_{0}, ϵ_{0}).

Note that T (H(ϵ)) = 0 with H(ϵ) = (μ, u) ∈ U for μ = (μ_{1}, …, μ_{n}) in a neighbourhood of λ = (λ_{1}, …, λ_{n}) and u = (u_{1}, …, u_{n}) in a neighbourhood of 0 = (0, …, 0) signifies that (μ, u) is a non trivial solution of the system (1.1).

Proof. As in [16], we introduce the following function h : ℝ × ℝ^{n} × V → V be defined as follows: For all (α, μ, w) ∈ ℝ × ℝ^{n} × V,

h(α, μ, w) := \begin{cases}\frac{1}{α}T (μ, αΦ_{1} + ⋯ + αΦ_{n} + αw) & if α ≠ 0 \\ T_{u}(μ, 0)(Φ_{1} + ⋯ + Φ_{n} + w) & if α = 0 \end{cases}  (3.8)

Since for i = 1, …, n, Φ_{i} ∈ N(T_{u}(λ, 0)), we deduce that h(0, λ, 0) = 0.

Let g : ℝ^{n} × V → V be given by g(μ, w) := T_{u}(μ, 0)(Φ_{1} + ⋯ + Φ_{n} + w) for all (μ, w) ∈ ℝ^{n} × V. For all ξ = (ξ_{1}, …, ξ_{n}) ∈ ℝ^{n} and v = (v_{1}, …, v_{n}) ∈ V we have that

Dg(μ, w)(ξ, v) = ∑_{i=1}^{n} ξ_{i}T_{μ_{i}u}(μ, 0)(Φ_{1} + ⋯ + Φ_{n} + w) + T_{u}(μ, 0)v,

where Dg is the Frechet derivative of g. It follows that

Dg(λ, 0)(ξ, v) = ∑_{i=1}^{n} ξ_{i}T_{μ_{i}u}(λ, 0)Φ_{i} + T_{u}(λ, 0)v since T_{μ_{i}u}(λ, 0)Φ_{j} = 0 (j ≠ i)

for all ξ ∈ ℝ^{n} and v ∈ V.

By Proposition 3.2, we deduce that Dg(λ, 0) is a linear homeomorphism from ℝ^{n} × V onto V. Hence, the implicit function theorem implies the existence of a neighbourhood U′ of (λ, 0), of a constant ϵ_{0} > 0 and of a function K : (−ϵ_{0}, ϵ_{0}) → U′ such that h(ϵ, μ, w) = 0 (see (3.8)) with ϵ ∈ (−ϵ_{0}, ϵ_{0}) and K(ϵ) = (μ, w) = (K_{1}ϵ, K_{2}ϵ) ∈ U′. Therefore for ϵ ∈ (−ϵ_{0}, ϵ_{0}), T (K_{1}ϵ, ϵ(Φ_{1} + ⋯ + Φ_{n} + K_{2}ϵ)) = 0 and if we set H(ϵ) := (K_{1}ϵ, ϵ(Φ_{1} + ⋯ + Φ_{n} + K_{2}ϵ)), then we have that T (H(ϵ)) = 0 for all ϵ ∈ (−ϵ_{0}, ϵ_{0}). □

Finally, we study the global nature of the continuum of solutions obtained by bifurcation from the (λ, 0) solution in a particular case. As in [13, 15], we follow a method developed in [10] and using the [23, Theorems 1.3 and 1.40], we obtain the following result.

Theorem 3.5. Assume (H1), (H2), (H10) are satisfied. Assume also that for all i, j, λ_{i} = λ_{j}. We denote by λ_{0} := λ_{i} for all i = 1, …, n. Then, there exists a continuum C of non trivial solutions for the system (1.1) obtained by bifurcation from the (λ_{0}, 0) solution, which is either unbounded or contains a point (μ, 0) where μ ≠ λ_{0} is the inverse of an eigenvalue of the operator A = (L_{1}, …, L_{n}) (where L_{i} is defined by for u = (u_{1}, …, u_{n}) ∈ V and ϕ ∈ V_{q_{i}}(ℝ^{N}), ⟨L_{i}u, ϕ⟩_{ρ_{i},q_{i}} = ∫_{ℝ^{N}} m_{i}u_{i}ϕ.) Since λ_{0} is simple, the continuum C has two connected subsets C^{+} and C^{−} which satisfy also the above alternatives.

Proof. First, we define an operator S by setting S(μ, u) = u − T (μ, u), S = (S^{1}, …, S^{n}) i.e. for all μ ∈ ℝ, for all u = (u_{1}, …, u_{n}) ∈ V, for all v_{i} ∈ V_{q_{i}}(ℝ^{N}),

⟨S^{i}(μ, u), v_{i}⟩_{ρ_{i},q_{i}} = ∫_{ℝ^{N}} [μm_{i}u_{i}v_{i} + g_{i}(x, u)v_{i}].

So u = (u_{1}, …, u_{n}) is a solution of the system (1.1) if and only if u = S(μ, u). We write S^{i}(μ, u) = μL_{i}u + H_{i}u where for all v_{i} ∈ V_{q_{i}}(ℝ^{N}),

⟨L_{i}u, v_{i}⟩_{ρ_{i},q_{i}} = ∫_{ℝ^{N}} m_{i}u_{i}v_{i} and ⟨H_{i}u, v_{i}⟩_{ρ_{i},q_{i}} = ∫_{ℝ^{N}} g_{i}(x, u)v_{i}.

So S(μ, u) = μAu + Hu with Au = (L_{1}u, …, L_{n}u) and Hu = (H_{1}u, …, H_{n}u).

To apply the results in [23], we must prove that S^{i} : ℝ × V → V_{q_{i}}(ℝ^{N}) is continuous and compact, that L_{i} : V → V_{q_{i}}(ℝ^{N}) is linear and compact, that H_{i}u = O(∥u∥_{V}) for u = (u_{1}, …, u_{n}) near 0 = (0, …, 0) uniformly on bounded intervals of μ and that \frac{1}{λ_{0}} is a simple eigenvalue of A (which is true because it is a simple eigenvalue of (L_{ρ_{i}} + q_{i})^{−1}M_{i}.)

Let ((μ_{p}, u_{p}))_{p} be a bounded sequence in ℝ × V, with u_{p} = (u_{1p}, …, u_{np}). Since the embedding of each V_{q_{i}}(ℝ^{N}) into L^{2}(ℝ^{N}) is compact, there exists a convergent subsequence, denoted also by ((μ_{p}, u_{p}))_{p} in ℝ × (L^{2}(ℝ^{N}))^{n}. We have:

∥S^{i}(μ_{p}, u_{p}) − S^{i}(μ_{m}, u_{m})∥_{ρ_{i},q_{i}}^{2}
= (μ_{p} − μ_{m}) ∫_{ℝ^{N}} m_{i}u_{ip}[S^{i}(μ_{p}, u_{p}) − S^{i}(μ_{m}, u_{m})]
+ μ_{m} ∫_{ℝ^{N}} m_{i}(u_{ip} − u_{im})[S^{i}(μ_{p}, u_{p}) − S^{i}(μ_{m}, u_{m})]
+ ∫_{ℝ^{N}} [g_{i}(x, u_{p}) − g_{i}(x, u_{m})][S^{i}(μ_{p}, u_{p}) − S^{i}(μ_{m}, u_{m})].

We deduce that (S^{i}(μ_{p}, u_{p}))_{p} is a Cauchy sequence and therefore a convergent sequence. So S^{i} is compact for all i = 1, …, n and S = (S^{1}, …, S^{n}) is also compact. Moreover we have for each i that

∥L_{i}u_{p} − L_{i}u_{m}∥_{ρ_{i},q_{i}}^{2} = ∫_{ℝ^{N}} m_{i}(u_{ip} − u_{im})[L_{i}u_{p} − L_{i}u_{m}]

and thus

∥L_{i}u_{p} − L_{i}u_{m}∥_{ρ_{i},q_{i}} ≤ const.∥u_{ip} − u_{im}∥_{L^{2}(Ω)}.

Therefore (L_{i}u_{p})_{p} is a Cauchy sequence, so L_{i} is compact and A is compact too. Finally we have:

∥H_{i}u∥_{ρ_{i},q_{i}}^{2} = ∫_{ℝ^{N}} g_{i}(x, u)H_{i}u ≤ const∥u∥_{V} ∥H_{i}u∥_{ρ_{i},q_{i}}.

So H_{i}u = O(∥u∥_{V}) and therefore Hu = O(∥u∥_{V}). □

4. Existence of positive solutions in ℝ^{N} for a particular case

In this section, we follow a method developed in [22] for the p-Laplacian in a bounded domain of ℝ^{N}, then in [13] for an equation defined in ℝ^{N} and involving a Schrödinger operator with a potential satisfying the hypothesis (H1) and a weight satisfying the hypothesis (H3) and in [15] for a system defined in ℝ^{N} and involving Schrödinger operators with potentials satisfying the hypothesis (H1) and weights satisfying the hypothesis (H3). We redefine the system (1.1) for this section.

We write (1.1) in the form

(L_{ρ_{i}} + q_{i})u_{i} = μ_{i}m_{i}u_{i} + ∑_{j=1;j≠i}^{n} a_{ij}u_{i}^{p}u_{j}^{q} + ∑_{j=1;j≠i}^{n} f_{ij}u_{j}^{p+q} in ℝ^{N}, i = 1, …, n (4.1)

where N = 3, 4; γ = 2^{∗} = \frac{2N}{N−2} = 6, 4 and p and q are positive integers be such that p + q < γ. We define for sufficiently large numbers C > 0 the sets

X_{q_{i},C} = {ϕ ∈ V_{q_{i}}(ℝ^{N}) : there exists a positive constant s such that

ϕ_{i} ≤ sϕ ≤ C a. e. } (4.2)

(the sets are non-empty by the properties of ϕ_{i}). We impose the following assumptions: (H11) For i, j = 1, …, n, a_{ij} ∈ L^{∞}(ℝ^{N}) and f_{ij} ∈ L^{∞}(ℝ^{N}). (H12) For i, j = 1, …, n, f_{ij} ≥ 0 a.e. (H13) For i = 1, …, n, there exists j_{i} ∈ {1, …, n} − {i} such that the following

items hold:

(1) Denote Ω_{i,+} := {x ∈ ℝ^{N}, a_{ij_{i}} > 0} and Ω_{i,−} := {x ∈ ℝ^{N}, a_{ij_{i}} < 0}. Then meas(Ω_{i,+}) ≠ 0, meas(Ω_{i,−}) ≠ 0.

(2) For each k ∈ {1, …, n} − {i, j_{i}}, a_{ik} is a nonnegative function, a_{ik} = 0 in D_{i} where D_{i} is a measurable subset of Ω_{i,−} with positive measure.

(3) For each k ∈ {1, …, n}, f_{ik} = 0 in D_{i}. (H14) There exists ϵ > 0 and l ≥ 1 such that for i = 1, …, n, a_{ij_{i}} ≥ −ϵm_{i} and

ϵ < \frac{μ_{i}}{p(lC)^{p+q−1}} (H15) For each i = 1, …, n, there exists a positive constant k_{ij_{i}} such that k_{ij_{i}} ≤

\frac{(p+q)}{lq(lC)^{p+q−1}} and a_{ij_{j}} ≥ −k_{ij_{i}}f_{ij_{i}}ϕ_{j_{i}}^{p+q−1} a.e.

We denote by

F_{i}(u_{1}, …, u_{n}) = ∫_{ℝ^{N}} [ ∑_{j=1,j≠i}^{n} a_{ij}u_{i}^{p+1}u_{j}^{q} + (p + 1) ∑_{j=1,j≠i}^{n} f_{ij}u_{j}^{p+q}u_{i}]  (4.3)

for i = 1, …, n and all (u_{1}, …, u_{n}) ∈ V_{q_{1}}(ℝ^{N}) × ⋯ × V_{q_{n}}(ℝ^{N}) and by

H_{μ_{i}}(v) = ∫_{ℝ^{N}} [ ∑_{k,j=1}^{N} ρ_{kj,i} \frac{∂v}{∂x_{j}} \frac{∂v}{∂x_{k}} + q_{i}v^{2} − μ_{i}m_{i}v^{2}]  (4.4)

for all i = 1, …, n and all v ∈ V_{q_{i}}(ℝ^{N}). Let

λ_{i}^{∗} = sup_{v_{i}∈V_{q_{i}} (ℝ^{N}),v_{i}≥0} \{ inf_{ϕ∈Φ_{v_{i}}} \{\frac{∫_{ℝ^{N}} ∑_{k,j=1}^{N} ρ_{kj,i} \frac{∂v_{i}}{∂x_{j}} \frac{∂ϕ}{∂x_{k}} + q_{i}v_{i}ϕ}{∫_{ℝ^{N}} m_{i}v_{i}ϕ}\}\}  (4.5)

and

λ_{i}^{∗∗} = sup_{v_{i}∈X_{q_{i},C}} \{ inf_{ϕ∈Φ_{v_{i}}} \{\frac{∫_{ℝ^{N}} ∑_{k,j=1}^{N} ρ_{kj,i} \frac{∂v_{i}}{∂x_{j}} \frac{∂ϕ}{∂x_{k}} + q_{i}v_{i}ϕ}{∫_{ℝ^{N}} m_{i}v_{i}ϕ}\}\}  (4.6)

where

Φ_{v_{i}} := {ϕ ∈ D(ℝ^{N}), ϕ ≥ 0, such that for j ≠ i, there exists

v_{j} ∈ V_{q_{j}}(ℝ^{N}), v_{j} ≥ 0 and \frac{∂F_{i}}{∂u_{i}} (v_{1}, …, v_{n})(ϕ) ≥ 0\}

(4.7)

and where \frac{∂F_{i}}{∂u_{i}} denotes the i-th partial derivative of F_{i}.

Note that the existences of λ_{i}^{∗} and λ_{i}^{∗∗} are due to the hypotheses (H4), (H11), (H12) and that λ_{i}^{∗∗} ≤ λ_{i}^{∗}. Assume the following hypotheses for i = 1, …, n: (H16) λ_{i}^{∗∗} < +∞. (H17) λ_{i}^{∗} < +∞. We proceed exactly as in [15] in this section; so that we will give only the steps of the proofs.

Lemma 4.1. (1) For i = 1, …, n, and all ϕ ∈ D(ℝ^{N}),

\frac{∂F_{i}}{∂u_{i}} (u_{1}, …, u_{n})(ϕ) = (p + 1) ∑_{j=1,j≠i}^{n} ∫_{ℝ^{N}} [a_{ij}u_{i}^{p}u_{j}^{q}ϕ + f_{ij}u_{j}^{p+q}ϕ],

H′_{μ_{i}}(v)(ϕ) = 2 ∫_{ℝ^{N}} [ ∑_{k,j=1}^{N} ρ_{kj,i} \frac{∂v}{∂x_{j}} \frac{∂ϕ}{∂x_{k}} + q_{i}vϕ − μ_{i}m_{i}vϕ].

(2) (u_{1}, …, u_{n}) ∈ V_{q_{1}}(ℝ^{N}) × ⋯ × V_{q_{n}}(ℝ^{N}) is a supersolution (resp. subsolution) of the system (4.1) if and only if for all ϕ ∈ D(ℝ^{N}), ϕ ≥ 0, for i = 1, …, n,

H′_{μ_{i}}(u_{i})(ϕ) ≥ \frac{2}{p + 1} \frac{∂F_{i}}{∂u_{i}} (u_{1}, …, u_{n})(ϕ) ( resp. ≤).

(3) For i = 1, …, n, all ϕ ∈ D(ℝ^{N}), for all t > 0,

\frac{∂F_{i}}{∂u_{i}} (tu_{1}, …, tu_{n})(ϕ) = t^{p+q} \frac{∂F_{i}}{∂u_{i}} (u_{1}, …, u_{n})(ϕ) and H′_{μ_{i}}(tu_{i})(ϕ) = tH′_{μ_{i}}(u_{i})(ϕ).

As in [13, 15, 22], we obtain the following lemma.

Lemma 4.2. For i = 1, …, n, we have λ_{i} ≤ λ_{i}^{∗∗}.

Proof. Suppose (for example) that λ_{1} > λ_{1}^{∗∗}. Because of the characterisation of λ_{1} (see (1.6), (1.7) and (4.4), we have H_{λ_{1}}(ϕ_{1}) = 0. By the definition of λ_{1}^{∗∗} (see (4.6)), there exists ϕ ∈ D(ℝ^{N}), ϕ ≥ 0, there exists (v_{2}, …, v_{n}) ∈ V_{q_{2}}(ℝ^{N})×⋯×V_{q_{n}}(ℝ^{N}), v_{i} ≥ 0, such that \frac{∂F_{1}}{∂u_{1}} (ϕ_{1}, v_{2}, …, v_{n})(ϕ) ≥ 0 and

\frac{∫_{ℝ^{N}} [∑_{k,j=1}^{N} ρ_{kj,1} \frac{∂ϕ_{1}}{∂x_{j}} \frac{∂ϕ}{∂x_{k}} + q_{1}ϕ_{1}ϕ]}{∫_{ℝ^{N}} m_{1}ϕ_{1}ϕ} ≤ λ_{1}^{∗∗} < λ_{1}.

So H′_{λ_{1}}(ϕ_{1})(ϕ) < 0. Moreover, we have for all η > 0,

H_{λ_{1}}(ϕ_{1} + ηϕ) = H_{λ_{1}}(ϕ_{1}) + ηH′_{λ_{1}}(ϕ_{1})(ϕ) + ∥ηϕ∥h(ηϕ) with h(ηϕ) → 0

as η → 0. Therefore, for η small enough, we have H_{λ_{1}}(ϕ_{1} + ηϕ) < 0 contradicting the definition of λ_{1}. □

We obtain now the main result of this section.

Theorem 4.3. Assume (H1)–(H4), (H11)-(H16) are satisfied. If for i = 1, …, n, λ_{i} + ϵ(lC)^{p+q−1} < μ_{i} < λ_{i}^{∗∗}, then the system (4.1) has at least one positive solution in X_{q_{1},C} × ⋯ × X_{q_{n},C}.

Proof. Since for all i, μ_{i} < λ_{i}^{∗∗}, due to the definition of λ_{i}^{∗∗} (see (4.6)), we can deduce the existence of v_{i}^{∗} ∈ X_{q_{i},C} such that for all ϕ ∈ Φ_{v_{i}^{∗}}, H′_{μ_{i}}(v_{i}^{∗})(ϕ) > 0.

We proceed exactly as in [15] to prove that there exists a real t ∈ (0, l) for which (tv_{1}^{∗}, …, tv_{n}^{∗}) is a supersolution of the system (4.1). Suppose that for all t ∈ (0, l), (tv_{1}^{∗}, …, tv_{n}^{∗}) is not a supersolution of the system (4.1). Then for all t ∈ (0, l), there exist i_{t} ∈ {1, …, n} and ψ_{i_{t}} ≥ 0 such that

H′_{μ_{i_{t}}} (tv_{i_{t}}^{∗})(ψ_{i_{t}}) < \frac{2}{p + 1} \frac{∂F_{i_{t}}}{∂u_{i_{t}}} (tv_{1}^{∗}, …, tv_{n}^{∗})(ψ_{i_{t}}).

Consider the sets

N_{t} = {i ∈ {1, …, n}, there exists ψ ∈ D(ℝ^{N}), ψ ≥ 0, such that

H′_{μ_{i}}(tv_{i}^{∗})(ψ) < \frac{2}{p + 1} \frac{∂F_{i}}{∂u_{i}} (tv_{1}^{∗}, …, tv_{n}^{∗})(ψ)\} (4.8)

and for i_{t} ∈ N_{t},

K_{i_{t}} = \{ψ ∈ D(ℝ^{N}), ψ ≥ 0, H′_{μ_{i_{t}}} (tv_{i_{t}}^{∗})(ψ) < \frac{2}{p + 1} \frac{∂F_{i_{t}}}{∂u_{i_{t}}} (tv_{1}^{∗}, …, tv_{n}^{∗})(ψ)\}. (4.9)

We prove that there exists t > 0, i_{t} ∈ N_{t}, ϕ ∈ K_{i_{t}}, and ψ ∈ K_{i_{t}} which satisfy

\frac{∂F_{i_{t}}}{∂u_{i_{t}}} (tv_{1}^{∗}, …, tv_{n}^{∗})(ϕ) < 0 and \frac{∂F_{i_{t}}}{∂u_{i_{t}}} (tv_{1}^{∗}, …, tv_{n}^{∗})(ψ) > 0.

So we have

H′_{μ_{i_{t}}} (v_{i_{t}}^{∗})(ϕ) < \frac{2}{p + 1}t^{p+q−1} \frac{∂F_{i_{t}}}{∂u_{i_{t}}} (v_{1}^{∗}, …, v_{n}^{∗})(ϕ) < 0,  (4.10)
0 < H′_{μ_{i_{t}}} (v_{i_{t}}^{∗})(ψ) < \frac{2}{p + 1}t^{p+q−1} \frac{∂F_{i_{t}}}{∂u_{i_{t}}} (v_{1}^{∗}, …, v_{n}^{∗})(ψ).  (4.11)

(Note that ψ ∈ Φ_{v_{i_{t}}^{∗}} (see (4.7).) Since \frac{∂F_{i_{t}}}{∂u_{i_{t}}} (v_{1}^{∗}, …, v_{n}^{∗}) is a continuous function, there exists α ∈ (0, 1) such that

\frac{∂F_{i_{t}}}{∂u_{i_{t}}} (v_{1}^{∗}, …, v_{n}^{∗})(αϕ + (1 − α)ψ) = 0.

Thus we deduce that αϕ + (1 − α)ψ ∈ Φ_{v_{i_{t}}^{∗}} and so H′_{μ_{i_{t}}} (v_{i_{t}}^{∗})(αϕ + (1 − α)ψ) > 0. But using (4.10) and (4.11) we have

0 < αH′_{μ_{i_{t}}} (v_{i_{t}}^{∗})(ϕ) + (1 − α)H′_{μ_{i_{t}}} (v_{i_{t}}^{∗})(ψ)

< \frac{2}{p + 1}t^{p+q−1}[α\frac{∂F_{i_{t}}}{∂u_{i_{t}}} (v_{1}^{∗}, …, v_{n}^{∗})(ϕ) + (1 − α)\frac{∂F_{i_{t}}}{∂u_{i_{t}}} (v_{1}^{∗}, …, v_{n}^{∗})(ψ)] = 0

and we get a contradiction. Therefore there exists t ∈ (0, l), for which (tv_{1}^{∗}, …, tv_{n}^{∗}) is a supersolution of the system (4.1). Note that for i = 1, …, n, tv_{i}^{∗} ≥ sϕ_{i} if 0 < s ≤ t.

Using (H11)–(H13), we can prove that (sϕ_{1}, …, sϕ_{n}) is a subsolution of system (4.1) with s > 0 be such that s ≤ t ≤ l and \frac{1}{l} ≤ s^{p+q−1} (which is possible with l ≥ 1). Let σ = [sϕ_{1}, tv_{1}^{∗}] × ⋯ × [sϕ_{n}, tv_{n}^{∗}] and the operator T be defined by T (u_{1}, …, u_{n}) = (v_{1}, …, v_{n}) with (v_{1}, …, v_{n}) solution of

(L_{ρ_{i}} + q_{i})v_{i} = μ_{i}m_{i}u_{i} + ∑_{j=1;j≠i}^{n} a_{ij}u_{i}^{p}u_{j}^{q} + ∑_{j=1;j≠i}^{n} f_{ij}u_{j}^{p+q} in ℝ^{N},  (4.12)

for i = 1, …, n. We want to prove that T (σ) ⊂ σ.

Let (u_{1}, …, u_{n}) ∈ σ and T (u_{1}, …, u_{n}) = (v_{1}, …, v_{n}). By (4.12) we can write for i = 1, …, n,

(L_{ρ_{i}} + q_{i})(v_{i} − sϕ_{i}) = μ_{i}m_{i}u_{i} + ∑_{j=1;j≠i}^{n} a_{ij}u_{i}^{p}u_{j}^{q} + ∑_{j=1;j≠i}^{n} f_{ij}u_{j}^{p+q} − sλ_{i}m_{i}ϕ_{i}.

Since for all k, u_{k} ≥ sϕ_{k}, using (H14) a_{ij_{i}} ≥ −ϵm_{i}, we can deduce that

(L_{ρ_{i}} + q_{i})(v_{i} − sϕ_{i}) ≥ [μ_{i} − λ_{i} − ϵs^{p+q−1}ϕ_{i}^{p−1}ϕ_{j_{i}}^{q}]m_{i}sϕ_{i}.  (4.13)

But ϕ_{i}^{p−1}ϕ_{j_{i}}^{q} ≤ C^{p+q−1}, s^{p+q−1} ≤ l^{p+q−1} and λ_{i} + ϵ(lC)^{p+q−1} ≤ μ_{i}, so we deduce from (4.13) that (L_{ρ_{i}} + q_{i})(v_{i} − sϕ_{i}) ≥ 0. By the Maximum Principle, we obtain that v_{i} ≥ sϕ_{i} for all i = 1, …, n. Moreover we have: For all i = 1, …, n,

(L_{ρ_{i}} + q_{i})(tv_{i}^{∗} − v_{i}) ≥ μ_{i}m_{i}(tv_{i}^{∗} − u_{i}) + ∑_{j=1;j≠i}^{n} a_{ij}[(tv_{i}^{∗})^{p}(tv_{j}^{∗})^{q} − u_{i}^{p}u_{j}^{q}]

+ ∑_{j=1;j≠i}^{n} f_{ij}[(tv_{j}^{∗})^{p+q} − u_{j}^{p+q}].

So we can rewrite this equation in form

(L_{ρ_{i}} + q_{i})(tv_{i}^{∗} − v_{i})

≥ μ_{i}m_{i}(tv_{i}^{∗} − u_{i}) + ∑_{j=1;j≠i}^{n} a_{ij}(tv_{i}^{∗} − u_{i})u_{j}^{q}[∑_{k=0}^{p−1}(tv_{i}^{∗})^{k}u_{i}^{p−1−k}]

+ ∑_{j=1;j≠i}^{n}(tv_{j}^{∗} − u_{j})[a_{ij}(tv_{i}^{∗})^{p}(∑_{k=0}^{q−1}(tv_{j}^{∗})^{k}u_{j}^{q−1−k}) + f_{ij}( ∑_{k=0}^{p+q−1}(tv_{j}^{∗})^{k}u_{j}^{p+q−1−k})].

Since (u_{1}, …, u_{n}) ∈ σ, we get

(L_{ρ_{i}} + q_{i})(tv_{i}^{∗} − v_{i})

≥ (tv_{i}^{∗} − u_{i})[μ_{i}m_{i} + a_{ij_{i}}u_{j_{i}}^{q}(∑_{k=0}^{p−1}(tv_{i}^{∗})^{k}u_{i}^{p−1−k})]

+ (tv_{j_{i}}^{∗} − u_{j_{i}})[a_{ij_{i}}(tv_{i}^{∗})^{p}(∑_{k=0}^{q−1}(tv_{j_{i}}^{∗})^{k}u_{j_{i}}^{q−1−k}) + f_{ij_{i}}( ∑_{k=0}^{p+q−1}(tv_{j_{i}}^{∗})^{k}u_{j_{i}}^{p+q−1−k})].

(4.14)

Since u_{j_{i}}^{q}(∑_{k=0}^{p−1}(tv_{i}^{∗})^{k}u_{i}^{p−1−k}) ≤ p(lC)^{p+q−1}, using (H14) we deduce that

μ_{i}m_{i} + a_{ij_{i}}u_{j_{i}}^{q}(∑_{k=0}^{p−1}(tv_{i}^{∗})^{k}u_{i}^{p−1−k}) ≥ 0.  (4.15)

Similarly, using (H15) and s^{p+q−1} ≥ \frac{1}{l}, we get

\frac{f_{ij_{i}}(∑_{k=0}^{p+q−1}(tv_{j_{i}}^{∗})^{k}u_{j_{i}}^{p+q−1−k})}{(tv_{i}^{∗})^{p}(∑_{k=0}^{q−1}(tv_{j_{i}}^{∗})^{k}u_{j_{i}}^{q−1−k})} ≥ \frac{(p + q)f_{ij_{i}}(sϕ_{j_{i}})^{p+q−1}}{q(lC)^{p+q−1}}

and so

a_{ij_{i}}(tv_{i}^{∗})^{p}(∑_{k=0}^{q−1}(tv_{j_{i}}^{∗})^{k}u_{j_{i}}^{q−1−k}) + f_{ij_{i}}( ∑_{k=0}^{p+q−1}(tv_{j_{i}}^{∗})^{k}u_{j_{i}}^{p+q−1−k}) ≥ 0.  (4.16)

Therefore, by (4.14), (4.15), (4.16), we obtain (L_{ρ_{i}} + q_{i})(tv_{i}^{∗} − v_{i}) ≥ 0 and so by the Maximum Principle we deduce that v_{i} ≤ tv_{i}^{∗} for all i = 1, …, n. We conclude that (v_{1}, …, v_{n}) ∈ σ. Finally, note that T is a continuous and compact operator (by the compact embedding of each V_{q_{i}}(ℝ^{N}) into L^{2}(ℝ^{N})).

Therefore the Schauder Fixed Point Theorem, implies the existence of at least one positive solution for the system (4.1). □

To complete this section, we give some conditions which assure the validity of the hypothesis (H17). First, we recall the following lemma (see [22]).

Lemma 4.4. For i = 1, …, n, for all u ∈ V_{q_{i}}(ℝ^{N}), u > 0, for all ϕ ∈ V_{q_{i}}(ℝ^{N}), ϕ ≥ 0, and all μ_{i} ∈ ℝ, H′_{μ_{i}}(u)((\frac{ϕ}{u})^{α}ϕ) − H′_{μ_{i}}(ϕ)((\frac{ϕ}{u})^{α}u) ≤ 0 with α ∈ ℕ, α > 0.

So we get the last theorem of this section.

Theorem 4.5. (i) Assume (H1)–(H4), (H11)-(H13) are satisfied. For i = 1, …, n, if Ω_{i,+} = {x ∈ ℝ^{N}, a_{ij_{i}}(x) > 0} is a nonempty, bounded domain of ℝ^{N} with a smooth boundary ∂Ω_{i,+}, then λ_{i}^{∗} < +∞.

(ii) Assume (H1)–(H4), (H11) are satisfied.

(1) We assume here that for all i, j, f_{ij} = 0. If there exists i ∈ {1, …, n} such that for j ≠ i, there exists u_{j} ≥ 0 which satisfies

F_{i}(u_{1}, …, u_{i−1}, ϕ_{i}, u_{i+1}, …, u_{n}) ≥ 0,

then λ_{i}^{∗} ≤ λ_{i} and since λ_{i}^{∗} ≥ λ_{i} is always satisfied, then λ_{i}^{∗} = λ_{i} < +∞.

(2) If there exists u_{1} ≥ 0, …, u_{n} ≥ 0, such that
F_{i}(u_{1}, …, u_{i−1}, ϕ_{i}, u_{i+1}, …, u_{n}) < 0,

then λ_{i} < λ_{i}^{∗}.

Note that the condition in Theorem 4.5(ii)(2) is verified if we assume also that the hypothesis (H13) is satisfied and if we take u_{j} ≥ 0 such that supp u_{j} ⊂ D_{i}.

Proof. (i) For i = 1, …, n consider the equation (L_{ρ_{i}} +q_{i})u = λ_{i}m_{i}u defined in Ω_{i,+} with Dirichlet condition on ∂Ω_{i,+}. We denote by λ_{i+} the first eigenvalue (which is simple and positive) and by ϕ_{i+} the first eigenfunction associated i.e:

(L_{ρ_{i}} + q_{i})ϕ_{i+} = λ_{i+}m_{i}ϕ_{i+} in Ω_{i,+},
ϕ_{i+} > 0 in Ω_{i+},
ϕ_{i+} = 0 on ∂Ω_{i+}.

(4.17)

Since supp ϕ_{i+} ⊂ Ω_{i+}, by the above lemma, we have

for all u_{i} ∈ D(ℝ^{N}), H′_{λ_{i+}}(u_{i})((\frac{ϕ_{i+}}{u_{i}})^{α}ϕ_{i+}) ≤ 0

i.e. for all u_{i} ∈ D(ℝ^{N}), u_{i} ≥ 0,

\frac{∫_{ℝ^{N}} [∑_{k,j=1}^{N} ρ_{kj,i} \frac{∂u_{i}}{∂x_{j}} ^{\frac{∂(( \frac{ϕ_{i+}}{u_{i}})^{α}ϕ_{i+})}{∂x_{k}}} + q_{i}u_{i}(\frac{ϕ_{i+}}{u_{i}})^{α}ϕ_{i+}]}{∫_{ℝ^{N}} m_{i}u_{i}(\frac{ϕ_{i+}}{u_{i}})^{α}ϕ_{i+}} ≤ λ_{i+} < +∞.  (4.18)

Moreover, for all u_{1}, …, u_{i−1}, u_{i+1}, …, u_{n} ≥ 0,

\begin{matrix}\frac{∂F_{i}}{∂u_{i}} (u_{1}, …, u_{n})((\frac{ϕ_{i+}}{u_{i}})^{α}ϕ_{i+}) \\ = (p + 1) ∑_{j=1;j≠i}^{n} ∫_{ℝ^{N}} [a_{ij}u_{i}^{p}u_{j}^{q}(\frac{ϕ_{i+}}{u_{i}})^{α}ϕ_{i+} + f_{ij}u_{j}^{p+q}(\frac{ϕ_{i+}}{u_{i}})^{α}ϕ_{i+}] ≥ 0 \end{matrix}  (4.19)

since supp ϕ_{i+} ⊂ Ω_{i+} and by the hypotheses (H11)–(H13). So by (4.18) and (4.19), for all u_{i} ∈ V_{q_{i}}(ℝ^{N}), u_{i} ≥ 0,

inf_{ϕ∈D(ℝ^{N})} \{\frac{∫_{ℝ^{N}} ∑_{k,j=1}^{N} ρ_{kj,i} \frac{∂u_{i}}{∂x_{j}} \frac{∂ϕ}{∂x_{k}} + q_{i}u_{i}ϕ}{∫_{ℝ^{N}} m_{i}u_{i}ϕ}, ϕ ≥ 0 such that for j = 1, …, n, j ≠ i,

there exists v_{j} ∈ V_{q_{j}}(ℝ^{N}), v_{j} ≥ 0 and \frac{∂F_{i}}{∂u_{i}} (v_{1}, …, u_{i}, …, v_{n})(ϕ) ≥ 0\}

≤ \frac{∫_{ℝ^{N}} [∑_{k,j=1}^{N} ρ_{kj,i} \frac{∂u_{i}}{∂x_{j}} ^{\frac{∂(( \frac{ϕ_{i+}}{u_{i}})^{α}ϕ_{i+})}{∂x_{k}}} + q_{i}u_{i}(\frac{ϕ_{i+}}{u_{i}})^{α}ϕ_{i+}]}{∫_{ℝ^{N}} m_{i}u_{i}(\frac{ϕ_{i+}}{u_{i}})^{α}ϕ_{i+}} ≤ λ_{i+} < +∞.

Therefore, λ_{i}^{∗} ≤ λ_{i+} < +∞. (ii) For the first claim, we assume that for i, j, f_{ij} = 0. So we have: for all u_{1}, …, u_{n},

F_{i}(u_{1}, …, u_{n}) = ∑_{j=1;j≠i}^{n} ∫_{ℝ^{N}} a_{ij}u_{i}^{p+1}u_{j}^{q}  (4.20)

and for all ϕ,

\frac{∂F_{i}}{∂u_{i}} (u_{1}, …, u_{n})(ϕ) = (p + 1) ∑_{j=1;j≠i}^{n} ∫_{ℝ^{N}} a_{ij}u_{i}^{p}u_{j}^{q}ϕ.  (4.21)

We suppose that: for j ≠ i there exists u_{j} ≥ 0 such that

F_{i}(u_{1}, …, u_{i−1}, ϕ_{i}, u_{i+1}, …, u_{n}) ≥ 0.  (4.22)

We want to prove that λ_{i}^{∗} ≤ λ_{i}. We use again the above lemma with α = p. We have H′_{λ_{i}}(ϕ_{i})((\frac{ϕ_{i}}{u_{i}})^{p}u_{i}) = 0 for all u_{i} > 0. So for all u_{i} ≥ 0, H′_{λ_{i}}(u_{i})((\frac{ϕ_{i}}{u_{i}})^{p}ϕ_{i}) ≤ 0 i.e.,

\frac{∫_{ℝ^{N}} [∑_{k,j=1}^{N} ρ_{kj,i} \frac{∂u_{i}}{∂x_{j}} \frac{∂(( \frac{ϕ_{i}}{u_{i}})^{p}u_{i})}{∂x_{k}} + q_{i}u_{i}(\frac{ϕ_{i}}{u_{i}})^{p}ϕ_{i}]}{∫_{ℝ^{N}} m_{i}u_{i}(\frac{ϕ_{i}}{u_{i}})^{p}ϕ_{i}} ≤ λ_{i} < +∞.  (4.23)

Moreover, using (4.20)–(4.22), for all u_{i} > 0, for all j ≠ i, there exists u_{j} ≥ 0,

\begin{matrix}\frac{∂F_{i}}{∂u_{i}} (u_{1}, …, u_{i−1}, u_{i}, u_{i+1}, …, u_{n})((\frac{ϕ_{i}}{u_{i}})^{p}ϕ_{i}) \\ = (p + 1) ∑_{j=1;j≠i}^{n} ∫_{ℝ^{N}} a_{ij}u_{i}^{p}u_{j}^{q}(\frac{ϕ_{i}}{u_{i}})^{p}ϕ_{i} \\ = (p + 1)F_{i}(u_{1}, …, u_{i−1}, ϕ_{i}, u_{i+1}, …, u_{n}) ≥ 0. \end{matrix}  (4.24)

Since

inf_{ϕ∈D(ℝ^{N})} \{\frac{∫_{ℝ^{N}} ∑_{k,j=1}^{N} ρ_{kj,i} \frac{∂u_{i}}{∂x_{j}} \frac{∂ϕ}{∂x_{k}} + q_{i}u_{i}ϕ}{∫_{ℝ^{N}} m_{i}u_{i}ϕ}, ϕ ≥ 0 such that

for j ≠ i, there exists u_{j} ∈ V_{q_{j}}(ℝ^{N}), u_{j} ≥ 0 and \frac{∂F_{i}}{∂u_{i}} (u_{1}, …, u_{n})(ϕ) ≥ 0\}

≤ \frac{∫_{ℝ^{N}} [∑_{k,j=1}^{N} ρ_{kj,i} \frac{∂u_{i}}{∂x_{j}} \frac{∂(( \frac{ϕ_{i}}{u_{i}})^{p}ϕ_{i})}{∂x_{k}} + q_{i}u_{i}(\frac{ϕ_{i}}{u_{i}})^{p}ϕ_{i}]}{∫_{ℝ^{N}} m_{i}u_{i}(\frac{ϕ_{i}}{u_{i}})^{p}ϕ_{i}} ≤ λ_{i} < +∞,

by (4.23) and (4.24), we get that λ_{i}^{∗} ≤ λ_{i} and therefore λ_{i}^{∗} = λ_{i}.

For the second claim, we assume that there exists u_{1} ≥ 0, …, u_{n} ≥ 0 such that

F_{i}(u_{1}, …, u_{i−1}, ϕ_{i}, u_{i+1}, …, u_{n}) < 0.  (4.25)

We denote by

λ_{i}^{−} = inf_{ϕ∈D(ℝ^{N}), ϕ≥0}\{\frac{∫_{ℝ^{N}} [∑_{k,j=1}^{N} ρ_{kj,i} \frac{∂ϕ}{∂x_{j}} \frac{∂ϕ}{∂x_{k}} + q_{i}|ϕ|^{2}]}{∫_{ℝ^{N}} m_{i}|ϕ|^{2}},
ϕ such that F_{i}(u_{1}, …, u_{i−1}, ϕ, u_{i+1}, …, u_{n}) > 0}.

(4.26)

Let

W_{i} = {ϕ ∈ V_{q_{i}}(ℝ^{N}), ϕ ≥ 0, F_{i}(u_{1}, …, u_{i−1}, ϕ, u_{i+1}, …, u_{n}) > 0}.

Since W_{i} ⊂ V_{q_{i}}(ℝ^{N}), we have λ_{i} ≤ λ_{i}^{−}. Since ϕ_{i} ∉ W_{i}, by the continuity of the function F_{i}, we deduce that λ_{i} < λ_{i}^{−} (see (1.7) and (4.26).

We prove now that λ_{i}^{−} ≤ λ_{i}^{∗}. As in [15], we prove that there exists u_{i}^{−} ∈ W_{i},

λ_{i}^{−} = \frac{∫_{ℝ^{N}} [∑_{k,j=1}^{N} ρ_{kj,i} \frac{∂u_{i}^{−}}{∂x_{j}} \frac{∂u_{i}^{−}}{∂x_{k}} + q_{i}|u_{i}^{−}|^{2}]}{∫_{ℝ^{N}} m_{i}|u_{i}^{−}|^{2}}.  (4.27)

After that, we prove that λ_{i}^{−} ≤ λ_{i}^{∗}. Suppose that λ_{i}^{−} > λ_{i}^{∗}. Then there exists ϕ ∈ Φ_{u_{i}^{−}} such that

\frac{∫_{ℝ^{N}} [∑_{k,j=1}^{N} ρ_{kj,i} \frac{∂u_{i}^{−}}{∂x_{j}} \frac{∂ϕ}{∂x_{k}} + q_{i}u_{i}^{−}ϕ]}{∫_{ℝ^{N}} m_{i}u_{i}^{−}ϕ} ≤ λ_{i}^{∗} < λ_{i}^{−}.  (4.28)

Therefore H′_{λ_{i}^{−}}(u_{i}^{−})(ϕ) < 0. Since F_{i}(u_{1}, …, u_{i−1}, u_{i}^{−}, u_{i+1}, …, u_{n}) > 0, by conti-

nuity we have F_{i}(u_{1}, …, u_{i−1}, u_{i}^{−} + ηϕ, u_{i+1}, …, u_{n}) > 0 for η > 0 small enough. Moreover, by (4.27) and (4.28) we have H′_{λ_{i}^{−}}(u_{i}^{−})(ϕ) < 0 and H_{λ_{i}^{−}}(u_{i}^{−}) = 0, we can

choose η > 0 small enough such that H_{λ_{i}^{−}}(u_{i}^{−} + ηϕ) < 0. So we obtain that:

\frac{∫_{ℝ^{N}} [∑_{k,j=1}^{N} ρ_{kj,i} \frac{∂(u_{i}^{−}+ηϕ)}{∂x_{j}} \frac{∂(u_{i}^{−}+ηϕ)}{∂x_{k}} + q_{i}(u_{i}^{−} + ηϕ)^{2}]}{∫_{ℝ^{N}} m_{i}(u_{i}^{−} + ηϕ)^{2}} < λ_{i}^{−}

and this contradicts the definition of λ_{i}^{−}. Therefore λ_{i}^{−} ≤ λ_{i}^{∗}. □

5. Remarks

We give several remarks to our conditions and results.

First, note that all the results presented in this paper are true even in a bounded domain Ω with Dirichlet boundary conditions. Indeed, if we assume that for all i, q_{i} ∈ L^{∞}(Ω) and q_{i} ≥ 0, if we still define the set V_{q_{i}}(Ω) as the completion of D(Ω) under the norm ∥u∥_{q_{i}} = ( ∫_{Ω}[|∇u|^{2} + q_{i}u^{2}])^{1/2}, we have V_{q_{i}}(Ω) = H_{0}^{1}(Ω). Therefore, we can define the eigenpair (λ_{i}, ϕ_{i}) by the Courant-Fischer formulas as above.

Second, the case where for all i, q_{i} = 0 in ℝ^{N} is quite different. Indeed, if we define the set V_{q_{i}}(ℝ^{N}) as the completion of D(ℝ^{N}) under the norm ∥u∥_{q_{i}} = ( ∫_{ℝ^{N}} |∇u|^{2})^{1/2} then V_{q_{i}}(ℝ^{N}) = D^{1,2} and we lose the compactness of the embedding of V_{q_{i}}(ℝ^{N}) into L^{2}(ℝ^{N}). So this case requires the introduction of other weightspaces depending on the weights m_{i} and therefore requires other hypotheses upon the weights m_{i}. For example, we recall from [10] that the equation

−Δu = λm_{i}u in ℝ^{N}
u(x) → 0 as |x| → +∞

admits a positive principle eigenvalue λ(m_{i}), associated with a positive eigenfunction ϕ_{m_{i}} such that

λ(m_{i}) ∫_{ℝ^{N}} m_{i}u^{2} ≤ ∫_{ℝ^{N}} |∇u|^{2} ( for all u ∈ D^{1,2})

under the hypotheses that N ≥ 3, 0 ≤ m_{i} ∈ L^{\frac{N}{2}} (ℝ^{N}), m_{i} ≠ 0. In [8], for example, there are results on the existence of a positive principle eigenvalue associated with a positive eigenfunction for the equation

−Δu = λm_{i}u in ℝ^{N}
u(x) → 0 as |x| → +∞

It is proved that such an eigenvalue exists if m_{i} is negative and bounded away from 0 at infinity, or if N ≥ 3 and |m_{i}| is sufficiently small at infinity (i.e. |m_{i}(x)| ≤ \frac{const}{(1+|x|^{2})^{α}} with α > 1) but does not exist if N = 1 or N = 2 and ∫_{ℝ^{N}} m_{i} > 0. When N ≥ 3 and the weight m_{i} satisfies |m_{i}(x)| ≤ \frac{const}{(1+|x|^{2})^{α}}, the variational space is

defined by V = \{u : ℝ^{N} → ℝ, ∫_{ℝ^{N}} [|∇u|^{2} + \frac{u^{2}}{1+|x|^{2}} ] < +∞\}. In this last case, under the hypotheses that N ≥ 3 and |m_{i}(x)| ≤ \frac{const}{(1+|x|^{2})^{α}}, our results are still valid with the additional conditions that all the parameters μ_{i} have to be positive (since the equation)

−Δu = λm_{i}u in ℝ^{N}
u(x) → 0 as |x| → +∞

(can admit negative eigenvalues).

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Laure Cardoulis CeReMath/UMR MIP, University of Toulouse 1, 21 allée de Brienne, 31000 Toulouse, France

   E-mail address: laure.cardoulis@univ-tlse1.fr