2002-Fez conference on Partial Differential Equations, Electronic Journal of Differential Equations, Conference 09, 2003, pp 171–182. http://ejde.math.swt.edu or http://ejde.math.unt.edu ftp ejde.math.swt.edu (login: ftp)

Existence and regularity of positive solutions for an elliptic system ^{∗}

Abdelouahed El Khalil, Mohammed Ouanan, & Abdelfattah Touzani

Abstract In this paper, we study the existence and regularity of positive solution for an elliptic system on a bounded and regular domain. The non linearities in this equation are functions of Caratheodory type satisfying some exponential growth conditions.

1 Introduction

In this work, we study the elliptic system

−Δ_{p}u = f(x, u, v) in Ω
−Δ_{p}v = g(x, u, v) in Ω
u = v = 0 on ∂Ω,

(1.1)

where Ω is a bounded regular domain in ℝ^{N}, 1 < p < +∞, and f and g are Carathéodory functions satisfying some growth conditions specified later.

In the recent years; the existence and non existence for the scalar case have been studied by several author’s by using various approaches [9, 5]. For the system case, we mention the recent work of Bechah [4]. He study the local and global behaviour of solutions of systems involving the p-Laplacian operator in unbounded domains with f , g functions satisfying some growth conditions of polynomial type. Also, we cite the work of Ahammou [2], where he studied the positive radial solutions of nonlinear elliptic systems (1.1) using the method of topological degree. There Ω is a ball in ℝ^{N} and f, g are positive functions satisfying f(x, 0, 0) = g(x, 0, 0) = 0 under some growth conditions of polynomial type.

Here we study the existence and regularity of positive solutions of (1.1) in a regular bounded domain and f, g are functions of Carathéodory type satisfying some growth conditions of exponential type. We extend the results of De Thelin [8] for the problem

Δ_{p}u + g(x, u) = 0

∗Mathematics Subject Classifications: 35J20, 35J45, 35J50, 35J70.

Key words: p-Laplacian operator, mountain pass Theorem Orlicz space.

\begin{matrix}c \\ ◯ \end{matrix}2002 Southwest Texas State University.

Published December 28, 2002.

in the case when the growth of g(x, .) is allowed to be of exponential type.

The rest of this paper is organized as follows: In section 2 we introduce the assumptions and some results preliminaries. In section 3 we introduce the main results of this paper.

2 Assumptions and preliminaries

Let X be a closed subspace of W_{0}^{1,p}(Ω); f and g be two positives Carathéodory functions satisfying the growth conditions:

(H1) For all K > 0, there exists m > 0 such that for all (ξ, η) ∈ ℝ×ℝ, satisfying |ξ| + |η| ≤ K and for almost every where x ∈ Ω we have

f(x, ξ, η) ≤ m and g(x, ξ, η) ≤ m.

(H2) There exist σ_{0} > 2p − 1, θ_{0} > 2p − 1 and R > 0 such that for all (ξ, η) ∈ ℝ^{+} × ℝ^{+} satisfying ξ + η ≥ R we have

ξf(x, ξ, η) ≥ (σ_{0} + 1)G(x, ξ, η) a.e x ∈ Ω  (2.1)
ηg(x, ξ, η) ≥ (θ_{0} + 1)G(x, ξ, η) a.e x ∈ Ω  (2.2)

where \frac{∂G(x,ξ,η)}{∂ξ} = f(x, ξ, η), and \frac{∂G(x,ξ,η)}{∂η} = g(x, ξ, η).

Definition We say that (u, v) is a weak solution of elliptic system (1.1) if for all (ϕ, ψ) ∈ (W_{0}^{1,p}(Ω))^{2} we have

∫_{Ω} |∇u|^{p−2}∇u∇ϕ = ∫_{Ω} f(x, u, v)ϕ
∫_{Ω} |∇v|^{p−2}∇v∇ψ = ∫_{Ω} g(x, u, v)ψ

Theorem 2.1 (Mountain Pass [3]) Let I be a C^{1}-differentiable functional on a Banach space E and satisfying the Palais-Smale condition (PS), suppose that there exists a neighbourhood U of 0 in E and a positive constant α satisfying the following conditions:

(I1) I(0) = 0.

(I2) I(u) ≥ α on the boundary of U.

(I3) There exists an e ∈ E\U such that I(e) < α.

Then

c = inf_{γ∈Γ} sup_{y∈[0,1]} I(γ(y))

is a critical value of I with Γ = {g ∈ C([0, 1]); g(0) = 0, g(1) = e}.

3 Main result

The case p ≠ N.

Set

J(u, v) = \frac{1}{p} ∫_{Ω}(|∇u|^{p} + |∇v|^{p}) dx − ∫_{Ω} G(x, u, v) dx

J is well define in (W_{0}^{1,p}(Ω))^{2}. In this subsection we have the following result

Theorem 3.1 Let f and g are two Carathéodory functions satisfying (H1), (H2) and suppose that

i) X ⊂ L^{∞}(Ω).

ii) There exist some r_{0} > 0, σ > p − 1, θ > p − 1 and c > 0 such that, for almost every where x ∈ Ω and for all |ξ| + |η| < r_{0} we have

G(x, ξ, η) ≤ c(ξ^{σ+1} + η^{θ+1}).

Then, there is at least one positive solution (u, v) ∈ (X ∩ C^{1,ν}(Ω̄))^{2} of (1.1).

Remark. The condition i) is true for X = W_{0}^{1,p}(Ω) where Ω is an open bounded domain in ℝ^{N} and p > N.

The following proposition gives another interesting example of the space X with p > 1.

Proposition 3.2 ([8]) Let 0 < ρ < R < +∞ and Ω = {x ∈ ℝ^{N} : ρ < |x| < R} an annulus in ℝ^{N}. Let X be the set of radially symmetric functions in W_{0}^{1,p}(Ω). Then, there exist a positive constant c(N, ρ, p, R) > 0 such that, for all u ∈ X and for almost every where x ∈ Ω we have

|u(x)| ≤ c(N, ρ, p, R)∥∇u∥_{p}.

To prove Theorem 3.1 we prove some preliminary lemmas.

Lemma 3.3 Let u ∈ X. Suppose that f and g satisfy (H1) and (H2). Then, any sequence {(u_{j}, v_{j})}_{j≥0} ∈ X × X satisfying the following two hypotheses:

|J(u_{j}, v_{j})| ≤ K  (3.1)

and for all ϵ > 0 there exist j_{0} ∈ ℕ^{∗} such that ∀j ≥ j_{0},

|⟨J′(u_{j}, v_{j}), (u_{j}, v_{j})⟩| ≤ ϵ∥(u_{j}, v_{j})∥,  (3.2)

is bounded in X × X.

Proof. Set ∥(u, v)∥ = (∥∇u∥_{p}^{p} + ∥∇v∥_{p}^{p})^{1/p}. This is a norm in the product space X × X, and ∥∇u∥_{p} = ∥u∥_{X}. Now we proceed by contradiction. Suppose that a subsequent denoted by {(u_{j}, v_{j})}_{j≥0} be such that

lim_{j→+∞} ∥(u_{j}, v_{j})∥ = +∞,

In virtue (3.1), we get

\frac{−K}{∥(u_{j}, v_{j})∥^{p}} ≤ \frac{1}{p} − \frac{∫_{Ω} G(x, u_{j}, v_{j})dx}{∥(u_{j}, v_{j})∥^{p}} ≤ \frac{K}{∥(u_{j}, v_{j})∥^{p}} .

By passing to limit we deduce that

lim_{j→+∞} \frac{∫_{Ω} G(x, u_{j}, v_{j})dx}{∥(u_{j}, v_{j})∥^{p}} = \frac{1}{p}.  (3.3)

On the other hand, (3.2) implies

\frac{−ε}{∥(u_{j}, v_{j})∥^{p−1}} ≤ 1 − \frac{∫_{Ω}(u_{j}f(x, u_{j}, v_{j}) + v_{j}g(x, u_{j}, v_{j}))dx}{∥(u_{j}, v_{j})∥^{p}} ≤ \frac{ε}{∥(u_{j}, v_{j})∥^{p−1}} .

By passing to limit, we obtain

lim_{j→+∞} \frac{∫_{Ω}(u_{j}f(x, u_{j}, v_{j}) + v_{j}g(x, u_{j}, v_{j}))dx}{∥(u_{j}, v_{j})∥^{p}} = 1.  (3.4)

Combining (2.1), (2.2), (3.3) and (3.4) we deduce that

\frac{1}{p} ≤ \frac{1}{σ_{0} + 1} + \frac{1}{θ_{0} + 1} < \frac{1}{p}.

A contradiction, whence ∥(u_{j}, v_{j})∥_{X} is bounded. □

Lemma 3.4 Let f and g be two Carathéodory functions satisfying the hypothesis of Theorem 3.1 and let {(u_{j}, v_{j})}_{j≥0} be a sequence in X × X such that (u_{j}, v_{j}) ⇀ (u, v) weakly in X × X. Then

lim_{j→+∞} ∫_{Ω} f(x, u_{j}, v_{j})(u_{j} − u) = 0, quad lim_{j→+∞} ∫_{Ω} g(x, u_{j}, v_{j})(v_{j} − v) = 0.

Proof. By using Hölder’s inequality we obtain

| ∫_{Ω} f(x, u_{j}, v_{j})(u_{j} − u)| ≤ ∥f(x, u_{j}, v_{j})∥_{p′}∥u_{j} − u∥_{p}.

In virtue i), (H1) and by using the imbedding Sobolev space we have (u_{j}, v_{j}) → (u, v) strongly in L^{p}(Ω) × L^{p}(Ω). Then, by Lebesgue’s theorem, as j → +∞,

lim_{j→+∞} ∫_{Ω} f(x, u_{j}, v_{j})(u_{j} − u) = 0.

The proof of the second limit in this Theorem is the same. □

Lemma 3.5 Under the hypothesis of Theorem 3.1, J ∈ C^{1}(X×X) and satisfies the Palais-Smale condition.

Proof. In virtue of the preceding lemma we have J ∈ C^{1}(X × X). Let {(u_{j}, v_{j})}_{j≥0} be a sequence of element in X × X satisfying the conditions (3.1) and (3.2). Hence, by Lemma 3.3 the sequence (u_{j}, v_{j}) is bounded, then, there exist a subsequent still denoted {(u_{j}, v_{j})}_{j≥0} weakly convergent to (u, v) ∈ X × X and strongly in L^{p}(Ω) × L^{p}(Ω) . On the other hand, since

⟨−Δ_{p}u_{j}, u_{j} − u⟩ = ⟨J′(u_{j}, v_{j}), (u_{j}, v_{j}) − (u, v)⟩ − ∫_{Ω} f(x, u_{j}, v_{j})(u_{j} − u).

As j → +∞, in virtue of Lemma 3.4 and (3.2) we have

lim_{j→+∞}⟨−Δ_{p}u_{j}, u_{j} − u⟩ = 0.

Or the p-Laplacian operator satisfies the condition (S_{+}), thus

u_{j} → u strongly in X.

The same way, we prove that v_{j} → v strongly in X. □

Proof of Theorem 3.1 It suffices to prove that the functional J satisfies the conditions for the Pass-Mountain lemma [3]: J satisfies condition of Palais-Smale and J(0) = 0 (see Lemma 3.5). For ∥(u, v)∥ = r sufficiently small, we have J(u, v) ≥ α > 0.

We prove this second conditions first. By i), for all x ∈ Ω, there exist c′ > 0 such that |u(x)|+|v(x)| ≤ c′∥(u, v)∥; and for ∥(u, v)∥ ≤ \frac{r_{0})}{c′}. Using ii) we deduce that

G(x, u(x), v(x)) ≤ c(|u(x)|^{σ+1} + |v(x)|^{θ+1})
≤ c[(c′)^{σ+1}∥(u, v)∥^{σ+1} + (c′)^{θ+1}∥(u, v)∥^{θ+1}]
≤ c′′(∥(u, v)∥^{σ+1} + ∥(u, v)∥^{θ+1}).

Then

J(u, v) ≥ \frac{1}{p}∥(u, v)∥^{p} − c′′(∥(u, v)∥^{σ+1} + ∥(u, v)∥^{θ+1})
≥ \frac{1}{p}∥(u, v)∥^{p} − 2c′′∥(u, v)∥^{l}

with l = min(σ + 1, θ + 1). It suffices to take r ≤ min(\frac{r_{0})}{c′}, (\frac{1}{2pc′′})^{\frac{1}{l−p}}). Finally, for ∥(u, v)∥ ≤ r, we have

J(u, v) ≥ α = \frac{r^{p}}{p} > 0.

Now, we prove the first condition. Let (u_{0}, v_{0}) ∈ X × X such that for almost every where x ∈ Ω_{0} with meas(Ω_{0}) > 0 we have u_{0}(x) + v_{0}(x) > α_{0} > 0with some α_{0} > 0. For t large enough we have tu_{0} > ξ_{0}, tv_{0} > η_{0} with ξ_{0} + η_{0} > R. From (2.1) and (2.2) we get

ξ → \frac{G(x, ξ, η)}{|ξ|^{σ_{0}+1}} and η → \frac{G(x, ξ, η)}{|η|^{θ_{0}+1}}

are increasing, then

∫_{Ω} G(x, tu_{0}, tv_{0}) ≥ ∫_{Ω_{0}} G(x, tu_{0}, tv_{0}) ≥ β(t^{σ_{0}+1} + t^{θ_{0}+1}),

with

β = \frac{1}{2} inf (\frac{1}{ξ_{0}^{σ_{0}+1}} ∫_{Ω_{0}} G(x, ξ_{0}, η_{0})|u_{0}(x)|^{σ_{0}+1},

\frac{1}{η_{0}^{θ_{0}+1}} ∫_{Ω_{0}} G(x, ξ_{0}, η_{0})|v_{0}(x)|^{θ_{0}+1}).

Consequently,

J(tu_{0}, tv_{0}) ≤ \frac{t^{p}}{p} ∥(u, v)∥^{p} − β(t^{σ_{0}+1} + t^{θ_{0}+1}).

by passing to the limit, as t → +∞ we have lim_{t→+∞} J(tu_{0}, tv_{0}) = −∞. Then, there exist some (e_{1}, e_{2}) ∈ X ×X, with e_{1} ≠ 0 and e_{2} ≠ 0, such that J(e_{1}, e_{2}) < 0.

By the Pass-Mountain theorem, there exists (u_{0}, v_{0}) ∈ X ×X u_{0} ≠ 0, v_{0} ≠ 0, such that J′(u_{0}, v_{0}) = 0, i.e for all (ϕ, ψ) ∈ W_{0}^{1,p}(Ω) × W_{0}^{1,p}(Ω),

∫_{Ω} |∇u_{0}|^{p−2}∇u_{0}∇ϕ − ∫_{Ω} f(x, u_{0}, v_{0})ϕ = 0,
∫_{Ω} |∇v_{0}|^{p−2}∇v_{0}∇ψ − ∫_{Ω} g(x, u_{0}, v_{0})ψ = 0.

In virtue of Tolksdorf regularity [10], (u_{0}, v_{0}) ∈ C^{1,ν}(Ω̄) × C^{1,ν}(Ω̄) and by Vazquez’s maximum principle [11], u_{0} > 0 and v_{0} > 0. □

Example Let f(x, ξ, η) = ξ^{σ} exp(ξ^{q} + η^{r}), g(x, ξ, η) = η^{θ}exp(ξ^{q} + η^{r}), σ > 2p − 1, θ > 2p − 1, r, q > 0. The functions f and g satisfy the hypotheses (H1), (H2), (ii), and X the space defined in Proposition 3.2. Hence, for σ, θ > 1;

−Δ_{p}u = u^{σ} exp(u^{q} + v^{r}) in Ω
−Δ_{p}v = v^{θ} exp(u^{q} + v^{r}) in Ω
u = v = 0 on ∂Ω,

has a positive solution (u, v) ∈ (X × C^{1,ν}(\overline{Ω}))^{2}.

The case p = N

Recall that a Young function is an even convex function from ℝ into ℝ^{+}, such that

lim_{ξ→0} \frac{M(ξ)}{ξ} = 0 and lim_{ξ→+∞} \frac{M(ξ)}{ξ} = +∞.

The conjugate function of M is defined as

M^{∗}(ξ) = sup_{s∈ℝ}[ξs − M(s)].

The Orlicz space L_{M}(Ω) is the set of measurable function u defined on ℝ such that, there is some λ > 0 with

∫_{Ω} M(\frac{u}{λ}) < +∞.

This is a Banach space for the norm

∥u∥_{M} = Inf \{λ > 0 : ∫_{Ω} M(\frac{u}{λ}) < 1\} .

Let E_{M}(Ω) be the closure of C_{0}^{∞}(Ω) in L_{M}(Ω). We say that M is super-homogenous of degree (σ + 1) [8] if there exists K > 0 such that

M(hξ) ≤ h^{σ+1}M(Kξ), ∀ξ ∈ ℝ, ∀h ∈ [0, 1] .

Let Ω be a bounded regular domain in ℝ^{N}. In this case W_{0}^{1,p}(Ω) ⊄ L^{∞}(Ω) but W_{0}^{1,p}(Ω) ⊂ E_{M_{1}}(Ω) [1] where

M_{1}(ξ) = exp(|ξ|^{p′}) − 1, or \frac{1}{p} + \frac{1}{p′} = 1.

So, we can get the following Theorem.

Theorem 3.6 Let f and g be two positive functions which are Caratheodory and satisfy (H1) and (H2). Assume also that there exists a Young function of exponential type M such that:

i) The imbedding W_{0}^{1,p}(Ω) ↪ E_{M}(Ω) is compact.

ii) M is super-homogeneous of degree σ_{1} + 1 > p.

iii) There are some c_{1} > 0 and K_{1} > 0 such that for a.e x ∈ Ω and for all (ξ, η) ∈ ℝ^{2},

ξf(x, ξ, η) ≤ c_{1}M(\frac{ξ}{K_{1}} ) and ηg(x, ξ, η) ≤ c_{1}M(\frac{η}{K_{1}} ).

iv) For all K > 0, we have

lim_{|ξ|+|η|→+∞} \frac{f(x, ξ, η)}{M′(\frac{ξ}{K})} = 0 and lim_{|ξ|+|η|→+∞} \frac{g(x, ξ, η)}{M′(\frac{η}{K})} = 0

almost every where in x ∈ Ω.

Then there is at least one positive solution (u, v) ∈ (W_{0}^{1,p}(Ω) ∩ C^{1,ν}(Ω̄))^{2} of (1.1).

The proof of this Theorem needs the following lemma.

Lemma 3.7 Under the hypotheses of Theorem 3.6, J ∈ C^{1}((W_{0}^{1,p}(Ω))^{2} and satisfies the Palais-Smale condition.

Proof. Let {(u_{j}, v_{j})}_{j≥0} be a bounded sequence in W_{0}^{1,p}(Ω) × W_{0}^{1,p}(Ω). By i) there exist some K > 0 such that

∫_{Ω} M(\frac{u_{j}}{K} ) ≤ 1, ∫_{Ω} M(\frac{v_{j}}{K} ) ≤ 1.

Let c > 0 be large enough such that M^{∗}(\frac{1}{c}) meas(Ω) < 1. From iv) for all (ξ, η) ∈ ℝ^{2} and for a.e x ∈ Ω we have

|f(x, ξ, η)| + |g(x, ξ, η)| ≤ \frac{c}{2} + \frac{1}{4}(M′(\frac{ξ}{K}) + M′(\frac{η}{K})),

or M^{∗} is a Young function satisfies the “Δ_{2}-condition”. Then

M^{∗}(\frac{f(x, u_{j}, v_{j})}{c^{2}}) ≤ \frac{1}{2}M^{∗}(\frac{1}{c}) + \frac{1}{2}M^{∗}(\frac{1}{2}M′(\frac{u_{j}}{c^{2}K}) + \frac{1}{2}M′(\frac{v_{j}}{c^{2}K}))
≤ \frac{1}{2}M^{∗}(\frac{1}{c}) + \frac{1}{4}(M(\frac{2u_{j}}{c^{2}K}) + M(\frac{2v_{j}}{c^{2}K}))
≤ \frac{1}{2}M^{∗}(\frac{1}{c}) + \frac{1}{4}(M(\frac{u_{j}}{K}) + M(\frac{v_{j}}{K})).

Hence

∫_{Ω} M^{∗}(\frac{f(x, u_{j}, v_{j})}{c^{2}}) ≤ 1.  (3.5)

In the same we obtain

∫_{Ω} M^{∗}(\frac{g(x, u_{j}, v_{j})}{c^{2}}) ≤ 1.  (3.6)

Let {(u_{j}, v_{j})}_{j≥0} be a subsequent of the least sequence of element in (W_{0}^{1,p}(Ω))^{2} converges to (u, v) ∈ (W_{0}^{1,p}(Ω))^{2}. For δ > 0 sufficiently small, for all ϵ > 0 and A ⊂ Ω such that meas(A) ≤ δ we have

∫AM^{∗}(\frac{f(x, u_{j}, v_{j})}{c^{2}})
≤\frac{1}{2}M^{∗}(\frac{1}{c}) meas(A) + \frac{1}{4} ∫_{A} [M(\frac{2u_{j}}{c^{2}K} ) + M(\frac{2v_{j}}{c^{2}K} )]
≤\frac{1}{2}M^{∗}(\frac{1}{c}) meas(A) + \frac{1}{8} ∫_{A} [M(\frac{u_{j} − u}{K}) + M(\frac{u}{K} ) + M(\frac{v_{j} − v}{K} ) + M(\frac{v}{K} )]
≤ϵ,

then M^{∗}(\frac{f(x,u_{j},v_{j})−f(x,u,v)}{c^{2}}) is equi-summable and

lim_{j→+∞} ∫_{Ω} M^{∗}(\frac{f(x, u_{j}, v_{j}) − f(x, u, v)}{c^{2}}) = 0

By ii) and since M^{∗} satisfies “Δ_{2}-condition” we have

lim_{j→+∞} ∥f(., u_{j}, v_{j}) − f(., u, v)∥_{M^{∗}} = 0.

In the same way we have

lim_{j→+∞} ∥g(., u_{j}, v_{j}) − g(., u, v)∥_{M^{∗}} = 0

Whence J ∈ C^{1}((W_{0}^{1,p}(Ω))^{2}. Let {(u_{j}, v_{j})}_{j≥0} be a sequence satisfying (3.1) and (3.2) then by lemma 3.3 the sequence {(u_{j}, v_{j})}_{j≥0} is bounded in (W_{0}^{1,p}(Ω))^{2} ,hence {(u_{j}, v_{j})}_{j≥0} converges weakly to (u, v) ∈ (W_{0}^{1,p}(Ω))^{2} and strongly in (E_{M}(Ω))^{2}. In view of (3.5) (3.6) we deduce that f(x, u_{j}, v_{j}), g(x, u_{j}, v_{j}) converge with σ(L_{M} ×L_{M}, E_{M} ×E_{M}). So the same proof of lemma 3.5 shows that the Palais-Smale condition is satisfied. □

Proof of Theorem 3.6 Let us show that for ∥(u, v)∥ = r sufficiently small, J(u, v) ≥ α > 0. By (H2) and iii), for a.e x ∈ Ω, for all ξ ∈ ℝ,and for all h ∈ [0, 1], we have

G(x, ξ, η) ≤ \frac{1}{2}[\frac{1}{σ_{0} + 1}ξf(x, ξ, η) + \frac{1}{θ_{0} + 1}ηg(x, ξ, η)]
≤ \frac{c_{1}}{2} [\frac{1}{σ_{0} + 1}M(\frac{ξ}{K_{1}} ) + \frac{1}{θ_{0} + 1}M(\frac{η}{K_{1}} )]
≤ \frac{c_{1}}{2} [h^{σ_{1}+1}M(\frac{Kξ}{hK_{1}} ) + h^{θ_{1}+1}M(\frac{Kη}{hK_{1}} )]

on the other hand, in virtue of i) there exists c > 0 such that for all (u, v) ∈ W_{0}^{1,p}(Ω)^{2} we have

∥u∥_{M} + ∥v∥_{M} ≤ c∥(u, v)∥.

Whence for ∥(u, v)∥ = r ≤ \frac{K_{1}}{cK} and h = \frac{cKr}{K_{1}} we get

∫_{Ω} M(\frac{u}{cr}) ≤ 1 and ∫_{Ω} M(\frac{v}{cr}) ≤ 1.

Hence

∫_{Ω} G(x, u, v)dx ≤ \frac{c_{1}}{2} [h^{σ_{1}+1} + h^{θ_{1}+1}]
≤ c′ [∥(u, v)∥^{σ_{1}+1} + ∥(u, v)∥^{θ_{1}+1}]

The same proof as in Theorem 3.1 gives (u, v) ∈ (W_{0}^{1,p}(Ω))^{2}, u ≢ 0, v ≢ 0, solution of (1.1). The rest of the proof is a consequence of the following lemma. □

Lemma 3.8 Under the hypotheses of Theorem 3.6, if (u, v) is a solution of (1.1) then (u, v) ∈ C^{1,ν}(Ω̄) × C^{1,ν}(Ω̄).

Proof. This proof is inspired by the work of De Thélin [8] and Otani [6] (see also [7]). In view of iii), there exists s > 1 such that uf(x, u, v) ∈ L^{s}(Ω) and vg(x, u, v) ∈ L^{s}(Ω). Consider the following sequences:

q_{1} = 2ps^{∗} = \frac{2ps}{s − 1}, q_{k+1} = 2(p + q_{k})

m_{k} = s^{∗}q_{k}.

Multiplying the first equation of (1.1) by |u|^{q_{k}}u and the second equation by |v|^{q_{k}}v, we obtain:

∫_{Ω} |∇u|^{p−2}∇u∇(|u|^{q_{k}}u) = ∫_{Ω} uf(x, u, v)|u|^{q_{k}}
∫_{Ω} |∇v|^{p−2}∇v∇(|v|^{q_{k}}v) = ∫_{Ω} vg(x, u, v)|u|^{q_{k}}

by Hölder’s inequality we deduce that

\begin{matrix}(\frac{p}{p + q_{k}} )^{p} ∫_{Ω} |∇u^{\frac{p+q_{k}}{p}} |^{p} = & ∫_{Ω} f(x, u, v)|u|^{q_{k}}u \\ ≤ & ∥uf(., u, v)∥_{s}∥u^{q_{k}}∥_{s^{∗}} \\ ≤ & c∥u∥_{s^{∗}}^{q_{k}}. \end{matrix}  (3.7)

Since the imbedding W_{0}^{1,p}(Ω) ↪ L^{2ps^{∗}}(Ω) is compact, there exists K > 0 such that

∥u∥_{2s^{∗}(p+q_{k})}^{p+q_{k}} ≤ K^{p} ∫_{Ω} |∇u^{\frac{p}{p+q_{k}}} |^{p}.  (3.8)

By combining (3.7) and (3.8) we have

∥u∥_{2s^{∗}(p+q_{k})}^{m_{k+1}/(2s^{∗})} ≤ c(\frac{K(p + q_{k})}{p})^{p}∥u∥_{m_{k}}^{m_{k}/s^{∗}}.

Since p + q_{k} ≤ 4^{k}ps^{∗} we get

∥u∥_{m_{k+1}}^{m_{k}+1} ≤ c^{2s^{∗}}(4Ks^{∗})^{2ps^{∗}}4^{2(k−1)ps^{∗}}∥u∥_{m_{k}}^{2m_{k}}.

Set E_{k} = m_{k} log ∥u∥_{m_{k}}, a = 4^{2ps^{∗}}, b = log[c^{2s^{∗}}(2Ks^{∗})^{2ps^{∗}}] and r_{k} = b + (k − 1) log a. We obtain

E_{k+1} ≤ r_{k} + 2E_{k}

then, by the result’s of Otani [6] we deduce that

∥u∥_{∞} ≤ lim sup_{k→+∞} exp (\frac{E_{k}}{m_{k}} ) < +∞.

Finally, by the regularity of Tolksdorf’s results u ∈ C^{1,ν}(Ω̄). In the same way we have v ∈ C^{1,ν}(Ω̄). □

Example Let f(x, ξ, η) = ξ^{σ} exp(ξ^{q} − η^{r}), g(x, ξ, η) = η^{θ} exp(−ξ^{q} + η^{r}), σ > 2p − 1, θ > 2p − 1, N ≥ 2, 0 < r, q < \frac{p}{p−1}, and M(ξ) = |ξ|^{σ+θ+1−l}(e^{|ξ|^{l}} − 1) with max(p, r) < l < 2 hence the functions f and g satisfy the hypothesis (H1), (H2), (i), (ii), (iii) and (iv). Then

−Δ_{p}u = u^{σ} exp(u^{q} − v^{r}) in Ω
−Δ_{p}v = v^{θ} exp(−u^{q} + v^{r}) in Ω
u = v = 0 on ∂Ω,

has a positive solution (u, v) ∈ (W_{0}^{1,p}(Ω) × C^{1,ν}(Ω̄))^{2}.

References

[1] R. Adams, Sobolev spaces, Academic, New-York (1975).

[2] A. Ahammou, Positive radial solutions of nonlinear elliptic systems, New York J. Math. 7 ,pp 267-280, (2001).

[3] Ambrosetti, A., and Rabinowitz, P., Dual variational methods in critical point theory and applications, J. Funct. Anal, pp. 349-381, 1973.

[4] A. Bechah,Local and global estimates for solutions of systems involving the p-Laplacian in unbounded domains,Electronic J. Diff. Eqations, vol. 2001, N0. 19,pp 1-14, (2001).

[5] L. B occardo, J. Fleckinger and F. De Thelin; Elliptic sysstems with various growth, reaction diffusion systems, (Trieste 1995), Lecture notes in Pure and Applied Math., no. 194, Marcel Dekker, New York, p 59-66, Zbl 0892.35059, (1998).

[6] M. Ôtani, Existence and non-existence of non trivial solutions of some nonlinear degenerate elliptic equations, J. Funct. Analysis, vol.76 n. 1,pp 140-159, (1988).

[7] F. De Thelin Résultats d’existence et de non existence pour la solution positive et bornee d’une E.D.P. elliptique non lineare; C.R. Acad. Sc. Paris, t. 299, Serie I n. 18, pp 911-914, (1984).

[8] F. De Thelin Positive Solutions of an elliptic equation with strongly nonlinear lower order terms, 2, 3, (1989).

[9] F. De Thelin, and J. Velin, Existenceand non existence of nontrivial solutions for some non-linear elliptic systems, Matematica Univ. Compl. Madrid 6, pp 153-194, Zbl 0834.35042, (1993).

[10] P. Tolksdorf, Comm.P.D.E,8, pp.773-817, (1989).

[11] J. L Vazquez, A Strong Maximum Principle for some Quasilinear Elliptic Equations, Appl. Math. Optim., 12, pp 191-202, (1984).

Abdelouahed EL Khalil (e-mail: lkhlil@hotmail.com) Mohammed. Ouanan (e-mail: m ouanan@hotmail.com) Abdelfattah Touzani (e-mail: atouzani@iam.net.ma) Laboratoire d’analyse non lineaire (lanolif) Departement de Mathématiques Faculté des sciences DM B.P. 1796 Atlas-Fes, Fes 30000 Maroc