2005-Oujda International Conference on Nonlinear Analysis. Electronic Journal of Differential Equations, Conference 14, 2006, pp. 95–107. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu (login: ftp)

MAXIMUM AND ANTI-MAXIMUM PRINCIPLES FOR THE P-LAPLACIAN WITH A NONLINEAR BOUNDARY CONDITION

AOMAR ANANE, OMAR CHAKRONE, NAJAT MORADI

Abstract. In this paper we study the maximum and the anti-maximum principles for the problem Δ_{p}u = |u|^{p−2}u in the bounded smooth domain Ω ⊂ ℝ^{N}, with |∇u|^{p−2} \frac{∂u}{∂ν} = λ|u|^{p−2}u + h as a non linear boundary condition on ∂Ω which is supposed C^{2β} for some β in ]0, 1[, and where h ∈ L^{∞}(∂Ω). We will also examine the existence and the non existence of the solutions and their signs.

1. Introduction

In this work we consider the problem

\begin{matrix}Δ_{p}u = |u|^{p−2}u in Ω, \\ |∇u|^{p−2} \frac{∂u}{∂ν} = λ|u|^{p−2}u + h on ∂Ω, \end{matrix}  (1.1)

where Ω is a bounded domain in ℝ^{N}, with a C^{2,β} boundary where β ∈]0, 1[, h ∈ L^{∞}(∂Ω) and \frac{∂}{∂ν} is the outer normal derivative.

For h ≡ 0 in ∂Ω, Fernandez Bonder, Pinasco and Rossi [1] proved that the problem

△_{p}u = |u|^{p−2}u on Ω,

|∇u|^{p−2} \frac{∂u}{∂ν} = λ|u|^{p−2}u on ∂Ω

admits an infinite sequence of eigenvalues (λ_{n}) such that λ_{n} → ∞ as n → ∞. Martinez and Rossi [4] showed that the first eigenvalue given by

λ_{1} = inf { ∫_{Ω} |∇u|^{p} + ∫_{Ω} |u|^{p} : u ∈ W^{1,p}(Ω) and ∫_{∂Ω} |u|^{p} = 1}

is simple and isolated with the eigenfunctions do not change sign in Ω.

We will be interested in he case where h ≢ 0 in ∂Ω. The case where h ≡ 0, is treated by Godoy, Gossez and Paczka [3]; they have proved in that (i) The maximum principle holds if and only if λ_{−1}(m) < λ < λ_{1}(m) with λ_{−1}(m) andλ_{1}(m) are the principal eigenvalues of the p-Laplacien with the weight m, for the Dirichlet problem, (respectively 0 < λ < λ^{∗}(m), for the Neumann problem, with λ^{∗}(m) is the nontrivial principal eigenvalue). (ii) The anti-maximum principle holds at the right of λ_{1}(m) and at the left of λ_{−1}(m) (resp, at the right of λ^{∗}(m) and at the left of 0. Moreover it is nonuniform when p ≤ N and uniform when p > N).

2000 Mathematics Subject Classification. 35J65, 35J25.

Key words and phrases. Anti-maximum; p-laplacian; non linear boundary condition.

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

Published Septebmer 20, 2006.

In what follows one supposes that any solution of (1.1) is in C^{1,α}(\overline{Ω}) with α ∈ ]0, 1[.

2. The maximum principle

The following result will be proven.

Theorem 2.1. The maximum principle holds for problem (1.1) if and only if λ ≤ λ_{1}.

Proof. (i) Given λ ≤ λ_{1}, 0 ≨ h ∈ L^{∞}(∂Ω) and u be a solution of (1.1), let us show that u ≥ 0 in Ω. We recall that u is a solution of(1.1) if and only if

∫_{Ω} |∇u|^{p−2}∇u∇v + ∫_{Ω} |u|^{p−2}uv = λ ∫_{∂Ω} |u|^{p−2}uv + ∫_{∂Ω} hv  (2.1)

for all v ∈ W^{1,p}(Ω). Applying this equality to u^{−}, one finds

∫_{Ω} |∇u|^{p−2}∇u∇u^{−} + ∫_{Ω} |u|^{p−2}uu^{−} = λ ∫_{∂Ω} |u|^{p−2}uu^{−} + ∫_{∂Ω} hu^{−}.

Then

∫_{Ω} |∇u^{−}|^{p} + ∫_{Ω} |u^{−}|^{p} = λ ∫_{∂Ω} |u^{−}|^{p} − ∫_{∂Ω} hu^{−},

and

λ ∫_{∂Ω} |u^{−}|^{p} − ( ∫_{Ω} |∇u^{−}|^{p} + ∫_{Ω} |u^{−}|^{p}) = ∫_{∂Ω} hu^{−}.

However, λ ≤ λ_{1} = inf{∫_{Ω} |∇u|^{p} + ∫_{Ω} |u|^{p} : u ∈ W^{1,p}(Ω) and ∫_{∂Ω} |u|^{p} = 1}, so

∫_{∂Ω} hu^{−} = λ ∫_{∂Ω} |u^{−}|^{p} − ( ∫_{Ω} |∇u^{−}|^{p} + ∫_{Ω} |u^{−}|^{p})
≤ λ_{1} ∫_{∂Ω} |u^{−}|^{p} − ( ∫_{Ω} |∇u^{−}|^{p} + ∫_{Ω} |u^{−}|^{p}) ≤ 0.

Moreover h ≩ 0 and u^{−} ≥ 0 thus ∫_{∂Ω} hu^{−} ≥ 0 from where ∫_{∂Ω} hu^{−} = 0. Consequently λ ∫_{∂Ω} |u^{−}|^{p} = ∫_{Ω} |∇u^{−}|^{p} + ∫_{Ω} |u^{−}|^{p}; i-e, u^{−} is a positive eigenfunction associated with λ.

If λ ≠ λ_{1} thenu^{−} change sign on ∂Ω by [4, lemma 2.4], which is not possible, so u^{−} ≡ 0 on ∂Ω. Hence 0 = λ ∫_{∂Ω} |u^{−}|^{p} = ∫_{Ω} |∇u^{−}|^{p} + ∫_{Ω} |u^{−}|^{p} and thus u^{−} ≡ 0 in Ω.

If λ = λ_{1} then u^{−} is a positive eigenfunction associated to λ_{1} and it is in C^{1,α}(\overline{Ω}), so u^{−} > 0 or u^{−} ≡ 0 in\overline{Ω} by the following maximum principle.

Theorem 2.2 (Vasquez [5]). Let u ∈ C^{1}(Ω) be such that Δ_{p}u ∈ L_{loc}^{2}(Ω), u ≥ 0 a.e in Ω and Δ_{p}u ≤ β(u) a.e in Ω, with β : [0, +∞[→ ℝ is a increasing continuous function, β(0) = 0 and either β(s) = 0 for some s > 0 or (β(s) > 0 for all s > 0 and ∫_{0}^{1}(j(s))^{− \frac{1}{p}} ds = ∞ with j(s) = ∫_{0}^{s} β(t)dt). Then if u does not vanish identically on Ω, it is positive everywhere in Ω.

Moreover if u ∈ C^{1}(Ω ∪ {x_{0}}) for an x_{0} ∈ ∂Ω that satisfies an interior sphere condition and u(x_{0}) = 0 then \frac{∂u}{∂n}(x_{0}) > 0 where n is an interior normal at x_{0}.

Indeed, (u^{−} > 0 or u^{−} = 0) in Ω, and if there exists x_{0} ∈ ∂Ω such that u^{−}(x_{0}) = 0, then \frac{∂u^{−}}{∂ν} (x_{0}) < 0. However |∇u^{−}|^{p−2} \frac{∂u^{−}}{∂ν} (x_{0}) = λ_{1}|u^{−}(x_{0})|^{p−2}u^{−}(x_{0}) = 0 and thus ∇u^{−} ≡ 0 in \overline{Ω}; i.e., u^{−} = 0 in \overline{Ω}. Consequently u ≥ 0 in\overline{Ω}. (ii) Given λ > λ_{1}, and one supposes that the maximum principle is applicable to the problem (1.1); i.e., for all h ∈ L^{∞}(∂Ω), if h ≩ 0 then any solution is positive on Ω.

Considering Φ a positive eigenfunction associated to the first eigenvalue λ_{1}, and h = (λ − λ_{1})|Φ|^{p−2}Φ ≩ 0, one checks that Ψ = −Φ ≤ 0 is a solution of (1.1), but this contradicted the maximum principle. □

3. Existence and nonexistence of solutions for (1.1)

In this paragraph one shows stronger results existence and nonexistence. They are stated in the following theorem.

Theorem 3.1. Given h ∈ L^{∞}(∂Ω) such that h ≥ 0 in ∂Ω.

(i) Ifλ < λ_{1} and h ≩ 0 in ∂Ω, then the problem admits an unique solution u which is in C^{1}(\overline{Ω}) andu > 0 on \overline{Ω}.

(ii) If λ = λ_{1} and h ≩ 0 in ∂Ω, then the problem has no solution.

(iii) If λ > λ_{1} then the problem does not have any solution u such that u ≩ 0 on \overline{Ω}.

For the proof of this theorem we need the following lemma.

Lemma 3.1. Let u be a solution of (1.1) with u > 0 on \overline{Ω}, and h ≥ 0 in ∂Ω (or h ≤ 0 in ∂Ω). Then ∀φ ∈ C^{1}(\overline{Ω}) with φ ≥ 0 in \overline{Ω}, one has

λ ∫_{∂Ω} φ^{p} + ∫_{∂Ω} h\frac{φ^{p}}{u^{p−1}} ≤ ∫_{Ω} |∇φ|^{p} + ∫_{Ω} φ^{p}..  (3.1)

Moreover, the equality holds if and only if φ is a multiple of u.

Proof. Given u and φ in C^{1}(\overline{Ω}) with u > 0 and φ ≥ 0 on \overline{Ω}, we denote R(φ, u) = |∇φ|^{p} − |∇u|^{p−2}∇u∇(\frac{φ^{p}}{u^{p−1}} ), and we prove that R(φ, u) ≥ 0 and that R(φ, u) = 0 if and only if there exists c ≥ 0 such that φ = cu.

One has R(φ, u) = |∇φ|^{p} + (p − 1)\frac{φ^{p}}{u^{p}} |∇u|^{p} − p\frac{φ^{p−1}}{u^{p−1}} |∇u|^{p−2}∇u∇φ. Applying Minkovsky inequality to |∇φ| and \frac{φ^{p−1}}{u^{p−1}} |∇u|^{p−1}, one obtains

\frac{φ^{p−1}}{u^{p−1}} |∇u|^{p−1}|∇φ| ≤ \frac{1}{p}|∇φ|^{p} + \frac{1}{q}(\frac{φ^{p−1}}{u^{p−1}} |∇u|^{p−1})^{q}  (3.2)

where \frac{1}{q} + \frac{1}{p} = 1. This implies \frac{φ^{p−1}}{u^{p−1}} |∇u|^{p−1}|∇φ| ≤ \frac{1}{p}|∇φ|^{p} + \frac{p−1}{q}(\frac{φ^{p}}{u^{p}} |∇u|^{p−1}). And it follows that

R(φ, u) = |∇φ|^{p} + (p − 1)\frac{φ^{p}}{u^{p}} |∇u|^{p} − p\frac{φ^{p−1}}{u^{p−1}} |∇u|^{p−2}∇u∇φ ≥ 0.

The inequality (3.2) becomes an equality if and only if

|∇φ| = \frac{φ}{u}|∇u|.  (3.3)

Let ψ = \frac{φ}{u} ≥ 0. Then we have φ(x) = ψ(x)u(x) which implies ∇φ = ψ∇u + u∇ψ. Then (3.3) is equivalent to |ψ∇u + u∇ψ| = ψ|∇u|, which is true if and only if u^{2}|∇ψ|^{2} = −2uψ∇u∇ψ. We have also:

(R(φ, u) = 0) ⇔ |∇φ|^{p} = |∇u|^{p−2}|∇u∥∇(\frac{φ}{u})^{p}u|

⇔ |∇φ|^{p} = |∇u|^{p}(\frac{φ}{u})^{p} + u|∇u|^{p−2}|∇u|∇(ψ^{p}).

Since |∇φ| = ψ|∇u|, one has

R(φ, u) = 0 ⇔ |∇u|^{p−2}|∇u|∇(ψ^{p}) = |∇u|^{p−2}(∇u∇ψ)pψ^{p−1} = 0

If |∇u| ≡ 0, then (3.3) implies that |∇φ| = 0. And in this case u and φ are constants; therefore there exists c ≥ 0 such that φ = cu.

If |∇u| ≠ 0, on the set {|∇u| ≠ 0}, (3.3) is equivalent to ∇u∇ψ = 0. Therefore, one has also (3.3) is equivalent to −2ψ∇u∇(φψ) = u|∇ψ|^{2} Thus ∇u∇ψ = 0 if and only if |∇ψ| = 0, which is equivalent to ψ equals a constant.

Consequently R(φ, u) = 0 if and only there exists c ≥ 0 such that φ = cu. Conclusions:

(i) 0 ≤ ∫_{Ω} R(φ, u) = ∫_{Ω} |∇φ|^{p} −∫_{Ω} |∇u|^{p−2}∇u∇(\frac{φ^{p}}{u^{p−1}} ) for all(u, φ) ∈ (C^{1}(\overline{Ω}))^{2}

with u > 0 and φ ≥ 0 on \overline{Ω}.

(ii) ∫_{Ω} R(φ, u) = ∫_{Ω} |∇φ|^{p} − ∫_{Ω} |∇u|^{p−2}∇u∇(\frac{φ^{p}}{u^{p−1}} ) = 0 if and only there exists

c ≥ 0 such that φ = cu.

Then if u is a solution of (1.1) and φ ∈ C^{1}(\overline{Ω}) such that φ ≥ 0 on \overline{Ω}, we get

∫_{Ω} |∇u|^{p−2}∇u∇(\frac{φ^{p}}{u^{p−1}} ) + ∫_{Ω}(u)^{p−1}(\frac{φ^{p}}{u^{p−1}} ) = λ ∫_{∂Ω}(u)^{p−1}(\frac{φ^{p}}{u^{p−1}} ) + ∫_{∂Ω} h\frac{φ^{p}}{u^{p−1}} .

Consequently,

0 ≤ ∫_{Ω} |∇φ|^{p} − λ ∫_{∂Ω}(u)^{p−1}(\frac{φ^{p}}{u^{p−1}} ) − ∫_{∂Ω} h\frac{φ^{p}}{u^{p−1}} + ∫_{Ω}(u)^{p−1}(\frac{φ^{p}}{u^{p−1}} ),

then

0 ≤ ∫_{Ω} |∇φ|^{p} − λ ∫_{∂Ω} φ^{p} − ∫_{∂Ω} h\frac{φ^{p}}{u^{p−1}} + ∫_{Ω} φ^{p},

and

∫_{Ω} |∇φ|^{p} + ∫_{Ω} φ^{p} ≥ λ ∫_{∂Ω} φ^{p} + ∫_{∂Ω} h\frac{ϕ^{p}}{u^{p−1}} .  (3.4)

If there is equality in (3.4) then ∫_{Ω} R(φ, u) = 0 which is equivalent to R(φ, u) = 0 almost everywhere in Ω. So there exists c ≥ 0 such that φ = cu. □

Proof of Theorem 3.1. The case of λ < λ_{1}: (i) Existence of the solutions for the problem (1.1): One considers the function Φ : W^{1,p}(Ω) → ℝ defined by

Φ(v) = \frac{1}{p}(∫_{Ω} |∇v|^{p} + ∫_{Ω} |v|^{p}) − \frac{λ}{p} ∫_{∂Ω} |v|^{p} − ∫_{∂Ω} hv .

(a) Φ is C^{1} and weakly lower semi-continuous.

(b) Let us show that Φ is coercive: Let v ∈ W^{1,p}(Ω) such that ∥v∥_{W^{1,p}(Ω)} ≠ 0, and we consider u = \frac{v}{∥v∥_{W^{1,p}(Ω)}} and s = ∥v∥_{W^{1,p}(Ω)}. So we have

Φ(v) = Φ(su)

= \frac{1}{p}s^{p}[( ∫_{Ω} |∇u|^{p} + ∫_{Ω} |u|^{p}) − λ ∫_{∂Ω} |u|^{p}] − s ∫_{∂Ω} hu

≥ \frac{1}{p}s^{p}(1 − \frac{λ}{λ_{1}} )∥u∥_{W^{1,p}(Ω)}^{p} − s∥h∥_{L^{q}(∂Ω)}∥u∥_{L^{p}(∂Ω)}

≥ \frac{1}{p}s^{p}(1 − \frac{λ}{λ_{1}} ) − \frac{s}{(λ_{1})^{\frac{1}{p}}} ∥h∥_{L^{q}(∂Ω)}

Thus as s → +∞ one has Φ(su) → +∞; i. e. Φ(v) → +∞ as ∥v∥_{W^{1,p}(Ω)} → +∞, so Φ is coercive. The function Φ is weakly lower semi-continuous coercive, so it admits a critical point u which is solution of (1.1). (ii) One shows that u > 0 on \overline{Ω}. According to the regularity and the theorem 2.1, one has u ∈ C^{1}(\overline{Ω}) and u ⩾ 0 on Ω. And with the maximum principle by Vazquez [5], one gets u > 0 on Ω. Moreover ∂Ω is C^{2,β}, so u > 0 on \overline{Ω}, because if there exists x_{0} ∈ ∂Ω such that u(x_{0}) = 0, then \frac{∂u}{∂n}(x_{0}) > 0; i.e. \frac{∂u}{∂ν}(x_{0}) < 0. However, |∇u|^{p−2} \frac{∂u}{∂ν}(x_{0}) = λ|u|^{p−2}u(x_{0}) + h(x_{0}) = h(x_{0}), then |∇u|^{p−2} \frac{∂u}{∂ν}(x_{0}) = h(x_{0}) < 0. That is impossible since h ⩾ 0 in ∂Ω. So u > 0 on \overline{Ω}. (iii) Uniqueness of the solution: By contradiction one supposes that there exist two nontrivial solutions of (1.1) u and v with h ≩ 0 in ∂Ω. According to what precedes one has u > 0 and v > 0 on \overline{Ω}. And Applying lemma 3.1 at u and v one gets

λ ∫_{∂Ω} v^{p} + ∫_{∂Ω} h(\frac{v^{p}}{u^{p−1}} ) ≤ ∫_{Ω} |∇v|^{p} + ∫_{Ω} |v|^{p} = λ ∫_{∂Ω} v^{p} + ∫_{∂Ω} hv.  (3.5)

Then

λ ∫_{∂Ω} h(\frac{v^{p}}{u^{p−1}} ) − ∫_{∂Ω} hv ≤ 0, i-e ∫_{∂Ω} hv(\frac{v^{p−1} − u^{p−1}}{u^{p−1}} ) ≤ 0.  (3.6)

By exchanging the roles of u and v one has

∫_{∂Ω} hu(\frac{u^{p−1} − v^{p−1}}{u^{p−1}} ) ≤ 0.  (3.7)

Inequalities (3.6) and (3.7) imply

∫_{∂Ω} h[u((\frac{u^{p−1} − v^{p−1}}{u^{p−1}} ) + v((\frac{v^{p−1} − u^{p−1}}{u^{p−1}} )] ≤ 0;

i.e., ∫_{∂Ω} h[(u^{p−1} − v^{p−1})(\frac{v}{u^{p−1}} − \frac{u}{v^{p−1}} )] ≤ 0. Consequently

∫_{∂Ω} h\frac{(v^{p−1} − u^{p−1})(v^{p} − u^{p})}{u^{p−1}v^{p−1}} ≤ 0.

However,

\frac{(v^{p−1} − u^{p−1})(v^{p} − u^{p})}{u^{p−1}v^{p−1}} ≥ 0

and h ≥ 0 in ∂Ω. Then (3.6) and (3.7) are equalities, and one has equality in (3.5); that is true if and only if u is a multiple of v. Consequently there exists c ≥ 0 such that u = cv on Ω and since u > 0 andv > 0 we have c > 0. Replacing u by cv in equation (1.1), one obtains

Δ_{p}(cv) = |cv|^{p−2}cv in Ω,

|∇cv|^{p−2} \frac{∂(cv)}{∂ν} = λ|cv|^{p−2}cv + h on ∂Ω.

This is equivalent to

Δ_{p}(v) = |v|^{p−2}v in Ω,

|∇v|^{p−2} \frac{∂(v)}{∂ν} = λ|v|^{p−2}v + \frac{h}{c^{p−1}} on ∂Ω.

But v is a solution of (1.1) then \frac{h}{c^{p−1}} = h, and sinceh ≠ 0 one has c = 1 and u = v. So the uniqueness of the solution of (1.1) in the case λ < λ_{1} and h ≩ 0 is proved.

The case λ = λ_{1}: Suppose that there exists u ∈ W^{1,p}(Ω) a nontrivial solution of (1.1). One has u ∈ C^{1}(\overline{Ω}) and u > 0 on \overline{Ω}. And let φ_{1} be a positive eigenfunction associated to λ_{1}, then φ_{1} ∈ C^{1}(\overline{Ω}) and φ_{1} > 0 on \overline{Ω}. Applying lemma 3.1 at u and φ_{1} one gets

λ_{1} ∫_{∂Ω} φ_{1}^{p} + ∫_{∂Ω} h(\frac{φ_{1}^{p}}{u^{p−1}} ) ≤ ∫_{Ω} |∇φ_{1}|^{p} + ∫_{Ω} |φ_{1}|^{p}.

However ∫_{Ω} |∇φ_{1}|^{p} + ∫_{Ω} |φ_{1}|^{p} = λ_{1} ∫_{∂Ω} φ_{1}^{p}, then ∫_{∂Ω} h\frac{φ_{1}^{p}}{u^{p−1}} = 0 and h\frac{φ_{1}^{p}}{u^{p−1}} = 0σ a.e. in ∂Ω. But φ_{1} > 0 andu > 0 on \overline{Ω} and h ≠ 0 in ∂Ω a contradiction. Then if λ = λ_{1} and h ≩ 0 in ∂Ω, problem (1.1) has no solution in W^{1,p}(Ω).

The case of λ > λ_{1}: By contradiction we suppose that there is u ≩ 0 a solution of (1.1). Then u ∈ C^{1}(\overline{Ω}) and u > 0 on \overline{Ω}. Lemma 3.1 implies

λ ∫_{∂Ω} φ^{p} + ∫_{∂Ω} h\frac{φ^{p}}{u^{p−1}} ≤ ∫_{Ω} |∇φ|^{p} + ∫_{Ω} |∇φ|^{p}

This implies that λ ∫_{∂Ω} φ^{p} ≤ ∫_{Ω} |∇φ|^{p} + ∫_{Ω} |∇φ|^{p} for all φ ∈ C^{1}(\overline{Ω}) with φ ≥ 0. Also by density λ ∫_{∂Ω} φ^{p} ≤ ∫_{Ω} |∇φ|^{p} + ∫_{Ω} |∇φ|^{p}|^{p} for all φ ∈ W^{1,p}(Ω). Then

λ ≤ \frac{∫_{Ω} |∇φ|^{p} + ∫_{Ω} |∇φ|^{p}|^{p}}{∫_{∂Ω} φ^{p}} ∀φ ∈ W^{1,p}(Ω) with ∫_{∂Ω} φ^{p} ≠ 0

Consequently λ ≤ λ_{1}, which is a contradiction, and the result follows. □

4. The anti-maximum principle

In this part we study the anti-maximum principle for the problem (1.1).

Theorem 4.1. Given h ∈ L^{∞}(∂Ω) with h ≩ 0, there exists δ = δ(h) > 0 such that if λ_{1} < λ < λ_{1} + δ, and u is a solution of (1.1), then

u < 0 in \overline{Ω} and \frac{∂u}{∂v} < 0 on ∂Ω  (4.1)

It will thus be said that the antimaximum principle (AMP) holds on the right of λ_{1}.

Proof. By contradiction we suppose that for all k ∈ N^{∗} there exists (μ_{k})_{k} ⊂ ℝ) such that

λ_{1} < μ_{k} < λ_{1} + \frac{1}{k}

and there exists (u_{k})_{k} ⊂ W^{1,p}(Ω) so that

\begin{matrix}Δ_{p}u_{k} = |u_{k}|^{p−2}u_{k} + h in Ω, \\ |∇u_{k}|^{p−2} \frac{∂u_{k}}{∂ν} = μ_{k}|u_{k}|^{p−2}u_{k} + h on ∂Ω \end{matrix}  (4.2)

and u_{k} does not check (4.1). One has according to the regularity result u_{k} ∈ C^{1,α}(\overline{Ω}) for some α ∈]0, 1[. And ∥u_{k}∥_{∞} does not remain bounded. Indeed, if there exists M > 0 such that ∥u_{k}∥_{∞} ≤ M for all k, then ∥Δ_{p}u_{k}∥ = ∥|u_{k}|^{p−1}∥_{∞} ≤ M^{p−1} = M′, and we get also ∥u_{k}∥_{C^{1,α}(\overline{Ω})} ≤ K_{1} independently of k.

So since (u_{k}) ⊂ C^{1,α}(\overline{Ω})↪C^{1}(\overline{Ω}), compact, then for a subsequence u_{k} → u in C^{1}(\overline{Ω}). Moreover, if

Δ_{p}u_{k} = |u_{k}|^{p−2}u_{k} + h inΩ,

|∇u_{k}|^{p−2} \frac{∂u_{k}}{∂ν} = μ_{k}|u_{k}|^{p−2}u_{k} + h on ∂Ω,

then

Δ_{p}u = |u|^{p−2}u in Ω,

|∇u|^{p−2} \frac{∂u}{∂ν} = λ_{1}|u|^{p−2}u + h on ∂Ω.

This contradicts the result of theorem 3.1 which ensures the nonexistence of solution for (1.1) when λ = λ_{1} and h ≩ 0. Consequently ∥u_{k}∥_{∞} → +∞.

Let us consider v_{k} = \frac{u_{k}}{∥u_{k}∥_{∞}}, so ∥v_{k}∥_{∞} = 1 and as previously for a subsequence v_{k} → v in C^{1}(\overline{Ω}) with ∥v∥_{∞} = 1. Also v_{k} solves

Δ_{p}v_{k} = \frac{Δ_{p}u_{k}}{∥u_{k}∥_{∞}^{p−1}} = |v_{k}|^{p−2}v_{k} in Ω,

|∇v_{k}|^{p−2} \frac{∂v_{k}}{∂ν} = \frac{1}{∥u_{k}∥_{∞}^{p−1}} |∇u_{k}|^{p−2} \frac{∂u_{k}}{∂ν} = μ_{k}|v_{k}|^{p−2}v_{k} + \frac{h}{∥u_{k}∥_{∞}^{p−1}} on ∂Ω.

(4.3)

Then

Δ_{p}v = |v|^{p−2}v in Ω,

|∇v|^{p−2} \frac{∂v}{∂ν} = λ_{1}|v|^{p−2}v on ∂Ω.

So v is a eigenfunction associated to λ_{1} and v ≠ 0. Applying again the maximum principle one has (v > 0 in \overline{Ω}) or (v < 0 in \overline{Ω}).

(i) If v > 0 in \overline{Ω}, then for k sufficiently large we have v_{k} > 0 in\overline{Ω}, but v_{k} is a solution of (1.1) with λ = μ_{k} > λ_{1} and h′ = \frac{h}{∥u_{k}∥_{∞}^{p−1}} ≩ 0 on ∂Ω which leads to a contradiction with the theorem 3.1. (ii) If v < 0 in \overline{Ω}, one has |∇v|^{p−2} \frac{∂v}{∂ν} = λ_{1}|v|^{p−2}v < 0 in ∂Ω, then \frac{∂v}{∂ν} < 0 in ∂Ω. So for k sufficiently large, (v_{k} < 0 in \overline{Ω} and \frac{∂v_{k}}{∂ν} < 0 in ∂Ω). This means that u_{k} checks (4.1). Contradiction with the assumption. This completes the proof of the anti-maximum principle. □

Now we study the uniformity of this principle. One will show that the AMP is nonuniform on the right of λ_{1} when p ≤ N, and uniform when p > N. In the latter case one will characterize the interval of uniformity. Moreover one shows that the AMP still holds on the right of this interval but not uniformly. For this, one shows accesses the following result.

Lemma 4.1. If p ≤ N then λ_{1} = \overline{λ_{1}}. If p > N then λ_{1} < \overline{λ_{1}}, and the interval ]λ_{1}, \overline{λ_{1}}] does not contain any eigenvalue where \overline{λ_{1}} = inf_{u∈P} (∫_{Ω} |∇u|^{p} + ∫_{Ω} |u|^{p}) and

P = \{u ∈ W^{1,p}(Ω) : ∫_{∂Ω} |u|^{p} = 1 and u vanishes in some ball in \overline{Ω}\}.

Proof. It is clear that λ_{1} ≤ \overline{λ_{1}}.

(i) If p < N: As in Godoy, Gossez and Paczka [2], one defines a sequence (v_{k})_{k} as follows: For k ∈ ℕ^{∗},

\begin{cases}1 & if |x| ≥ 1/k \\ 2k|x| − 1 & if 1/(2k) < |x| < 1/k, \\ 0 & if |x| ≤ 1/2k \end{cases}

Note that v_{k}converges to the constant function 1 as k → +∞ in W_{loc}^{1,p}(R^{N}). Given x_{0} ∈ Ω, and u a eigenfunction associated to λ_{1}; i.e., λ_{1} = \frac{∫_{Ω} |∇u|^{p}+∫_{Ω} |u|^{p}}{∫_{∂Ω} |u|^{p}}, then the

sequence (w_{k}) defined by w_{k}(x) = u(x)v_{k}(x − x_{0}) vanishes in the ball B(x_{0}, \frac{1}{2k}) and

\overline{λ_{1}} ≤ \frac{∫_{Ω} |∇w_{k}|^{p} + ∫_{Ω} |w_{k}|^{p}}{∫_{∂Ω} |w_{k}|^{p}} → \frac{∫_{Ω} |∇u|^{p} + ∫_{Ω} |u|^{p}}{∫_{∂Ω} |u|^{p}} as k → +∞

So \overline{λ_{1}} ≤ λ_{1}, then \overline{λ_{1}} = λ_{1}. (ii) If p = N: One defines a sequence (v_{k})_{k} as follows: For k ∈ ℕ^{∗},

v_{k} = \begin{cases}1 − \frac{1}{2k} & if |x| ≥ \frac{1}{k} \\ |x|^{δ_{k}} − \frac{1}{k} & if (\frac{1}{k})^{1/δ_{k}} < |x| < \frac{1}{k} \\ 0 & if |x| ≤ (\frac{1}{k})^{1/δ_{k}}, \end{cases}

where δ_{k} satisfies (\frac{1}{k})^{δ_{k}} = 1 − \frac{1}{k}. (δ_{k} = 1 − \frac{ln(k+1)}{ln(k)} → 0 as k → +∞). And (w_{k}) is as previously, and we show that

\frac{∫_{Ω} |∇w_{k}|^{p} + ∫_{Ω} |w_{k}|^{p}}{∫_{∂Ω} |w_{k}|^{p}} → \frac{∫_{Ω} |∇u|^{p} + ∫_{Ω} |u|^{p}}{∫_{∂Ω} |u|^{p}} as k → +∞

Indeed

∫_{Ω} |w_{k}|^{p} = ∫_{|x−x_{0}|≥ \frac{1}{k}} |1 − \frac{1}{2k} |^{p}|u(x)|^{p} + ∫_{( \frac{1}{k})^{\frac{1}{δ_{k}}} ≤|x−x_{0}|≤ \frac{1}{k}} ∥x − x_{0}|^{δ_{k}} − \frac{1}{k} |^{p}|u(x)|^{p},

and

∫_{( \frac{1}{k})^{\frac{1}{δ_{k}}} ≤|x−x_{0}|≤ \frac{1}{k}} ∥x − x_{0}|^{δ_{k}} − \frac{1}{k} |^{p}|u(x)|^{p} ≤ (1 − \frac{2}{k})^{p} ∫_{( \frac{1}{k})^{\frac{1}{δ_{k}}} ≤|x−x_{0}|≤ \frac{1}{k}} |u(x)|^{p} → 0,

consequently

\overline{λ_{1}} ≤ \frac{∫_{Ω} |∇w_{k}|^{p} + ∫_{Ω} |w_{k}|^{p}}{∫_{∂Ω} |w_{k}|^{p}} → \frac{∫_{Ω} |∇u|^{p} + ∫_{Ω} |u|^{p}}{∫_{∂Ω} |u|^{p}} = λ_{1},

then \overline{λ_{1}} = λ_{1}. (iii) When p > N, we have

\overline{λ_{1}} = inf_{u∈G} ( ∫_{Ω} |∇u|^{p} + ∫_{Ω} |u|^{p})

where G = \{u ∈ W^{1,p}(Ω) : ∫_{∂Ω} |u|^{p} = 1 and ∃x_{0} ∈ \overline{Ω} : u(x_{0}) = 0\}. Since W^{1,p}(Ω) ↪_{cpt} C(\overline{Ω}), the minimum is achieved in a certain function \overline{u}, that one can suppose positive on \overline{Ω}. And there exists x_{0} ∈ \overline{Ω} such that \overline{u}(x_{0}) = 0. Let us show that it vanishes only once on \overline{Ω}.

One assumes, by contradiction, that there exists x_{1} ≠ x_{0} ∈ \overline{Ω} with \overline{u}(x_{1}) = 0. Setting F = {u ∈ W^{1,p}(Ω) : ∫_{∂Ω} |u|^{p} = 1 and u(x_{0}) = 0} and μ = inf_{u∈F} (∫_{Ω} |∇u|^{p}+ ∫_{Ω} |u|^{p}), one gets \overline{λ_{1}} ≤ μ , \overline{u} ∈ F and \overline{λ_{1}} = ∫_{Ω} |∇\overline{u}|^{p} + ∫_{Ω} |\overline{u}|^{p} then \overline{λ_{1}} = μ.

Applying the standard theorem on Lagrange multipliers, we ensure the existence of (α_{1}, α_{2}) ∈ R^{2} such that for all w ∈ W^{1,p}(Ω) with w(x_{0}) = 0 one has

Φ′(\overline{u}).w = α_{1}Ψ′_{1}(\overline{u}).w + α_{2}Ψ′_{2}(\overline{u}).w = α_{1}Ψ′_{1}(\overline{u}).w

where Ψ_{1}(u) = ∫_{∂Ω} |u|^{p} −1, Ψ_{2}(u) = u(x_{1}) and Φ(u) = ∫_{Ω} |∇u|^{p} +∫_{Ω} |u|^{p}. Then for all w ∈ W^{1,p}(Ω) : w(x_{0}) = 0 one has ∫_{Ω} |∇\overline{u}|^{p−2}∇\overline{u}∇w+∫_{Ω} \overline{u}^{p−1}w = α_{1} ∫_{∂Ω} \overline{u}^{p−1}w. Taking w = \overline{u} one gets α_{1} = \overline{λ_{1}} and

∫_{Ω} |∇\overline{u}|^{p−2}∇\overline{u}∇w + ∫_{Ω} \overline{u}^{p−1}w = \overline{λ_{1}} ∫_{∂Ω} \overline{u}^{p−1}w  (4.4)

for all w ∈ W^{1,p}(Ω) with w(x_{0}) = 0. By the same process, one has

∫_{Ω} |∇\overline{u}|^{p−2}∇\overline{u}∇w + ∫_{Ω} \overline{u}^{p−1}w = \overline{λ_{1}} ∫_{∂Ω} \overline{u}^{p−1}w

for all w ∈ W^{1,p}(Ω) with w(x_{1}) = 0. Knowing that for all w ∈ W^{1,p}(Ω), there exists (w_{0}, w_{1}) ∈ (W^{1,p}(Ω))^{2} such that w = w_{0} + w_{1}, w_{0}(x_{0}) = 0 and w_{1}(x_{1}) = 0, one arrives at the result: For all w ∈ W^{1,p}(Ω),

∫_{Ω} |∇\overline{u}|^{p−2}∇\overline{u}∇w + ∫_{Ω} \overline{u}^{p−1}w = \overline{λ_{1}} ∫_{∂Ω} \overline{u}^{p−1}w  (4.5)

Then \overline{λ_{1}} is a eigenvalue and \overline{u} is a positive eigenfunction associated. However by [4, Lemma 2.4], \overline{λ_{1}} = λ_{1} and \overline{u} > 0 on \overline{Ω} by the maximum principle. But this contradicted the assumption that \overline{u} vanishes in \overline{Ω}. Consequently \overline{u} vanishes only once on \overline{Ω}, and

\overline{λ_{1}} = ∫_{Ω} |∇\overline{u}|^{p} + ∫_{Ω} |\overline{u}|^{p} = inf_{u∈F} ( ∫_{Ω} |∇u|^{p} + ∫_{Ω} |u|^{p})

with \overline{u} > 0 in \overline{Ω}\\{x_{0}\} and \overline{u}(x_{0}) = 0.

Now one considers u_{ϵ}(x) = max(\overline{u}(x), ϵ) and B_{ϵ} = \{x ∈ \overline{Ω} : \overline{u}(x) < ϵ\} where ϵ > 0. One has u_{ϵ} → \overline{u}, as ϵ → 0, in W^{1,p}(Ω), and

∫_{∂Ω} |u_{ϵ}|^{p}∂σ = ∫_{∂Ω\B_{ϵ}} |\overline{u}|^{p}∂σ + ϵ^{p} ∫_{∂Ω∩B_{ϵ}} ∂σ →_{ϵ→0} ∫_{∂Ω} |\overline{u}|^{p}∂σ = 1

Then ∫_{∂Ω} |u_{ϵ}|^{p}∂σ > 0 for ϵ enough small. One hopes to show that

λ_{1} ≤ \frac{∫_{Ω} |∇u_{ϵ}|^{p} + ∫_{Ω} |u_{ϵ}|^{p}}{∫_{∂Ω} |u_{ϵ}|^{p}} < \overline{λ_{1}} = ∫_{Ω} |∇\overline{u}|^{p} + ∫_{Ω} |\overline{u}|^{p}

for ϵ enough small. We notice that

∫_{Ω} |u_{ϵ}|^{p} = ∫_{Ω} \overline{u}^{p} − ∫_{Ω∩B_{ϵ}} \overline{u}^{p} + ϵ^{p} meas(Ω ∩ B_{ϵ}),

∫_{∂Ω} |u_{ϵ}|^{p} = ∫_{∂Ω} \overline{u}^{p} − ∫_{B_{ϵ}∩∂Ω} \overline{u}^{p} + ϵ^{p} meas_{σ}(B_{ϵ} ∩ ∂Ω),

∫_{Ω} |∇u_{ϵ}|^{p} = ∫_{Ω} |∇\overline{u}|^{p} − ∫_{Ω∩B_{ϵ}} |∇\overline{u}|^{p}

and we conclude that

A_{ε} = \frac{∫_{Ω} |∇u_{ϵ}|^{p} + ∫_{Ω} |u_{ϵ}|^{p}}{∫_{∂Ω} |u_{ϵ}|^{p}} − ∫_{Ω} |∇\overline{u}|^{p} − ∫_{Ω} |\overline{u}|^{p}

= \frac{∫_{Ω} |∇u_{ϵ}|^{p} + ∫_{Ω} |u_{ϵ}|^{p} − ∫_{∂Ω} |u_{ϵ}|^{p}(∫_{Ω} |∇\overline{u}|^{p} + ∫_{Ω} \overline{u}^{p})}{∫_{∂Ω} |u_{ϵ}|^{p}}

= ( ∫_{Ω} |∇\overline{u}|^{p} + ∫_{Ω} |\overline{u}|^{p})C_{ϵ}

where

C_{ϵ} = \frac{( ∫_{B_{ϵ}∩∂Ω} \overline{u}^{p} − ϵ^{p} meas_{σ}(B_{ϵ} ∩ ∂Ω)) − ( ∫_{Ω∩B_{ϵ}} |∇\overline{u}|^{p} + ∫_{Ω∩B_{ϵ}} \overline{u}^{p} − ϵ^{p}mes(B_{ϵ}))}{1 − ∫_{B_{ϵ}∩∂Ω} |\overline{u}|^{p} + ϵ^{p} meas_{σ}(B_{ϵ} ∩ ∂Ω)}.

We will prove that A_{ε}/ε converges towards a non positive value. For that we recall that \overline{u} checks the property (4.4), and considering the sequence v_{ϵ} = min(\overline{u}, ϵ) we get

∫_{Ω∩B_{ϵ}} |∇\overline{u}|^{p} + ∫_{Ω∩B_{ϵ}} \overline{u}^{p} = \overline{λ_{1}} ∫_{∂Ω∩B_{ϵ}} \overline{u}^{p} + ϵ(P_{ε}),

where P_{ε} = \overline{λ_{1}} ∫_{∂Ω\B_{ϵ}} \overline{u}^{p−1} − ∫_{Ω\B_{ϵ}} \overline{u}^{p−1} → \overline{λ_{1}} ∫_{∂Ω} \overline{u}^{p−1} − ∫_{Ω} \overline{u}^{p−1} as ε → 0.

If \overline{λ_{1}} ∫_{∂Ω} \overline{u}^{p−1} = ∫_{Ω} \overline{u}^{p−1}, then (4.4) holds for w ≡ 1; consequently it holds also for all φ in W^{1,p}(Ω) because φ = ψ +φ(x_{0})w with ψ = φ−φ(x_{0}). So \overline{u} is a positive solution of the problem

Δ_{p}v = |v|^{p−2}v in Ω,

|∇v|^{p−2} \frac{∂v}{∂ν} = \overline{λ_{1}}|v|^{p−2}v on ∂Ω,

then \overline{u} > 0 on \overline{Ω}, absurdity since \overline{u}(x_{0}) = 0. So we deduce that \overline{λ_{1}} ∫_{∂Ω} \overline{u}^{p−1} ≠ ∫_{Ω} \overline{u}^{p−1}.

If \overline{λ_{1}} ∫_{∂Ω} \overline{u}^{p−1} < ∫_{Ω} \overline{u}^{p−1}, then since ∫_{∂Ω∩B_{ϵ}} \overline{u}^{p} = o(ϵ^{p}) one has

\frac{1}{ϵ}(∫_{Ω∩B_{ϵ}} |∇\overline{u}|^{p} + ∫_{Ω∩B_{ϵ}} \overline{u}^{p}) →_{ϵ→0} \overline{λ_{1}} ∫_{∂Ω} \overline{u}^{p−1} − ∫_{Ω} \overline{u}^{p−1} < 0.

This implies that for ϵ enough small ∫_{Ω∩B_{ϵ}} |∇\overline{u}|^{p} + ∫_{Ω∩B_{ϵ}} \overline{u}^{p} < 0, which is not true. Then \overline{λ_{1}} ∫_{∂Ω} \overline{u}^{p−1} > ∫_{Ω} \overline{u}^{p−1}. Moreover 1 − ∫_{B_{ϵ}∩∂Ω} \overline{u}^{p} + ϵ^{p} meas_{σ}(B_{ϵ} ∩ ∂Ω) → 1 as ϵ → 0 and ∫_{B_{ϵ}∩∂Ω} \overline{u}^{p} − ϵ^{p} meas_{σ}(B_{ϵ} ∩ ∂Ω) = o(ϵ^{p}), then

\frac{1}{ϵ}(\frac{∫_{Ω} |∇u_{ϵ}|^{p} + ∫_{Ω} |u_{ϵ}|^{p}}{∫_{∂Ω} |u_{ϵ}|^{p}} − ∫_{Ω} |∇\overline{u}|^{p} − ∫_{Ω} |\overline{u}|^{p}) →_{ϵ→0} \overline{λ_{1}} ∫_{∂Ω} \overline{u}^{p−1} − ∫_{Ω} \overline{u}^{p−1} < 0

So for ϵ enough small λ_{1} ≤ \frac{∫_{Ω} |∇u_{ϵ}|^{p}+∫_{Ω} |u_{ϵ}|^{p}}{∫_{∂Ω} |u_{ϵ}|^{p}} < ∫_{Ω} |∇\overline{u}|^{p} + ∫_{Ω} |\overline{u}|^{p} = \overline{λ_{1}}. Conse-

quently, λ_{1} < \overline{λ_{1}}. To complete the proof, one shows that there is no eigenvalue in ]λ_{1}, \overline{λ_{1}}]. Let us suppose by absurdity that there exists an eigenvalue μ in ]λ_{1}, \overline{λ_{1}}], with associated eigenfunction v. By [4, Lemma 2.4], v changes sign on ∂Ω, consequently it vanishes somewhere in \overline{Ω}. Then v ∈ G, so

\overline{λ_{1}} ∫_{∂Ω} |v|^{p} ≤ ∥∇v∥_{L^{p}(Ω)}^{p} + ∥v∥_{L^{p}(Ω)}^{p} = μ ∫_{∂Ω} |v|^{p}

Since ∫_{∂Ω} |v|^{p} ≠ 0, one concluded that \overline{λ_{1}} = μ . In this case v satisfies (4.5), and it vanishes only once on \overline{Ω}. Applying (4.5) to w = v^{−} one obtains

\overline{λ_{1}} ∫_{∂Ω} |v^{−}|^{p} = ∥∇v^{−}∥_{L^{p}(Ω)}^{p} + ∥v^{−}∥_{L^{p}(Ω)}^{p},

then v^{−} vanishes only once on \overline{Ω}. Contradiction since v changes sign on ∂Ω. □

Theorem 4.2. If p > N

(i) For all h ∈ L^{∞}(∂Ω), such that h ≩ 0 in \overline{Ω}, if λ ∈]λ_{1}, \overline{λ_{1}}], then any solution u of (1.1) satisfies u < 0 in \overline{Ω}.

(ii) \overline{λ_{1}} is the largest number such that the preceding implication holds.

(iii) Given h ∈ L^{∞}(∂Ω), h ≩ 0 in ∂Ω, there exists δ = δ(h) > 0 such that if \overline{λ_{1}} < λ < \overline{λ_{1}} + δ, and u is a solution of (1.1) then u < 0 on \overline{Ω}.

Theorem 4.3. The AMP is not uniform on the right of \overline{λ_{1}} for all 1 < p < ∞; i.e., there is no δ independent of h satisfying (i).

The proof of this result is a consequence of the following theorem.

Theorem 4.4. Given ϵ > 0, there exists h ∈ L^{∞}(∂Ω), such that h ≩ 0 in ∂Ω and if \overline{λ_{1}} + ϵ < λ then problem (1.1) does not admit any non-positive solution.

Proof of theorem 4.2 (i) Given h ∈ L^{∞}(∂Ω), such that h ≩ 0 in ∂Ω, and λ ∈]λ_{1}, \overline{λ_{1}}], then if u is a solution of (1.1), it does not satisfy u ≩ 0 on \overline{Ω} by the Theorem 3.1. Taking u^{−} ≠ 0 as testing function in (1.1), one gets

∫_{Ω} |∇u^{−}|^{p} + ∫_{Ω} |u^{−}|^{p} = λ ∫_{∂Ω} |u^{−}|^{p} − ∫_{∂Ω} hu^{−} ≤ λ ∫_{∂Ω} |u^{−}|^{p}.

If ∫_{∂Ω} |u^{−}|^{p} = 0, then ∫_{Ω} |∇u^{−}|^{p} + ∫_{Ω} |u^{−}|^{p} = 0, consequently u^{−} ≡ 0, this is not true, so ∫_{∂Ω} |u^{−}|^{p} > 0. One deduces that

\frac{∫_{Ω} |∇u^{−}|^{p} + ∫_{Ω} |u^{−}|^{p}}{∫_{∂Ω} |u^{−}|^{p}} ≤ λ.

If λ < \overline{λ_{1}} then

\frac{∫_{Ω} |∇u^{−}|^{p} + ∫_{Ω} |u^{−}|^{p}}{∫_{∂Ω} |u^{−}|^{p}} ≤ λ < \overline{λ_{1}} = inf_{u∈G}( ∫_{Ω} |∇u|^{p} + ∫_{Ω} |u|^{p}),

so u^{−} ∉ G i-e u^{−} does not vanish any where on \overline{Ω}. Consequently u < 0 on \overline{Ω}. If λ = \overline{λ_{1}} then \frac{∫_{Ω} |∇u^{−}|^{p}+∫_{Ω} |u^{−}|^{p}}{∫_{∂Ω} |u^{−}|^{p}} ≤ \overline{λ_{1}}. Two cases are distinguished: If u^{−} ∉ G then

u < 0 on \overline{Ω}. But if u^{−} ∈ G then \frac{∫_{Ω} |∇u^{−}|^{p}+∫_{Ω} |u^{−}|^{p}}{∫_{∂Ω} |u^{−}|^{p}} = \overline{λ_{1}}, and u^{−} vanishes only once

on \overline{Ω}. In this case one has the equality

∫_{∂Ω} hu^{−} = λ ∫_{∂Ω} |u^{−}|^{p} − ∫_{Ω} |∇u^{−}|^{p} + ∫_{Ω} |u^{−}|^{p} = 0,

so u^{−} vanishes on the set where h ≠ 0 which is of positive measure. Absurdity. So u^{−} ∉ G and as previously u < 0 on \overline{Ω}. (ii) This part is a consequence of Theorem 4.4.

(iii) This result is similar as Theorem 4.1. As in his proof we suppose, by contradiction, that for all k ∈ N^{∗} there exists (μ_{k})_{k} ⊂ ℝ) such that \overline{λ_{1}} < μ_{k} < \overline{λ_{1}} + \frac{1}{k}, and there exists (u_{k})_{k} ⊂ W^{1,p}(Ω) holding

Δ_{p}u_{k} = |u_{k}|^{p−2}u_{k} + h in Ω,

|∇u_{k}|^{p−2} \frac{∂u_{k}}{∂ν} = μ_{k}|u_{k}|^{p−2}u_{k} + h in ∂Ω,

where u_{k} is not non-positive on \overline{Ω}, and u_{k} ∈ C^{1,α}(\overline{Ω}) for some α ∈]0, 1[. One distinguishes two cases, either ∥u_{k}∥_{∞} remains bounded, or ∥u_{k}∥_{∞} → +∞. In the first case, one has for a subsequence u_{k} → u in C^{1}(\overline{Ω}), and u solves

Δ_{p}u = |u|^{p−2}u in Ω,

|∇u|^{p−2} \frac{∂u}{∂ν} = \overline{λ_{1}}|u|^{p−2}u + h on ∂Ω.

By part (i) one gets u < 0 on \overline{Ω}. Contradiction. In the second case one considers v_{k} = \frac{u_{k}}{∥u_{k}∥_{∞}}, so ∥v_{k}∥_{∞} = 1 and as previously for a subsequence v_{k} → v in C^{1}(\overline{Ω}) with ∥v∥_{∞} = 1, and v solves

Δ_{p}v = |v|^{p−2}v in Ω,

|∇v|^{p−2} \frac{∂v}{∂ν} = \overline{λ_{1}}|v|^{p−2}v on ∂Ω.

But this yields a contradiction since by Lemma 4.1, \overline{λ_{1}} is not an eigenvalue.

Proof of theorem 4.4. Assume by contradiction that there exists ε > 0 such that for any h ≩ 0 there exists λ(h) with λ(h) ≥ \overline{λ_{1}}+ε such that (1.1) has a solution u(h) < 0 in \overline{Ω}. We consider φ ∈ C^{1}(\overline{Ω}) satisfying ∫_{∂Ω} |φ|^{p} ≠ 0 and φ vanishes in some ball in \overline{Ω}, and we choose h ∈ L^{∞}(∂Ω), such that h ≩ 0 in ∂Ω and supp φ∩supph ∩∂Ω = ∅. Applying Lemma 3.1 to |φ| and v = −u(h) > 0 in \overline{Ω}, we obtain

λ(h) ∫_{∂Ω} |φ|^{p} ≤ ∫_{Ω} |∇φ|^{p} + ∫_{Ω} |φ|^{p},

Then

\overline{λ_{1}} + ε ≤ λ(h) ≤ \frac{∫_{Ω} |∇φ|^{p} + ∫_{Ω} |φ|^{p}}{∫_{∂Ω} |φ|^{p}} ,

for all φ ∈ W^{1,p}(Ω) satisfying ∫_{∂Ω} |φ|^{p} ≠ 0 and φ vanishes in some ball in \overline{Ω}, which contradicts the definition of \overline{λ_{1}}. □

References

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[2] T. Godoy, J. P. Gossez, S. Paczka; The anti-maximum principle for elliptic problems with weight, Electronic Journal of Differential Equations, Vol. 1999 (1999), No. 22, pp 1-15.

[3] T. Godoy, J. P. Gossez, S.Paczka; The anti-maximum principle for the p-Laplacien with indefinite weight, Anal, TMA, Vol 51, 2002, pp 449-467.

[4] S. Martinez, J.D.Rossi; Isolation and simplicity for the first eigenvalue of the p-Laplacian with nonlinear boundary conditions, Abstr. Appl. Vol 7 (5), 2002, pp 287-293.

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Aomar Anane Département de Mathématiques et Informatique, Faculté des Sciences, Université Mohammed 1er, Oujda, Maroc

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

Omar Chakrone Département de Mathématiques et Informatique, Faculté des Sciences, Université Mohammed 1er, Oujda, Maroc

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

Najat Moradi Département de Mathématiques et Informatique, Faculté des Sciences, Université Mohammed 1er, Oujda, Maroc

E-mail address: najat moradi@yahoo.fr