2002-Fez conference on Partial Differential Equations, Electronic Journal of Differential Equations, Conference 09, 2002, pp 161–170. http://ejde.math.swt.edu or http://ejde.math.unt.edu ftp ejde.math.swt.edu (login: ftp)

On the spectrum of the p-biharmonic operator ^{∗}

Abdelouahed El Khalil, Siham Kellati & Abdelfattah Touzani

Abstract This work is devoted to the study of the spectrum for p-biharmonic

operator with an indefinite weight in a bounded domain.

1 Introduction

Let Ω be a bounded domain in ℝ^{N}, N ≥ 1, not necessary regular; 1 < p < ∞ and ρ ∈ L^{r}(Ω), ρ ≠ 0, an unbounded weight function which can change its sign, with r = r(N, p) satisfying the conditions

r \begin{cases}> \frac{N}{2p} & for \frac{N}{p} ≥ 2 \\ = 1 & for \frac{N}{p} < 2. \end{cases}

We assume that |Ω_{ρ}^{+}| ≠ 0, where Ω_{ρ}^{+} = {x ∈ Ω; ρ(x) > 0} and λ ∈ ℝ. We consider the eigenvalue problem

\begin{matrix}Δ_{p}^{2}u = λρ(x)|u|^{p−2}u in Ω \\ u ∈ W_{0}^{2,p}(Ω). \end{matrix}  (1.1)

Here Δ_{p}^{2} := Δ(|Δu|^{p−2}Δu), the operator of fourth order called the p-biharmonic operator. For p = 2, the linear operator Δ_{2}^{2} = Δ^{2} = Δ.Δ is the iterated Laplacian that multiplied with positive constant appears often in Navier-Stokes equations as being a viscosity coefficient. Its reciprocal operator denoted (Δ^{2})^{−1} is the celebrated Green’s operator [5].

It is important to indicate that here we don’t suppose any boundary conditions on the high order partial derivatives \frac{∂^{2}u}{∂x_{i}∂x_{j}} on the boundary set ∂Ω of the domain Ω. The particular case ρ ≡ 1 and u = Δu = 0 on ∂Ω was considered recently by Drábek and Ôtani [2]. There the authors proved the existence, the simplicity, and the isolation of the first eigenvalue of (1.1) by using a transformation of a problem to a known Poisson’s problem, and using the well-known advanced regularity of Agmon-Douglis-Niremberg [3]. Note that this transformation processus is not applicable to our situation because the quantity Δu does not necessary vanished on ∂Ω and the eventual regularity is not required in any bounded domain.

∗Mathematics Subject Classifications: 35P30, 34C23.

Key words: p-biharmonic operator, Duality mapping, Palais-Smale condition,

unbounded weight.

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

Published December 28, 2002.

The main objective of this work is to show that problem (1.1) has at least one non-decreasing sequence of positive eigenvalues (λ_{k})_{k≥1}, by using the Ljusternichschnirelmann theory on C^{1} manifolds, see e.g. [6]. Our approach is based only on some properties of the considered operator. So that we give a direct characterization of λ_{k} involving a minimax argument over sets of genus greater than k.

We set

λ_{1} = inf {∥Δv∥_{p}^{p}, v ∈ W_{0}^{2,p}(Ω); ∫_{Ω} ρ(x)|v|^{p}dx = 1},

where ∥.∥_{p} denotes the L^{p}(Ω)-norm. It is not difficult to show that ∥Δu∥_{p} defines a norm in W_{0}^{2,p}(Ω) and W_{0}^{2,p}(Ω) equipped with this norm is a uniformly convex Banach space for 1 < p < +∞. The norm ∥Δ.∥_{p} is uniformly equivalent on W_{0}^{2,p}(Ω) to the usual norm of W_{0}^{2,p}(Ω) [3].

This paper is organized as follows: In section 2, we establish some definitions and show certain basic lemmas. In section 3, we use a variational technique to prove the existence of a sequence of the positive eigenvalues of p-biharmonic operator with any unbounded weight.

2 Preliminaries

Throughout this paper, all solutions are weak, i.e, u ∈ W_{0}^{2,p}(Ω) is a solution of (1.1), if for all φ ∈ C_{0}^{∞}(Ω),

∫_{Ω} |Δu|^{p−2}ΔuΔφ dx = λ ∫_{Ω} ρ(x)|u|^{p−2}uφ dx.

If u ∈ W_{0}^{2,p}(Ω)\{0}, then u shall be called an eigenfunction of the p-biharmonic operator (or of (1.1)) associated to the eigenvalue λ. The following proposition states some fundamental properties of the p-biharmonic operator.

Proposition 2.1 For any bounded domain Ω and 1 < p < +∞, Δ_{p}^{2} satisfies the following:

(i) Δ_{p}^{2} is an hemicontinuous operator from W_{0}^{2,p}(Ω) into W^{−2,p′}(Ω).

(ii) Δ_{p}^{2} is a bounded monotonous, and coercive operator.

(iii) Δ_{p}^{2} : W_{0}^{2,p}(Ω) → W^{−2,p′}(Ω) is a bicontinuous operator. Here p′ = \frac{p}{p−1}.

Proof (i) Define on W_{0}^{2,p}(Ω) the potential functional

A(u) = \frac{1}{p}∥Δu∥_{p}^{p}.

This functional is convex and of class C^{1} on W_{0}^{2,p}(Ω). Further its derivative operator is A′ = Δ_{p}^{2}. So this yields the hemicontinuity. (ii) By a simple calculation we can show that ∥Δ_{p}^{2}u∥_{∗} = ∥Δu∥_{p}^{p−1}, where ∥.∥_{∗} is the dual norm associated to ∥Δ.∥_{p}. This implies that Δ_{p}^{2} is bounded and is a monotonous operator. The continuity and coercivity are obvious . (iii) The fact that, for any u, v ∈ W_{0}^{2,p}(Ω), ∥Δu∥_{p} = ∥Δv∥_{p} if Δ_{p}^{2}u = Δ_{p}^{2}(v) and (W_{0}^{2,p}(Ω), ∥Δ.∥_{p}) is a uniformly convex space completes the proof.

Definition Let X be a real reflexive Banach space and let X^{∗} stand for its dual with respect to the pairing ⟨., .⟩. T a mapping acting from X into X^{∗}. T is said to belong to the class (S^{+}), if for any sequence {u_{n}} in X with u_{n} converges weakly to u ∈ X and lim sup_{n→+∞}⟨T u_{n}, u_{n} − u⟩ ≤ 0. It follows that u_{n} converges strongly to u in X. We write T ∈ (S^{+}).

3 Main results

We will use Ljusternick-Schnirelmann theory on C^{1}-manifolds [6]. Consider the following two functionals defined on W_{0}^{2,p}(Ω):

A(u) = \frac{1}{p}∥Δu∥_{p}^{p}, B(u) = \frac{1}{p} ∫_{Ω} ρ(x)|u|^{p}dx.

We set M = {u ∈ W_{0}^{2,p}(Ω); pB(u) = 1}.

Lemma 3.1 (i) A and B are even, and of class C^{1} on W_{0}^{2,p}(Ω). (ii) M is a closed C^{1}-manifold.

Proof (i) It is clear that B is of class C^{1} on W_{0}^{2,p}(Ω). M = B^{−1}\{\frac{1}{p}\} so B is closed. Its derivative operator B′ satisfies B′(u) ≠ 0 ∀u ∈ M (i.e., B′(u) is onto ∀u ∈ M), so B is a submersion, then M is a C^{1}-manifold. □

Remark 3.2 Observe that J : W_{0}^{2,p}(Ω) → W^{−2,p′}(Ω),

J(u) = \begin{cases}∥Δu∥_{p}^{2−p}Δ_{p}^{2}u & if u ≠ 0 \\ 0 & if u = 0 \end{cases}

is the duality mapping of (W_{0}^{2,p}(Ω), ∥Δ.∥_{p}).

The following lemma is the key to show existence.

Lemma 3.3 (i) B′ : W_{0}^{2,p}(Ω) → W^{−2,p′}(Ω) is completely continuous. (ii) The functional A satisfies the Palais-Smale condition on M, i.e., for {u_{n}} ⊂ M, if A(u_{n}) is bounded and

ϵ_{n} := A′(u_{n}) − g_{n}B′(u_{n}) → 0 as n → +∞,  (3.1)

where g_{n} = ⟨A′(u_{n}), u_{n}⟩/⟨B′(u_{n}), u_{n}⟩. Then {u_{n}}_{n≥1} has a convergent subsequence in W_{0}^{2,p}(Ω).

Proof (i) Step 1: Definition of B′. First case: \frac{N}{p} > 2 and r > \frac{N}{2p}. Let u, v ∈ W_{0}^{2,p}(Ω). By Hölder’s inequality, we have

| ∫_{Ω} ρ(x)|u(x)|^{p−2}u(x)v(x)dx| ≤ ∥ρ∥_{r}∥u∥_{s}^{p−1}∥v∥_{p_{2}}

where \frac{1}{p_{2}} = \frac{1}{p} − \frac{2}{N} and s is given by

\frac{p − 1}{s} + \frac{1}{p_{2}} + \frac{1}{r} = 1.  (3.2)

Therefore,

\frac{p − 1}{s} = 1 − \frac{1}{r} − \frac{1}{p_{2}} > 1 − \frac{2p}{N} − \frac{1}{p_{2}} = \frac{p − 1}{p_{2}}.

Then it suffices to take

max(1, p − 1) < s < p_{2}  (3.3)

so that B′ is well defined. Second case: \frac{N}{p} = 2 and r > \frac{N}{2p}. In this case W_{0}^{2,p}(Ω) ↪ L^{q}(Ω), for any q ∈ [p, +∞[. There is q ≥ p such that \frac{1}{q} + \frac{1}{r} + \frac{p−1}{p} = \frac{1}{q} + \frac{1}{r} + \frac{1}{p′} = 1. We obtain that

\frac{1}{q} = \frac{1}{p} − \frac{1}{r} ≤ \frac{1}{p}.  (3.4)

By Hölder’s inequality, we arrive at

| ∫_{Ω} ρ(x)|u(x)|^{p−2}u(x)v(x)dx| ≤ ∥ρ∥_{r}∥u∥_{p}^{p−1}∥v∥_{q},

for any u, v ∈ W_{0}^{2,p}(Ω). Then B′ is also well defined. Third case: \frac{N}{p} < 2 and r = 1. In this case W_{0}^{2,p}(Ω) ↪ C(\overline{Ω}) ∩ L^{∞}(Ω). Then for any u, v ∈ W_{0}^{2,p}(Ω), we have

| ∫_{Ω} ρ(x)|u(x)|^{p−2}u(x)v(x)dx| < ∞,

with ρ ∈ L^{1}(Ω), and B′ is well defined.

Step 2. B′ is completely continuous. Let (u_{n}) ⊂ W_{0}^{2,p}(Ω) be a sequence such that u_{n} → u weakly in W_{0}^{2,p}(Ω). We have to show that B′(u_{n}) → B′(u) strongly in W_{0}^{2,p}(Ω), i.e.,

sup_{v∈W_{0}^{2,p}(Ω) ∥Δv∥_{p}≤1} | ∫_{Ω} ρ[|u_{n}|^{p−2}u_{n} − |u|^{p−2}u]v dx| → 0, as n → +∞.

For this end, we distinguish three cases as in step 1 above. For \frac{N}{p} > 2, and r > \frac{N}{2p}. Let s be as in (3.3). Then

sup_{v∈W_{0}^{2,p}(Ω), ∥Δv∥_{p}≤1} | ∫_{Ω} ρ[|u_{n}|^{p−2}u_{n} − |u|^{p−2}u]vdx|

≤ sup_{v∈W_{0}^{2,p}(Ω), ∥Δv∥_{p}≤1} [∥ρ∥_{r}∥|u_{n}|^{p−2}u_{n} − |u|^{p−2}u∥_{\frac{s}{p−1}} ∥v∥_{p_{2}}]

≤ c∥ρ∥_{r}∥|u_{n}|^{p−2}u_{n} − |u|^{p−2}u∥_{\frac{s}{p−1}} ,

where c is the constant of Sobolev’s embedding [1]. On the other hand, the Nemytskii’s operator u ↦ |u|^{p−2}u is continuous from L^{s}(Ω) into L^{\frac{s}{p−1}} (Ω), and u_{n} → u weakly in W_{0}^{2,p}(Ω). So, we deduce that u_{n} → u strongly in L^{s}(Ω) because s < p_{2}. Hence,

∥|u_{n}|^{p−2}u_{n} − |u|^{p−2}u∥_{\frac{s}{p−1}} → 0, as n → +∞.

This completes the proof of the claim. If \frac{N}{p} = 2 then

| ∫_{Ω} ρ[|u_{n}|^{p−2}u_{n} − |u|^{p−2}u]v dx| ≤ ∥ρ∥_{r}∥|u_{n}|^{p−2}u_{n} − |u|^{p−2}u∥_{p}^{p−1}∥v∥_{q},

where q is given by (3.4). By Sobolev’s embedding there exist c > 0 such that

∥v∥_{q} ≤ c∥Δv∥_{p}, ∀v ∈ W_{0}^{2,p}(Ω).

Thus

sup_{∥Δv∥_{p}≤1^{v∈W{2,p \atop 0}(Ω)}} | ∫_{Ω} ρ[|u_{n}|^{p−2}u_{n} − |u|^{p−2}u]v dx| ≤ c∥ρ∥_{r}∥|u_{n}|^{p−2}u_{n} − |u|^{p−2}u∥_{p}^{p−1}.

From the continuity of u ↦ |u|^{p−1}u from L^{p}(Ω) into L^{p′}(Ω), and from the compact embedding of W_{0}^{2,p}(Ω) in L^{p}(Ω), we have the desired result. If \frac{N}{p} < 2 and r = 1, W_{0}^{2,p}(Ω) ⊂ C(\overline{Ω}), then we obtain

sup_{∥Δv∥_{p}≤1^{v∈W{2,p \atop 0}(Ω)}} | ∫_{Ω} ρ[|u_{n}|^{p−2}u_{n} − |u|^{p−2}u]v dx| ≤ c∥ρ∥_{1} sup_{\overline{Ω}} ||u_{n}|^{p−2}u_{n} − |u|^{p−2}u|,

where c is the constant given by embedding of W_{0}^{2,p}(Ω) in C(\overline{Ω}) ∩ L^{∞}(Ω). It is clear that

sup_{\overline{Ω}} ||u_{n}|^{p−2}u_{n} − |u|^{p−2}u| → 0, as n → +∞.

Hence B′ is completely continuous, also in this case. (ii) {u_{n}} is bounded in W_{0}^{2,p}(Ω). Hence without loss of generality, we can assume that u_{n} converges weakly in W_{0}^{2,p}(Ω) for some function u ∈ W_{0}^{2,p}(Ω) and ∥Δu_{n}∥_{p} → c. For the rest we distinct two cases: If c = 0 then u_{n} converges strongly to 0 in W_{0}^{2,p}(Ω).

If c ≠ 0, then we argue as follows:

⟨Δ_{p}^{2}u_{n}, u_{n} − u⟩ = ∥Δu_{n}∥_{p}^{p} − ⟨Δ_{p}^{2}(u_{n}), u⟩.

Applying ϵ_{n} of (3.1) to u, we deduce that

Θ_{n} := ⟨Δ_{p}^{2}(u_{n}), u⟩ − ∥Δu∥_{p}^{p}⟨B′(u_{n}), u⟩ → 0 as n → +∞.  (3.5)

Thus

⟨Δ_{p}^{2}u_{n}, u_{n} − u⟩ = ∥Δu_{n}∥_{p}^{p} − Θ_{n} − ∥Δu_{n}∥_{p}^{p}⟨B′(u_{n}), u⟩.

That is,

⟨Δ_{p}^{2}u_{n}, u_{n} − u⟩ = ∥Δu_{n}∥_{p}^{p}(1 − ⟨B′(u_{n}), u⟩) − Θ_{n}.

Hence,

lim sup_{n→+∞}⟨Δ_{p}^{2}u_{n}, u_{n} − u⟩ ≤ c^{p}lim sup_{n→+∞}(1 − ⟨B′(u_{n}), u⟩).

On the other hand, from (i) B′(u_{n}) → B′(u) in W^{−2,p′}(Ω) and pB(u) = 1, because pB(u_{n}) = 1 for all n ∈ ℕ^{∗}. So pB(u) = ⟨B′(u), u⟩ = 1. This yields that

1 − ⟨B′(u_{n}), u⟩ =⟨B′(u), u > − < B′(u_{n}), u⟩ ≤ ∥B′(u) − B′(u_{n})∥_{∗}∥Δu∥_{p},

where ∥.∥_{∗} is the dual norm associated with ∥Δ.∥_{p}.

From (i) again B′(u_{n}) → B′(u) in W_{0}^{−2,p′}(Ω), we deduce that

lim sup_{n→+∞}⟨Δ_{p}^{2}u_{n}, u_{n} − u⟩ ≤ 0  (3.6)

We can write Δ_{p}^{2}u_{n} = ∥Δu_{n}∥_{p}^{p−2}J(u_{n}), since ∥Δu_{n}∥_{p} ≠ 0 for n large enough. Therefore,

lim sup_{n→+∞}⟨Δ_{p}^{2}u_{n}, u_{n} − u⟩ = c^{p−2} lim sup_{n→+∞}⟨Ju_{n}, u_{n} − u⟩.

According to (3.5) we conclude that

lim sup_{n→+∞}⟨Ju_{n}, u_{n} − u⟩ ≤ 0

J being a duality mapping, thus it satisfies the condition S^{+}. Therefore, u_{n} → u strongly in W_{0}^{2,p}(Ω). This achieves the proof of the lemma. □

Remark 3.4 A′ is continuous, odd, (p-1)-homogeneous, continuously invertible and ∥A′(u)∥_{∗} = ∥Δu∥_{p}^{p−1}, ∀u ∈ W_{0}^{2,p}(Ω).

Remark 3.5 We can give another method to prove that the functional A satisfies the Palais-Smale condition on M.

Indeed, {u_{n}} is bounded in W_{0}^{2,p}(Ω), we can assume for a subsequence if necessary that u_{n} converges weakly in W_{0}^{2,p}(Ω). The claim is to prove that u_{n} is of Cauchy in W_{0}^{2,p}(Ω). Set

G(u_{n}, u_{m}) = ∫_{Ω}(|Δu_{n}|^{p−2}Δu_{n} − |Δu_{m}|^{p−2}Δu_{m}).Δ(u_{n} − u_{m}).

From (ii) of the proposition 2.1, Δ_{p}^{2} is a monotonous operator on W_{0}^{2,p}(Ω). So that

0 ≤ G(u_{n}, u_{m}) =⟨Δ_{p}^{2}u_{n} − Δ_{p}^{2}u_{m}, u_{n} − u_{m}⟩
=⟨ϵ_{n} − ϵ_{m}, u_{n} − u_{m}⟩ + ⟨h_{n} − h_{m}, u_{n} − u_{m}⟩,

with ϵ_{n} defined as in (3.1) and h_{n} = ∥Δu_{n}∥_{p}^{p}B′(u_{n}).

G(u_{n}, u_{m}) ≤ ∥ϵ_{n} − ϵ_{m}∥_{∗}∥Δu_{n} − Δu_{m}∥_{p} + ∥h_{n} − h_{m}∥_{∗}∥Δu_{n} − Δu_{m}∥_{p}.

Or h_{n} converges for a subsequence if necessary in W_{0}^{2,p}(Ω). Indeed, from (i) of Lemma 3.3 B′ : W_{0}^{2,p}(Ω) → W^{−2,p′}(Ω) is completely continuous. On the other hand, for a subsequence if necessary ∥Δu_{n}∥ → c ≥ 0. It follows that (h_{n})_{n≥0} is convergent in W^{−2,p′}(Ω). Then

G(u_{n}, u_{m}) → 0, as n → +∞.  (3.7)

From [4], we have the following inequality

|t_{1} − t_{2}|^{p} ≤ c\{(|t_{1}|^{p−2}t_{1} − |t_{2}|^{p−2}t_{2}).(t_{1} − t_{2})\}^{\frac{γ}{2}} (|t_{1}|^{p} + |t_{2}|^{p})^{1− \frac{γ}{2}} ,

for any t_{1}, t_{2} ∈ ℝ, with γ = p if 1 < p < 2 and γ = 2 if p ≥ 2. By applying Hölder’s inequality, we deduce that

∥Δu_{n} − Δu_{m}∥_{p}^{p} ≤ c\{G(u_{n}, u_{m})\}^{\frac{γ}{2}} (∥Δu_{n}∥_{p}^{p} + ∥Δu_{m}∥_{p}^{p})^{1− \frac{γ}{2}}  (3.8)

for some positive constante c independent of n and m. According to (3.7), (3.8) shows that (u_{n})_{n} is a Cauchy’s sequence in W_{0}^{2,p}(Ω). This proves the claim. □

Set

Γ_{k} = {K ⊂ M : K is symmetric,compact and γ(K) ≥ ∥},

where γ(K) = k is the genus of k, i.e., the smallest integer k such that there exists an odd continuous map from K to ℝ^{k} − {0}.

Now, by the Ljusternick-Schnirelmann theory, see e.g. [6], we have our main result formulated as follows.

Theorem 3.6 For any integer k ∈ ℕ^{∗},

λ_{k} := inf_{K∈Γ_{k}} max_{u∈K} pA(u)

is a critical value of A restricted on M. More precisely, there exists u_{k} ∈ K_{k} ∈ Γ_{k} such that

λ_{k} = pA(u_{k}) = sup_{u∈K_{k}} pA(u)

and (λ_{k}, u_{k}) is a solution of (1.1) associated with the positive eigenvalue λ_{k}. Moreover,

λ_{k} → +∞, as k → +∞.

Proof We need only to prove that for any k ∈ ℕ^{∗}, Γ_{k} ≠ ∅ and the least assertion. Indeed, since W_{0}^{2,p}(Ω) is separable, there exist (e_{i})_{i≥1} linearly dense in W_{0}^{2,p}(Ω) such that supp e_{i} ∩ supp e_{j} = ∅ if i ≠j. We can assume that e_{i} ∈ M. Let k ∈ ℕ^{∗}, denote F_{k} = span{e_{1}, e_{2}, …, e_{k}}. F_{k} is a vectorial subspace and dim F_{k} = k. If v ∈ F_{k}, then there exist α_{1}, … α_{k} in ℝ such that v = ∑_{i=1}^{k} α_{i}e_{i}. Thus B(v) = ∑_{i=1}^{k} |α_{i}|^{p}B(e_{i}) = \frac{1}{p} ∑_{i=1}^{k} |α_{i}|^{p}. It follows that the map v ↦ (pB(v))^{1/p} := ∥v∥ defines a norm on F_{k}. Consequently, there is a constant c > 0 such that

c∥Δu∥_{p} ≤ ∥v∥ ≤ \frac{1}{c}∥Δu∥_{p}.

This implies that the set

V = F_{k} ∩ \{v ∈ W_{0}^{2,p}(Ω) : B(v) ≤ \frac{1}{p}\}

is bounded. Thus V is a symmetric bounded neighbourhood of 0 ∈ F_{k}. By (f) in [6, Prop. 2.3], γ(F_{k} ∩ M) = ∥. Because F_{k} ∩ M is compact and Γ_{k} ≠ ∅. Now, we claim that λ_{k} → +∞, as k → +∞. Indeed let be (e_{n}, e_{j}^{∗})_{n,j} a bi-orthogonal system such that e_{n} ∈ W_{0}^{2,p}(Ω) and e_{j}^{∗} ∈ W^{−2,p′}(Ω), the e_{n} are linearly dense in W_{0}^{2,p}(Ω); and the e_{j}^{∗} are total for W^{−2,p′}(Ω), see e.g. [6]. For k ∈ ℕ^{∗}, set

F_{k} = span{e_{1}, …, e_{k}}, F_{k}^{⊥} = span{e_{k+1,e_{k+2},...}}.

By (g) of Proposition 2.3 in [6], we have for any A∈ Γ_{k}, A ∩ F_{k−1}^{⊥} ≠ ∅. Thus

t_{k} := inf_{A∈Γ_{k}} sup_{u∈A∩F_{k−1}^{⊥}} pA(u) → +∞.

Indeed, if not, for k is large, there exists u_{k} ∈ F_{k−1}^{⊥} with ∥u_{k}∥_{p} = 1 such that

t_{k} ≤ pA(u_{k}) ≤ M,

for some M > 0 independent of k. Thus ∥Δu_{k}∥_{p} ≤ M. This implies that (u_{k})_{k} is bounded in W_{0}^{2,p}(Ω). For a subsequence of {u_{k}} if necessary, we can assume that {u_{k}} converge weakly in W_{0}^{2,p}(Ω) and strongly in L^{p}(Ω). By our choice of F_{k−1}^{⊥} , we have u_{k} ↪ 0 weakly in W_{0}^{2,p}(Ω). Because ⟨e_{n}^{∗}, e_{k}⟩ = 0, ∀k ≥ n. This contradicts the fact that ∥u_{k}∥_{p} = 1 ∀k. Since λ_{k} ≥ t_{k}, the claim is proved. This completes the proof. □

Corrolary 3.7 (i) λ_{1} = inf{∥Δv∥_{p}^{p}, v ∈ W_{0}^{2,p}(Ω); ∫_{Ω} ρ(x)|v|^{p}dx = 1}. (ii) 0 < λ_{1} ≤ λ_{2} ≤ ⋯ ≤ λ_{n} → +∞.

Proof (i) For u ∈ M, we put K_{1} = {u, −u}. It is clear that γ(K_{1}) = 1, that A is even and that

pA(u) = max_{K_{1}} pA ≥ inf_{K∈Γ_{1}} max_{K} pA.

Hence

inf_{u∈M} pA(u) ≥ inf_{K∈Γ_{1}} max_{K} pA = λ_{1}.

On the other hand, ∀K ∈ Γ_{1}, ∀u ∈ K,

sup_{K} pA ≥ pA(u) ≥ inf_{u∈M} pA(u).

So

inf_{K∈Γ_{1}} max_{K} pA = λ_{1} ≥ inf_{u∈M} pA(u).

Thus

λ_{1} = inf_{u∈M} pA(u) = inf{∥Δv∥_{p}^{p}, v ∈ W_{0}^{2,p}(Ω) : ∫_{Ω} ρ(x)|v|^{p}dx = 1}.

(ii) For all i ≥ j, Γ_{i} ⊂ Γ_{j}. From the definition of λ_{i}, i ∈ ℕ^{∗}, we have λ_{i} ≥ λ_{j}. λ_{n} → +∞ is already proved in Theorem 3.6. Which completes the proof. □

Corrolary 3.8 Assume that |Ω_{ρ}^{−}| ≠ 0 with Ω_{ρ}^{−} = {x ∈ Ω : ρ(x) < 0}. Then Δ_{p}^{2} has a decreasing sequence of the negative eigenvalues (λ_{−n})(ρ)_{n≥0}, such that lim_{n→+∞} λ_{−n} = −∞.

Proof First, remark that Ω_{ρ}^{−} = Ω_{(−ρ)}^{+}, so |Ω_{(−ρ)}^{+}| = |Ω_{ρ}^{−}| ≠ 0. From Theorem 3.6, Δ_{p}^{2} has an increasing sequence of the positive eigenvalues λ_{n}(−ρ), such that lim_{n→+∞} λ_{n}(−ρ) = +∞. Note that λ_{n}(−ρ) satisfies

Δ_{p}^{2}u = λ_{n}(−ρ)(−ρ)|u|^{p−2}u = −λ_{n}(−ρ)ρ|u|^{p−2}u,

for u ∈ W_{0}^{2,p}(Ω). Put λ_{−n}(ρ) := −λ_{n}(−ρ) then λ_{n}(−ρ)_{n≥0} is an increasing positive sequence so (λ_{−n})(ρ)_{n≥0} is a negative decreasing sequence. On the other hand, lim_{n→+∞} λ_{n}(−ρ) = +∞. So

lim_{n→+∞} λ_{−n}(ρ) = −∞.

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Abdelouahed El Khalil (e-mail: lkhlil@hotmail.com) Siham Kellati (e-mail: siham360@caramail.com) Abdelfattah Touzani (e-mail: atouzani@iam.net.ma) Department of Mathematics and Informatic, Faculty of Sciences Dhar-Mahraz, P.O. Box 1796 Atlas-Fez, Morocco