2005-Oujda International Conference on Nonlinear Analysis. Electronic Journal of Differential Equations, Conference 14, 2006, pp. 231–240. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu (login: ftp)

EXISTENCE AND UNIQUENESS OF A POSITIVE SOLUTION FOR A NON HOMOGENEOUS PROBLEM OF FOURTH ORDER

WITH WEIGHTS

MOHAMED TALBI, NAJIB TSOULI

Abstract. In this work we study the existence of a positive solutions to the non homogeneous equation

Δ(|Δu|^{p−2}Δu) = m|u|^{q−2}u with Navier boundary conditions, where 1 < p, q < p_{2}^{∗} and m ∈ L^{∞}(Ω) \ {0}, m ≥ 0. In the case p > q and m ∈ C(\overline{Ω}), we prove the uniqueness of this solution.

1. Introduction

We consider the following problem with Navier boundary conditions

Δ_{p}^{2}u = m|u|^{q−2}u in Ω,
u > 0 in Ω,
u = Δu = 0 on ∂Ω.

(1.1)

Here Ω is a smooth domain in ℝ^{N} (N ≥ 1), Δ_{p}^{2} is the p-biharmonic operator defined by Δ_{p}^{2}u = Δ(|Δu|^{p−2}Δu), m ∈ L^{∞}(Ω) \ {0}, m ≥ 0 and p, q ∈]1, p_{2}^{∗}[, p ≠ q where

p_{2}^{∗} = \begin{cases}\frac{Np}{N−2p} & if p < N/2, \\ +∞ & if p ≥ N/2. \end{cases}

In [9], we proved that the problem (1.1), without the second condition, has an infinity of solutions in the case p > q by using the fundamental multiplicity theorem, but for p < q we have applied the mountain-pass lemma to prove the existence of nontrivial solution. Finally we have studied the regularity of these solutions. In this work we are interested by the existence of a positive solution then in the case p > q we prove the uniqueness of this solution. Notice that our approach does not use the fundamental multiplicity theorem and the mountain-pass lemma. We can refer the reader to [6] for the existence of a positive solution and to [8] for the uniqueness.

2000 Mathematics Subject Classification. 35J60, 35J30, 35J65.

Key words and phrases. Ekeland’s principle; p-biharmonic operator; Palais-Smale condition.

\begin{matrix}c \\ ◯ \end{matrix}2006 Texas State University - San Marcos.

Published September 20, 2006.

Similar results as ours, but with p-Laplacian operator, were studied by authors [8, 2].

2. Preliminaries

In this paper, we consider the transformation of Poisson problem used by Drábek and Ôtani [3]. We recall some properties of the Dirichlet problem for the Poisson equation

\begin{matrix}−Δu = f in Ω, \\ u = 0 on ∂Ω. \end{matrix}  (2.1)

It is well known that (2.1) is uniquely solvable in W^{2,p}(Ω) ∩ W_{0}^{1,p}(Ω) for all f ∈ L^{p}(Ω) and for any p ∈]1, +∞[.

We denote by: X = W^{2,p}(Ω) ∩ W_{0}^{1,p}(Ω), ∥u∥_{p} = (∫_{Ω} |u|^{p}dx)^{1/p} the norm in L^{p}(Ω), ∥u∥_{2,p} = (∥Δu∥_{p}^{p} + ∥u∥_{p}^{p})^{1/p} the norm in X, ∥u∥_{∞} the norm in L^{∞}(Ω), and ⟨·, ·⟩ is the duality bracket between L^{p}(Ω) and L^{p′}(Ω), where p′ = p/(p − 1). Denote by Λ the inverse operator of −Δ : X → L^{p}(Ω). The following lemma gives us some properties of the operator Λ (c.f. [3, 7]).

Lemma 2.1. (i) (Continuity): There exists a constant c_{p} > 0 such that

∥Λf∥_{2,p} ≤ c_{p}∥f∥_{p}

holds for all p ∈]1, +∞[ and f ∈ L^{p}(Ω). (ii) (Continuity) Given k ∈ ℕ^{∗}, there exists a constant c_{p,k} > 0 such that

∥Λf∥_{W^{k+2,p}} ≤ c_{p,k}∥f∥_{W^{k,p}}

holds for all p ∈]1, +∞[ and f ∈ W^{k,p}(Ω). (iii) (Symmetry) The equality

∫_{Ω} Λu · vdx = ∫_{Ω} u · Λvdx

holds for all u ∈ L^{p}(Ω) and v ∈ L^{p′}(Ω) with p ∈]1, +∞[. (iv) (Regularity) Given f ∈ L^{∞}(Ω), we have Λf ∈ C^{1,α}(Ω̄) for all α ∈]0, 1[;

moreover, there exists c_{α} > 0 such that

∥Λf∥_{C^{1,α}} ≤ c_{α}∥f∥_{∞}.

(v) (Regularity and Hopf-type maximum principle) Let f ∈ C(Ω̄) and f ≥ 0 then w = Λf ∈ C^{1,α}(Ω̄), for all α ∈]0, 1[ and w satisfies: w > 0 in Ω, \frac{∂w}{∂n} < 0 on ∂Ω.

(vi) (Order preserving property) Given f, g ∈ L^{p}(Ω) if f ≤ g in Ω, then Λf < Λg in Ω.

Note that for all u ∈ X and all v ∈ L^{p}(Ω), we have v = −Δu if and only if u = Λv.

Let us denote N_{p} the Nemytskii operator defined by

N_{p}(v)(x) = \begin{cases}|v(x)|^{p−2}v(x) & if v(x) ≠ 0 \\ 0 & if v(x) = 0. \end{cases}

Then for all v ∈ L^{p}(Ω) and all w ∈ L^{p′}(Ω), we have N_{p}(v) = w if and only if v = N_{p′}(w).

For v = −Δu which means that u = Λv. As X ↪ L^{q}(Ω), then Λv ∈ L^{q}(Ω)∀v ∈ L^{p}(Ω). We define the functionals F, G : L^{p}(Ω) → ℝ as follows:

F (v) = \frac{1}{p}∥v∥_{p}^{p} and G(v) = \frac{1}{q} ∫_{Ω} m|Λv|^{q}dx.

Then it is clear that F and G are well defined on L^{p}(Ω), and are of class C^{1} on L^{p}(Ω) and for all v ∈ L^{p}(Ω) we have F′(v) = N_{p}(v) and G′(v) = Λ(mN_{q}(Λv)) in L^{p′}(Ω).

The operator Λ enables us to transform problem (1.1) to another problem which we shall study in the space L^{p}(Ω).

Definition 1. We say that u ∈ X \ {0} is a solution of problem (1.1), if v = −Δu is a solution of the problem: Find v ∈ L^{p}(Ω) \ {0}, v > 0, such that

N_{p}(v) = Λ(mN_{q}(Λv)) in L^{p′}(Ω).  (2.2)

3. Existence of a positive solution

For solutions of (2.2) we understand critical points of the associated Euler-Lagrange functional E ∈ C^{1}(L^{p}(Ω)), which are given by

E(v) = F (v) − G(v).

As in [4, 10], we introduce the modified Euler-Lagrange functional defined on ℝ × L^{p}(Ω) by

A(t, v) = E(tv).

If v is an arbitrary element of L^{p}(Ω), ∂_{t}A(., v) (resp. ∂_{tt}A(., v))are the first (resp. second) derivative of the real valued function: t ↦ A(t, v). Since the functional A is even in t and that we are interested by the positive solutions, we limit our study for t > 0.

Theorem 3.1. Problem (1.1) has a positive solution.

To prove theorem 3.1, we need the following preliminary results.

Case p > q: Let v be an arbitrary element of L^{p}(Ω) \ {0}. It is clair that the real valued function t ↦ A(t, v) is decreasing on ]0, t(v)[, increasing on ]t(v), +∞[ and attains its unique minimum for t = t(v), where

t(v) = (\frac{qG(v)}{pF (v)})^{\frac{1}{p−q}} .  (3.1)

On the other hand, a direct computation gives

A(t(v), v) = (\frac{1}{p} − \frac{1}{q})\frac{(qG(v))^{\frac{p}{p−q}}}{(pF (v))^{\frac{q}{p−q}}} < 0.

Furthermore we have proved in [9] that E is bounded bellow and coercive. We deduce that A is also bounded bellow and if

α = inf_{v∈L^{p}(Ω)\{0}} A(t(v), v),  (3.2)

we get −∞ < α < 0. Let (v_{n}) ⊂ L^{p}(Ω) \ {0} be a minimizing sequence of (3.2). Put V_{n} = t(v_{n})v_{n}. Since E is coercive the sequence (V_{n}) is bounded.

Lemma 3.2. The sequence (V_{n}) satisfies

lim inf_{n→+∞} ∥V_{n}∥_{p} > 0.

Proof. Suppose that there is a subsequence of (V_{n}), still denoted by (V_{n}) such that lim_{n→+∞} ∥V_{n}∥ = 0. It follows that lim_{n→+∞} E(V_{n}) = 0; i.e. α = 0, which is impossible since A(t(v_{n}), v_{n}) < 0. □

Lemma 3.3. If 𝕊 is the unit sphere of L^{p}(Ω), we have

α = inf_{v∈𝕊,v≥0} A(t(v), v).

Proof. For every v ∈ L^{p}(Ω), we have |Λv| ≤ Λ|v| and since p > q, we get

A(t(v), v) ≥ (\frac{1}{p} − \frac{1}{q})\frac{(qG(|v|))^{\frac{p}{p−q}}}{(pF (|v|))^{\frac{q}{p−q}}} = A(t(|v|), |v|).

On the other hand the relation (3.1) implies that ∀r > 0 and ∀v ∈ L^{p}(Ω) \ {0}, t(v) = \frac{1}{r}t(\frac{v}{r}). We deduce that

α = inf_{v∈𝕊,v≥0} A(t(v), v),  (3.3)

where 𝕊 is the unit sphere of L^{p}(Ω). □

Note that the minimizing sequences considered up to here are in 𝕊 and are nonnegative.

Lemma 3.4. Let (v_{n}) ⊂ 𝕊 be a minimizing sequence of (3.3), then (V_{n}) := (t(v_{n})v_{n}) is Palais-Smale sequence for the functional E.

Proof. We have E(V_{n}) → α. We show that

E′(V_{n}) → 0 in L^{p′}(Ω).

Note that for every v ∈ L^{p}(Ω) \ {0}, we have ∂_{t}A(t(v), v) = 0 and ∂_{tt}A(t(v), v) ≠ 0. The implicit function theorem implies that v → t(v) is C^{1} since A is. Let us introduce the C^{1} functional B defined on 𝕊 by

B(v) = A(t(v), v) = E(t(v)v).

Then

α = inf_{v∈𝕊,v≥0} B(v) and lim_{n→+∞} B(v_{n}) = α

Using the Ekeland variational principle on the complete manifold (𝕊, ∥ · ∥_{p}) to the functional B, we conclude that

|B′(v)(φ)| ≤ \frac{1}{n}∥φ∥_{p}, for every φ ∈ T_{u_{n}}𝕊,

where T_{v_{n}}𝕊 is the tangent space to 𝕊 at the point v_{n}. Moreover, for every φ ∈ T_{v_{n}}𝕊, one has

B′(v_{n})(φ) = ∂_{t}A(t(v_{n}), v_{n})t′(v_{n})(φ) + ∂_{v}A(t(v), v)(φ)
= ∂_{v}A(t(v), v)(φ),

since ∂_{t}A(t(v), v) = 0, where t′(v) denotes the derivative of v ↦ t(v) at the point v. Furthermore, let P : L^{p}(Ω) \ {0} → ℝ × 𝕊,

v ↦ (P_{1}(v), P_{2}(v)) = (∥v∥_{p}, \frac{v}{∥v∥_{p}}).

Applying Hölder’s inequality, for every (v, φ) ∈ L^{p}(Ω) \ {0} × L^{p}(Ω) we have

∥P′_{2}(v)(φ)∥_{p} ≤ 2\frac{∥φ∥_{p}}{∥v∥_{p}} .

From lemma 3.2 and by the fact that ∥V_{n}∥_{p} = t(v_{n}), there is a positive constant C such that

t(v_{n}) ≥ C, ∀n ∈ ℕ.

Then for every φ ∈ L^{p}(Ω) we get

|E′(V_{n})(φ)| = |∂_{t}A(P_{1}(V_{n}), P_{2}(V_{n}))P′_{1}(V_{n})(φ) + ∂_{v}A(P_{1}(V_{n}), P_{2}(V_{n}))P′_{2}(V_{n})(φ)|

= |∂_{v}A(t(v_{n}), v_{n})P′_{2}(V_{n})(φ)|

= |B′(v_{n})P′_{2}(V_{n})(φ)|

≤ \frac{1}{n}∥P′_{2}(V_{n})(φ)∥_{p}

≤ \frac{2}{n} \frac{∥φ∥_{p}}{C} .

We easily conclude that lim_{n→+∞} E′(V_{n}) = 0 in L^{p′}(Ω). □

Case p < q: If v is an arbitrary element of L^{p}(Ω) \ {0}, the real valued function t ↦ A(t, v) is increasing on ]0, t(v)[, decreasing on ]t(v), +∞[ and attains its unique maximum for t = t(v), where

t(v) = (\frac{pF (v)}{qG(v)})^{\frac{1}{q−p}} .  (3.4)

Lemma 3.5. If p < q, there exists a positive constant c(p, q, Ω, m) which depends uniquely of p, q, Ω and m such that A(t(v), v) ≥ c(p, q, Ω, m).

Proof. A direct computation gives

A(t(v), v) = (\frac{1}{p} − \frac{1}{q})\frac{(pF (v))^{\frac{q}{q−p}}}{(qG(v))^{\frac{p}{q−p}}} .

Hence

A(t(v), v) ≥ (\frac{1}{p} − \frac{1}{q})\frac{1}{∥m∥∞^{\frac{p}{q−p}}} (\frac{∥v∥_{p}}{∥Λv∥_{q}})^{\frac{pq}{q−p}} .

The assertion (i) of Lemma 2.1 and the fact that X ↪ L^{q}(Ω) imply that there exists positive constants c_{q} and c such that

A(t(v), v) ≥ (\frac{1}{p} − \frac{1}{q})\frac{1}{(c_{q}c)^{\frac{pq}{q−p}} ∥m∥∞^{\frac{p}{q−p}}} (\frac{∥v∥_{p}}{∥v∥_{p} + ∥Λv∥_{p}})^{\frac{pq}{q−p}} .

Finally the assertion (i) of lemma2.1 implies that there exits a positive constant c_{p} such that

A(t(v), v) ≥ (\frac{1}{p} − \frac{1}{q})\frac{1}{(c_{q}c_{p}c)^{\frac{pq}{q−p}} ∥m∥∞^{\frac{p}{q−p}}} .

We take c_{(}p, q, Ω, m) = (\frac{1}{p} − \frac{1}{q})_{\frac{1}{(c_{q}c_{p}c)^{\frac{pq}{q−p}} ∥m∥ \frac{p}{q−p} ; ∞}} . □

Put

α = inf_{v∈L^{p}(Ω)\{0}} A(t(v), v).

Then Lemma 3.5 implies α > 0.

Lemma 3.6. If 𝕊 is the unit sphere of L^{p}(Ω), we have

α = inf_{v∈𝕊,v≥0} A(t(v), v).

Proof. For every v ∈ L^{p}(Ω) \ {0}, we have

A(t(v), v) = (\frac{1}{p} − \frac{1}{q})\frac{(pF (v))^{\frac{q}{q−p}}}{(qG(v))^{\frac{p}{q−p}}} .

Since |Λv| ≤ Λ|v|, we get

A(t(v), v) ≥ (\frac{1}{p} − \frac{1}{q})\frac{pF (|v|)^{\frac{q}{q−p}}}{qG(|v|)^{\frac{p}{q−p}}} = A(t(|v|), |v|).

On the other hand, the relation (3.4) implies that for every r > 0 and for every v ∈ L^{p}(Ω) \ \{0\}, t(v) = \frac{1}{r}t(\frac{v}{r}). Hence

α = inf_{v∈𝕊,v≥0} A(t(v), v).  (3.5)
□

Let (v_{n}) be a minimizing sequence of (3.5), as in the case p > q, we put

V_{n} = t(v_{n})v_{n}.

The proof of the following lemmas can be done like in the previous case.

Lemma 3.7. lim inf_{n→+∞} ∥V_{n}∥_{p} > 0.

Lemma 3.8. Let (v_{n}) ⊂ 𝕊 be a minimizing sequence of (3.3). Then (V_{n}) := (t(v_{n})v_{n}) is Palais-Smale sequence for the functional E.

Proof of theorem 3.1. In our paper [9] we showed that E verifies the Palais-Smale condition. Then by lemma 3.4 and lemma 3.8, we deduce that there is a subsequence of (V_{n}), still noted by (V_{n}) such that V_{n} → V, V ∈ L^{p}(Ω)\{0} and V ≥ 0. Moreover, since E′(V_{n}) → 0, then E′(V) = 0. i.e. V is a nonnegative solution of problem (2.2). Hence

N_{p}(V) = Λ(mN_{q}(ΛV)).  (3.6)

The assertion (vi) of lemma 2.1, the relation (3.6) and the fact that m ∈ L^{p}(Ω)\{0}, m ≥ 0 enable us to claim that N_{p}(V) > 0 and V > 0. Furthermore U = ΛV is a positive solution of problem (1.1). □

4. Uniqueness of the positive solution

Theorem 4.1. If m ∈ C(\overline{Ω}), m ≥ 0 and p > q, then (1.1) has a unique nonnegative solution.

Problem (2.2) is equivalent to the problem: Find v ∈ L^{p}(Ω) \ {0}, v > 0 such that

N_{p}(v) = ∥m^{1/q}Λv∥_{q}^{q−p}∥m^{1/q}Λv∥_{q}^{p−q}Λ(mN_{q}(Λv)) in L^{p′}(Ω).  (4.1)

To prove that problem (2.2) has a unique nonnegative solution, we will study the principal positive eigenvalue of the eigenvalue problem: Find v ∈ L^{p}(Ω) \ {0} × ℝ_{+}^{∗} such that

N_{p}(v) = λ∥m^{1/q}Λv∥_{q}^{p−q}Λ(mN_{q}(Λv)) in L^{p′}(Ω).  (4.2)

Consider the functionals f and g defined on L^{p}(Ω) by

f(v) = \frac{1}{p}∥v∥_{p} and g(v) = \frac{1}{p}(∫_{Ω} m|Λv|^{q}dx)^{\frac{p}{q}} .

Hence problem (4.2) is equivalent to the problem: Find (v, λ) ∈ L^{p}(Ω) \ {0} × ℝ_{+}^{∗} such that

f′(v) = λg′(v) in L^{p′}(Ω).  (4.3)

Define

λ_{1} = inf_{v∈M} f(v),

where M = {v ∈ L^{p}(Ω)/g(v) = 1}. We need the preliminary results.

Lemma 4.2. (i) λ_{1} is the first positive eigenvalue of problem (4.2). Moreover v_{1} is an eigenfunction associated with λ_{1} if and only if

f(v_{1}) − λ_{1}g(v_{1}) = 0 = inf_{v∈L^{p}(Ω)\{0}} f(v) − λ_{1}g(v).

(ii) Every eigenfunction associated with λ_{1} is positive or negative.

Proof. (i) The functional f is weakly semi-continuous below and coercive on M. Since g is weakly continuous, then M is weakly closed. Hence there is v_{1} ∈ M such that f(v_{1}) = λ_{1} = λ_{1}g(v_{1}).

The p-homogeneity of f and g implies that λ_{1} is an eigenvalue of problem (4.2) if and only if

∀v ∈ L^{p}(Ω) \ \{0\}, λ_{1} ≤ \frac{f(v)}{|g(v)|}

if and only if for all v ∈ L^{p}(Ω) \ {0},

f(v) − λ_{1}g(v) ≥ f(v) − λ_{1}|g(v)| ≥ 0 = f(v_{1}) − λ_{1}g(v_{1}).

Now we show that λ_{1} is the first positive eigenvalue: Suppose on the contrary that there exits λ ∈]0, λ_{1}[ and v ∈ L^{p}(Ω) \ {0} such that f(v) − λg(v) = 0. Then we get

0 = f(v_{1}) − λ_{1}g(v_{1}) ≤ f(v) − λ_{1}g(v) < f(v) − λg(v) = 0,

which is a contradiction. (ii) Let v be an eigenfunction associated with λ_{1}. From the assertion (i) and by the fact that |Λv| ≤ Λ|v|, we get

0 = f(v) − λ_{1}g(v) ≤ f(|v|) − λ_{1}g(|v|) ≤ f(v) − λ_{1}g(v) = 0.

Therefore, |v| an is eigenfunction associated with λ_{1}. From the assertion in lemma 2.1(vi) and by the fact that

N_{p}(|v|) = λ_{1}Λ(mN_{q}(|v|),

we deduce that |v| > 0 in Ω. Hence v is positive or negative in Ω. □

Lemma 4.3. If v and w are positive eigenfunctions of (2.2) associated with λ_{1}, then the functions max and min defined in Ω by max(x) = max(v(x), w(x)) and min(x) = min(u(x), w(x)) are also solutions of (2.2) associated with λ_{1}.

To prove lemma 4.3 we need the following results.

Lemma 4.4. Let a, b, c and p be reals such that a ≥ 0, b ≥ 0 and p > 1. If c ≥ max{b − a, 0}, then

|a + c|^{p} + |b − c|^{p} ≥ a^{p} + b^{p}.

For the proof of the above lemma see for example [3].

Lemma 4.5. Let a, b, c and d be in ℝ_{+} such that a ≥ max(c, d). If a + b ≥ c + d, then for every p ∈ [1, +∞[, a^{p} + b^{p} ≥ c^{p} + d^{p}.

Proof. If b ≥ min(c, d) or a ≥ c + d it is evident. Else, set α = a − d and β = c − b. We can suppose that d ≤ c. Since a < c + d and a + b ≥ c + d we deduce that α < c and β ≤ α. Then

a^{p} + b^{p} = |d + α|^{p} + |c − β|^{p} ≥ |d + α|^{p} + |c − α|^{p}.

As α ≥ c − d, then from lemma 4.4 we conclude that a^{p} + b^{p} ≥ c^{p} + d^{p}. □

Proof of lemma 4.3. If u and v are two positive eigenfunctions associated with λ_{1}, we claim that

( ∫_{Ω} m|Λ max(u, v)|^{q}dx)^{\frac{p}{q}} + ( ∫_{Ω} m|Λ min(u, v)|^{q}dx)^{\frac{p}{q}}
≥ ( ∫_{Ω} m|Λu|^{q}dx)^{\frac{p}{q}} + ( ∫_{Ω} m|Λv|^{q}dx)^{\frac{p}{q}} .

(4.4)

Indeed, we have

max(u, v) = u + \frac{v − u + |v − u|}{2}.

Then the fact that for every w ∈ L^{p}(Ω), Λ|w| ≥ |Λw| enables us to deduce that

Λ max(u, v) ≥ Λu + \frac{Λv − Λu + |Λv − Λu|}{2} = max(Λu, Λv).

Hence

∫_{Ω} m|Λ max(u, v)|^{q}dx) ≥ ∫_{Ω} m| max(Λu, Λv)|^{q}dx
≥ max(∫_{Ω} m|Λu|^{q}dx, ∫_{Ω} m|Λv|^{q}dx).

Therefore, from lemma 4.5 we conclude inequality (4.4). If we put

ϕ(w) = f(w) − λ_{1}g(w) ∀w ∈ L^{p}(Ω),

from (4.4) and from lemma 4.2, we deduce that

0 ≤ ϕ(max(u, v)) + ϕ(min(u, v) ≤ ϕ(u) + ϕ(v) = 0

and ϕ(max(u, v)) = ϕ(min(u, v)) = 0. Thus, min(u, v) and max(u, v) are eigenfunctions associated with λ_{1}. □

Lemma 4.6. Every eigenfunction of problem (2.2) is in C(\overline{Ω}).

Proof. If v is an eigenfunction of problem (2.2) associated with a positive eigenvalue λ, then

v = λ^{1/(p−1)}N_{p′}(∥m^{1/q}Λw∥_{q}^{p−q}Λ(mN_{q}(Λv))).  (4.5)

Since |Λv| ≤ Λ|v|, we get

|v| ≤ λ^{1/(p−1)}∥m∥∞^{\frac{1}{p−1}} ∥m^{1/q}Λw∥q^{\frac{p−q}{p−1}} N_{p′}(ΛN_{q}(|Λv|)).  (4.6)

We showed in our paper [9] that N_{p′}(ΛN_{q}(|Λv|)) ∈ C(\overline{Ω}). Hence from (4.6) we deduce that v ∈ L^{∞}(Ω) and from (4.5) and the assertion in lemma 2.1(iv) it follows that v ∈ C(\overline{Ω}). □

Proposition 4.7. The eigenvalue λ_{1} is simple and every positive eigenfunction is associated with λ_{1}.

Proof. Let v and w be two positive eigenfunctions associated with λ_{1}. For x_{0} ∈ Ω set k = v(x_{0})/w(x_{0}) and max_{k}(x) = max(v(x), kw(x)). Lemma 4.3 enables us to claim that max_{k} is a solution of problem (2.2) associated with λ_{1}. Since

N_{p}(v) = λ_{1}Λ(mN_{p}(Λv)),
N_{p}(w) = λ_{1}Λ(mN_{p}(Λw)),
N_{p}(max_{k}) = λ_{1}Λ(mN_{p}(Λ max_{k})),

Lemma 4.6 and lemma 2.1 imply that N_{p}(v), N_{p}(w), N_{p}(max_{k}) ∈ C^{1,α}(\overline{Ω}) and N_{p}(v), N_{p}(w) are positive in Ω. Then

N_{p}(v)/N_{p}(w) ∈ C^{1}(Ω).

For any unit vector e, we have

N_{p}(v)(x_{0} + te) − N_{p}(v)(x_{0}) ≤ N_{p}(max_{k})(x_{0} + te) − N_{p}(max_{k})(x_{0})

and

N_{p}(kw)(x_{0} + te) − N_{p}(kw)(x_{0}) ≤ N_{p}(max_{k})(x_{0} + te) − N_{p}(max_{k})(x_{0}).

Dividing these inequalities by t > 0 and t < 0 and letting t tend to 0^{±}, we get

∇N_{p}(v)(x_{0}) = ∇N_{p}(max_{k})(x_{0}) = k^{p−1}∇N_{p}(w)(x_{0}).

Thus

∇(\frac{N_{p}(v)}{N_{p}(w)})(x_{0}) = ∇(\frac{N_{p}(v)}{N_{p}(w)})(x_{0})

= \frac{(∇(N_{p}(v))(x_{0})N_{p}(w)(x_{0}) − N_{p}(v)(x_{0})∇(N_{p}(w))(x_{0}))}{(N_{p}(w)(x_{0}))^{2}} = 0.

Hence

N_{p}(\frac{v}{w}) = \frac{N_{p}(v)}{N_{p}(w)} = const = k^{p−1} in Ω

and

\frac{v}{w} = k in Ω.

Now we show that every positive eigenfunction is associated with λ_{1}: Let λ > λ_{1}, suppose that problem (2.2) has a positive eigenfunction w associated with λ and let v be a positive solution of problem (2.2) associated with λ_{1}, we have

N_{p}(v) = λ_{1}Λ(mN_{p}(Λv)) and N_{p}(w) = λΛ(mN_{p}(Λw)).

Then from the assertion in lemma 2.1(v) we deduce that N_{p}(v) and N_{p}(w) are in C^{1,α}(\overline{Ω}), and

∂(N_{p}(v))/∂n < 0, ∂(N_{p}(w))/∂n < 0 on ∂Ω.

It follows that N_{p}(v)/N_{p}(w) is in C(\overline{Ω}). Set

a = max_{x∈\overline{Ω}} N_{p}(v)(x)/N_{p}(w)(x).

We deduce that N_{p}(v) ≤ aN_{p}(w). The monotonicity of N_{p′} implies

v ≤ a^{\frac{1}{p−1}} w.

Since problem (2.2) is homogeneous, a^{\frac{1}{p−1}} w is also a solution of problem (2.2), we may assume without loss of generality that v ≤ w. Then, from the assertion of lemma 2.1(vi) and by the monotonicity of N_{q}, we get

N_{p}(v) = λ_{1}∥m^{1/q}Λv∥_{q}^{p−q}Λ(mN_{q}(Λv))
≤ ∥m^{1/q}Λw∥_{q}^{p−q}λ_{1}Λ(mN_{q}(Λw))
= λ∥m^{1/q}Λcw∥_{q}^{p−q}Λ(mN_{q}(Λcw))
= N_{p}(cw),

where

c = (λ_{1}/λ)^{1/(p−1)} < 1.

Hence it follows by the monotonicity of N_{p′} that v < cw. Repeating this argument n times, we obtain 0 ≤ v ≤ c^{n}w. Therefore by letting n tend to infinity, we deduce that v ≡ 0. This is a contradiction. □

Proof of theorem 4.1. Let v and w be two positive solutions of problem (4.1). Then v and w are eigenfunctions associated with the eigenvalues ∥m^{1/q}Λv∥_{q}^{q−p} and ∥m^{1/q}Λw∥_{q}^{q−p} respectively. From proposition 4.7 we deduce that

∥m^{1/q}Λv∥_{q}^{q−p} = ∥m^{1/q}Λw∥_{q}^{q−p} = λ_{1}

and there is k > 0 such that w = kv. It follows that v = w. □

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Mohamed Talbi Département de Mathématiques et Informatique Faculté des Sciences, Université Mohamed 1, Oujda, Maroc

  E-mail address: talbi md@yahoo.fr

Najib Tsouli Département de Mathématiques et Informatique Faculté des Sciences, Université Mohamed 1, Oujda, Maroc

  E-mail address: tsouli@sciences.univ-oujda.ac.ma