2005-Oujda International Conference on Nonlinear Analysis. Electronic Journal of Differential Equations, Conference 14, 2006, pp. 155–162. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu (login: ftp)

NOTE ON THE NODAL LINE OF THE P-LAPLACIAN

ABDEL R. EL AMROUSS, ZAKARIA EL ALLALI, NAJIB TSOULI

Abstract. In this paper, we prove that the length of the nodal line of the eigenfunctions associated to the second eigenvalue of the problem

−Δ_{p}u = λρ(x)|u|^{p−2}u in Ω with the Dirichlet conditions is not bounded uniformly with respect to the weight.

1. Introduction

In this paper we consider the nonlinear elliptic boundary-value problem

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

where Δ_{p}u = div(|∇u|^{p−2}∇u) is the p-Laplacian operator, Ω is a bounded and smooth domain in ℝ^{N} (1 < p < +∞) and ρ ∈ L^{∞}(Ω) is an indefinite weight such that

meas(Ω_{ρ}^{+}) ≠ 0 with Ω_{ρ}^{+} = {x ∈ Ω | ρ(x) > 0}.

Several authors have been studied the spectrum σ(−Δ_{p}) of p-Laplacian, precisely around of the first and the second eigenvalue. In particular Anane [1] proved that the spectrum σ(−Δ_{p}) contains a positive non-decreasing sequence of eigenvalues (λ_{n})_{n∈ℕ^{∗}} such that λ_{n} → +∞ by using the Ljusternik-Schnirelmann, where

λ_{n}^{−1} = λ_{n}(Ω, ρ)^{−1} = sup_{K∈A_{n}} inf_{v∈K} ∫_{Ω} ρ(x)|v|^{p}dx

and

A_{n} = {K ⊂ W_{0}^{1,p}(Ω) : K is symmetrical compact and γ(K) ≥ n}.

Moreover, he showed that the first eigenvalue is simple and isolated, and that the first eigenfunction corresponding to λ_{1} does not change the sign in Ω. In [2] they have showed that the second eigenvalue of the spectrum σ(−Δ_{p}) is exactly λ_{2}. The complete determination of this spectrum remains unanswered question. It is useful to announce that in the linear case (p = 2), the spectrum is perfectly given [4, 6].

2000 Mathematics Subject Classification. 58E05, 35J65, 56J20.

Key words and phrases. Nonlinear eigenvalue problem; p-Laplacian; nodal line.

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

Published September 20, 2006.

Let us consider a solution (u, λ) of problem P (Ω, ρ). We denote by

Z(u) = {x ∈ Ω : u(x) = 0}

the nodal set of u, N (u) is the number of connected components of Ω\Z(u), C(u) is the set of connected components of Ω\Z(u),

N (λ) = max{N (u) | (u, λ) solution of P (Ω, ρ)} and N (λ_{n}) = N (n).

Recently Cuesta, de Figueiredo and Gossez proved that N (2) = 2 [3].

The main result in this paper is the generalization of the work of Kappeler and Ruf [5], in which they affirmed that the length of the nodal lines is not bounded uniformly with respect to the weights in dimension N = 2 and p = 2. In this work, we locate in the case p > N and we prove that for all real number L > 0, there exists a weights ρ ∈ L^{∞}(Ω) and an eigenfunction associated to the second eigenvalue of (1.1) such that the length of Z(u) is largest than L. The proof is relatively simpler than that given by Kappeler and Ruf; in which they use the uniform convergence of the gradient.

2. On the measure of nodal sets

In this section, we will extend the result of Kappeler and Ruf [5] in the case p ≠ 2.

2.1. Main result. We consider the case p > N. Let Γ be a surface of class C^{1} which subdivides Ω in two nodal components Ω_{1} and Ω_{2} such that

μ_{1} ≤ ν_{1}  (2.1)

where μ_{1} (respectively ν_{1}) is the first eigenvalue of P (Ω_{1}, 1) (respectively P (Ω_{2}, 1). Let v_{1} (respectively w_{1}) the associated eigenfunction. For n ∈ ℕ^{∗}, let

Ω′_{n} = \{x ∈ Ω_{1} : dist(x, Γ) < \frac{1}{n + 1}\},

Ω_{n} = Ω_{1}\Ω′_{n}

where Γ ⊂ ∂Ω′_{n} of class C^{1}. Then, we denote by v_{1}^{n} the eigenfunction associated to μ_{1}^{n} the first eigenvalue of (1.1) with ρ = 1.

Let (a_{n})_{n∈ℕ^{∗}} be a sequence of decreasing positive real numbers such that

a_{n} = \frac{μ_{1}^{n}}{ν_{1}}  (2.2)

which tends to the limit a ∈ ℝ_{+}^{∗} (0 < a = \frac{μ_{1}}{ν_{1}} ≤ 1). Let (ρ_{n})_{n∈ℕ^{∗}} be a sequence of weight functions defined by

ρ_{n}(x) = −r_{n}1_{Ω′_{n}}(x) + a_{n}1_{Ω_{n}}(x) + 1_{Ω_{2}}(x),  (2.3)

for all x ∈ Ω, where r_{n} > 0 such that lim_{n→+∞} \frac{c_{n}d_{n}}{r_{n}^{(p−1)/p^{2}}} = 0 with c_{n} and d_{n} are

strictly positive constants of immersion and interpolation.

Let us denote u_{2}^{n} the eigenfunction associated to the second eigenvalue λ_{2}^{n} of (1.1).

Theorem 2.1. There exists a subsequence of (u_{2}^{n})_{n∈ℕ^{∗}} still denoted by (u_{2}^{n})_{n∈ℕ^{∗}} such that (i) The sequence (u_{2}^{n})_{n∈ℕ^{∗}} converges weakly to (α\overline{v_{1}} + β\overline{w_{1}}) in W_{0}^{1,p}(Ω), for some scalars α , β not all null, where \overline{v_{1}} (respectively \overline{w_{1}} ) is the extension of v_{1} (respectively w_{1}) by zero in \overline{Ω}.

(ii) If U and V are two opens of ℝ^{N} such that \overline{U} ⊂ Ω_{1} and \overline{V} ⊂ Ω_{2}. Then for n enough large, we have

\overline{U} ∩ Z(u_{2}^{n}) = \overline{V} ∩ Z(u_{2}^{n}) = ∅

and u_{2}^{n} change the sign on U ∪ V

To prove this result, we need the following preliminary lemmas.

Lemma 2.2. The following inequalities are true independently of (r_{n})_{n∈ℕ^{∗}}:

(i) 0 < b^{−1}λ_{2}(Ω, 1) ≤ λ_{2}^{n} ≤ ν_{1} , where b = \begin{cases}1 & if μ_{1} < ν_{1} \\ a_{1} & if μ_{1} = ν_{1}, \end{cases}

(ii) ∥Δ_{p}u_{2}^{n}∥_{L^{p′} (Ω_{2})}^{p′} ≤ Mν_{1}^{p′},

(iii) ∥Δ_{p}u_{2}^{n}∥_{L^{p′} (Ω_{n})}^{p′} ≤ M(a_{1}ν_{1})^{p′}, where M is the Sobolev-Poincaré constant.

Proof. (i) Let F_{2} be the vector subspace of W_{0}^{1,p}(Ω) spanned by \{\overline{v_{1}^{n}}, \overline{w_{1}}\}, where \overline{v_{1}^{n}} (resp. \overline{w_{1}}) is the extension by zero of v_{1}^{n} (resp. w_{1}) in Ω \ Ω_{n} (resp. Ω \ Ω_{1}). Let S_{2} denote the unit sphere of F_{2}. For all v ∈ S_{2} such that v = α\overline{v_{1}^{n}} + β\overline{w_{1}}, we have

|α|^{p} + |β|^{p} = 1 and ∫_{Ω} ρ_{n}(x)|v(x)|^{p}dx = |α|^{p}a_{n} \frac{1}{μ_{1}^{n}} + |β|^{p}ν_{1}^{−1}.

Using (2.3), we get

∫_{Ω} ρ_{n}(x)|v(x)|^{p}dx = \frac{1}{ν_{1}}

In particular

\frac{1}{ν_{1}} ≤ inf_{v∈S_{2}}\{∫_{Ω} ρ_{n}(x)v(x)dx\} ≤ \frac{1}{λ_{2}^{n}} .

Since ρ_{n}(x) ≤ b,

\frac{λ_{2}(Ω, b)}{b} = λ_{2}(Ω, b) ≤ λ_{2}^{n}(Ω, ρ_{n}) = λ_{2}^{n}

(ii) It is sufficient to notice that

−Δ_{p}u_{2}^{n} = λ_{2}^{n}|u_{2}^{n}|^{p−2}u_{2}^{n} a.e. on Ω_{2}

and by using the Sobolev-Poincaré inequality, we have

∫_{Ω_{2}} | − Δ_{p}u_{2}^{n}|^{p′} ≤ M(λ_{2}^{n})^{p′}∥∇u_{2}^{n}∥_{L^{p}(Ω)}^{p} ≤ Mν_{1}^{p′}.

(iii) Using −Δ_{p}u_{2}^{n} = λ_{2}^{n}a_{n}|u_{2}^{n}|^{p−2}u_{2}^{n} a.e. on Ω_{n} and with the same argument as above, one gets

∫_{Ω_{n}} | − Δ_{p}u_{2}^{n}|^{p′} ≤ M(a_{1}ν_{1})^{p′}.
□

Remark 2.1. (1) By the lemma 2.2 we can choose the sequence (ρ)_{n∈ℕ^{∗}} such that (λ_{2}^{n})_{n∈ℕ^{∗}} converges to the positive limit λ_{2}. For all p > N , u_{2}^{n} con-

verges weakly in W_{0}^{1,p}(Ω) and strongly in C(\overline{Ω}) to u_{2} ∈ W_{0}^{1,p}(Ω).
(2) ρ_{n}(x) ≤ b for all x ∈ Ω.

(3) u_{2} ≠ 0 in L^{p}(Ω).

Lemma 2.3. With the above notation, λ_{2} = ν_{1}

Proof. To prove this lemma, we proceed in three steps: First step: We show that

−Δ_{p}u_{2} = λ_{2}a|u_{2}|^{p−2}u_{2} a.e. on Ω_{1}  (2.4)

Indeed; for m ≥ 1 and by the lemma 2.2 , we have

∥Δ_{p}u_{2}^{n}∥_{L^{p′} (Ω_{m})}^{p′} ≤ ∥Δ_{p}u_{2}^{n}∥_{L^{p′} (Ω_{n})}^{p′} ≤ M(a_{1}ν_{1})^{p′} ∀n ≥ m

hence

∥Δ_{p}u_{2}^{n}∥_{(W^{1,p}(Ω_{m}))′}^{p′} ≤ M(a_{1}ν_{1})^{p′} ∀n ≥ m.

It follows that there exists a subsequence, still denoted by (−Δ_{p}u_{2}^{n}), such that −Δ_{p}u_{2}^{n} ⇀ T_{m} weakly in the sapce (W^{1,p}(Ω_{m}))′. By remark 2.1, u_{2}^{n} ⇀ u_{2} weakly in W^{1,p}(Ω_{m}). Since

−Δ_{p}u_{2}^{n} = λ_{2}^{n}a_{n}|u_{2}^{n}|^{p−2}u_{2}^{n} a.e. on Ω_{m} for all n ≥ m,

we have

lim_{n→+∞}⟨−Δ_{p}u_{2}^{n}, u_{2}^{n}⟩_{m} = ⟨T_{m}, u_{2}⟩_{m}

where ⟨·, ·⟩_{m} is the duality bracket between W^{1,p}(Ω_{m}) and its dual (W^{1,p}(Ω_{m}))′. However, −Δ_{p} is an operator of type (M), consequently T_{m} = −Δ_{p}u_{2} is in the space (W^{1,p}(Ω_{m}))′. We deduce that

−Δ_{p}u_{2} = λ_{2}a|u_{2}|^{p−2}u_{2} a.e. on Ω_{m} ∀m ≥ 1

hence (2.4). Similarly, we prove that

−Δ_{p}u_{2} = λ_{2}|u_{2}|^{p−2}u_{2} a.e. on Ω_{2}  (2.5)

Second step: We show that

u_{2_{/∂Ω_{i}}} = 0 in L^{p}(∂Ω_{i}) for i = 1, 2.

Indeed, it follows from (2.2) and since ∂Ω′_{n} is of C^{1} and (Ω ∩ ∂Ω_{2}) ⊂ ∂Ω′_{n}, we have

∥u_{2_{/Ω∩∂Ω_{2}}}^{n} ∥_{L^{p}(Ω∩∂Ω_{2})} ≤ c_{n}d_{n}∥u_{2}^{n}∥_{W^{1,p}(Ω′_{n})}^{σ}∥u_{2}^{n}∥_{L^{p}(Ω′_{n})}^{1−σ},  (2.6)

where c_{n} is the constant of the immersion W^{σ,p}(Ω′_{n}) ↪ L^{p}(∂Ω′_{n}); σ = \frac{1}{p} and d_{n} is the constant of the interpolation of the inequality

∥u∥_{W^{σ,p}(Ω′_{n})} ≤ d_{n}∥u∥_{W^{1,p}(Ω′_{n})}^{σ}∥u∥_{L^{p}(Ω′_{n})}^{1−σ} for all u ∈ W^{σ,p}(Ω′_{n}).

The two norms of the second member in (2.6) can be estimated as follows:

∥u_{2}^{n}∥_{W^{1,p}(Ω′_{n})}^{p} ≤ ∥u_{2}^{n}∥_{L^{p}(Ω′_{n})}^{p} + ∥∇u_{2}^{n}∥_{L^{p}(Ω′_{n})}^{p} ≤ M + 1,  (2.7)

where M is the constant of the Sobolev-Poincaré of lemma 2.2. Moreover, since (u_{2}^{n}, λ_{2}^{n}) is a solution of P (Ω, ρ_{n}), by (2.3) and lemma 2.2, we get

r_{n} ∫_{Ω′_{n}} |u_{2}^{n}|^{p}dx ≤ ∫_{Ω_{2}} |u_{2}^{n}|^{p}dx + a_{n} ∫_{Ω_{n}} |u_{2}^{n}|^{p}dx .

Since b ≥ 1, we deduce that

∫_{Ω′_{n}} |u_{2}^{n}|^{p}dx ≤ \frac{b}{r_{n}} ∫_{Ω} |u_{2}^{n}|^{p}dx ≤ \frac{b}{r_{n}} M.  (2.8)

Thus, by (2.6), (2.7) and (2.8), we have

∥u_{2_{/Ω∩∂Ω_{2}}}^{n} ∥_{L^{p}(Ω∩∂Ω_{2})} ≤ c_{n}d_{n}(M + 1)^{\frac{σ}{p}} (\frac{bM}{r_{n}})^{\frac{1−σ}{p}} .

However, lim_{n→+∞} \frac{c_{n}d_{n}}{r_{n}^{(1−σ)/p}} = 0 and u_{2} ∈ W_{0}^{1,p}(Ω), consequently u_{2_{/∂Ω_{2}}} = 0 in L^{p}(∂Ω_{2}). Similarly, we have u_{2_{/∂Ω_{1}}} = 0 in L^{p}(∂Ω_{1}) because (Ω ∩ ∂Ω_{1}) ⊂ ∂Ω′_{n}. Third step: We establish that λ_{2} = ν_{1} = η_{1}. Indeed, since u_{2} = 0 in L^{p}(∂Ω_{1}) and L^{p}(∂Ω_{2}) with u_{2} ∈ W_{0}^{1,p}(Ω), we have u_{2} ∈ W_{0}^{1,p}(Ω_{1}) and u_{2} ∈ W_{0}^{1,p}(Ω_{2}). Moreover, if we use (2.4), (2.5) and the remark 2.1, then (u_{2_{/Ω_{1}}} , λ_{2}) and (u_{2_{/Ω_{2}}} , λ_{2}) are respectively solutions of problems (1.1) with Ω = Ω_{1} and with Ω = Ω_{2}. We have by lemma 2.2,

λ_{2} = lim_{n→+∞} λ_{2}^{n} ≤ ν_{1},

where ν_{1} is the first eigenvalue of (1.1) with Ω = Ω_{2}. We conclude that λ_{2} = ν_{1} = η_{1}.

Lemma 2.4. The sequence (f_{n})_{n∈ℕ^{∗}} admits a subsequence which converges weakly in W_{0}^{1,p}(Ω) to \overline{v_{1}}, where

f_{n} = \frac{(u_{2}^{n})^{+}}{(\frac{a}{p})^{1/p}∥(u_{2}^{n})^{+}∥_{L^{p}(Ω)}} .

with (u_{2}^{n})^{+} = max{0, u_{2}^{n}}.

Proof. It is known that

∫_{Ω} |∇(u_{2}^{n})^{+}|^{p}dx = λ_{2}^{n} ∫_{Ω} ρ_{n}|(u_{2}^{n})^{+}|^{p}dx .

If we multiply by ((\frac{a}{p})^{1/p}∥(u_{2}^{n})^{+}∥_{L^{p}(Ω)}^{p})^{−p}, then

∫_{Ω} |∇f_{n}|^{p}dx = λ_{2}^{n} ∫_{Ω} ρ_{n}|f_{n}|^{p}dx ≤ λ_{2}^{n}b ∫_{Ω} |f_{n}|^{p}dx = bλ_{2}^{n} \frac{p}{a}.  (2.9)

Thus, by lemmas 2.2 and 2.3, we have

∫_{Ω} |∇f_{n}|^{p}dx ≤ \frac{p}{a}bλ_{2} < +∞ ∀n ∈ ℕ^{∗}.

So, for a subsequence of the sequence (f_{n})_{n∈ℕ^{∗}}, still denoted (f_{n})_{n∈ℕ^{∗}}, we have f_{n} ⇀ f weakly in W_{0}^{1,p}(Ω), then f_{n} → f strongly in C(\overline{Ω}) with p > N. Since ∥f_{n}∥_{L^{p}(Ω)} = (\frac{p}{a})^{1/p} then ∥f∥_{L^{p}(Ω)} ≠ 0. Hence f ≠ 0 on Ω, since u_{2_{/Ω_{2}}} = βw_{1} < 0,

f = 0 on Ω_{2}  (2.10)

a fortiori f = 0 on ∂Ω_{2} and f = 0 on ∂Ω_{1}. It results that f ∈ W_{0}^{1,p}(Ω_{1}). According to (2.9), we have

∫_{Ω} |∇f_{n}|^{p}dx = λ_{2}^{n}(a_{n} ∫_{Ω} |f_{n}|^{p}dx − r_{n} ∫_{Ω′_{n}} |f_{n}|^{p}dx + ∫_{Ω_{2}} |f_{n}|^{p}dx)
≤ λ_{2}^{n}(a_{n} ∫_{Ω_{n}} |f_{n}|^{p}dx + ∫_{Ω_{2}} |f_{n}|^{p}dx).

Hence, lim inf_{n→+∞} ∫_{Ω} |∇f_{n}|^{p}dx ≤ λ_{2}(a ∫_{Ω_{1}} |f|^{p}dx+∫_{Ω_{2}} |f|^{p}dx). From (2.10), we deduce

that

lim_{n→+∞} inf ∫_{Ω} |∇f_{n}|^{p}dx ≤ λ_{2}a ∫_{Ω} |f|^{p} dx.

Thus

∫_{Ω_{1}} |∇f|^{p}dx ≤ λ_{2}a ∫_{Ω} |f|^{p}dx = λ_{2}p.

We have λ_{2} = η_{1} being the first eigenvalue of (1.1) with Ω = Ω_{1} and ρ = a; consequently

λ_{2} = \frac{1}{p} ∫_{Ω_{1}} |∇f|^{p}dx and f = \overline{v_{1}}.

□

2.2. Proof of the main result. (i) From lemma 2.3, u_{2} is an eigenfunction associated to ν_{1} (resp. η_{1}). So, there exists α, β ∈ ℝ^{n} such that

u_{2} = αv_{1} + βw_{1} with |α|^{p} + |β|^{p} > 0.

(ii) We distinguish two possible cases: First case: If α ≠ 0 and β ≠ 0, we can assume that α > 0 and β < 0 (the other cases will be treated in the same way ). As u_{2} = αv_{1} on Ω_{1} and v_{1} is a positive eigenfunction of class C^{1} on Ω_{1}, then ∃x_{0} ∈ \overline{U} such that min\{u_{2}(x) : x ∈ \overline{U}\} = u_{2}(x_{0}) > 0. By lemma 2.2, u_{2}^{n} converges uniformly (p > N) to u_{2} in \overline{Ω}, consequently for ϵ = u_{2}(x_{0}) > 0 there exists n_{0}(\overline{U}) ∈ ℕ such that for all n ≥ n_{0}(\overline{U}), we have

u_{2}^{n}(x) > \frac{ϵ}{2} ∀x ∈ \overline{U}

i.e. (\overline{U} ∩ Z(u_{2}^{n})) = ∅ for all n ≥ n_{0}(\overline{U}). It is the same for (\overline{V} ∩ Z(u_{2}^{n})) = ∅ for all n ≥ n_{0}(\overline{V}). We announce here that according to the lemma 2.2 the case where αβ > 0 does not intervene. Second case: If α = 0 or β = 0. We consider now the case where α = 0 and β < 0. The other cases will be treated in the same way. By lemma 2.4, there exists a subsequence, still denoted (f_{n})_{n∈ℕ}, which converges uniformly to f = \overline{v_{1}} in \overline{Ω}. Moreover \overline{v_{1}} > 0 in \overline{U}, and there exists x_{0} ∈ \overline{U} such that

f(x_{0}) = min\{f(x) = \overline{v_{1}}(x) : x ∈ \overline{U}\} > 0.

Thus, for ϵ = f(x_{0}) > 0, there exists n_{0}(\overline{U}) ∈ ℕ^{∗} such that for all n ≥ n_{0}(\overline{U}) we have

f_{n}(x) > \frac{ϵ}{2} ∀x ∈ \overline{U}.

i.e for all n ≥ n_{0}(\overline{U}), \overline{U} ∩ Z(u_{2}^{n}) = ∅. Therefore, since β < 0, it is the same for \overline{V} ∩ Z(u_{2}^{n}) = ∅ for all n ≥ n_{0}(\overline{V}) .

We remark here that by (i) of the lemma 2.2 the case where α = β = 0 does not intervene. □

2.3. Consequences of the main result.

Corollary 2.5. If Γ is a surface of class C^{1} in Ω which subdivides Ω in two connected components, then for all neighborhood [Γ]_{ϵ} of Γ, there exists a weight ρ_{ϵ} ∈ L^{∞}(Ω) and an eigenfunction u associated to the second eigenvalue of P (Ω, ρ_{ϵ}) such that Z(u) ⊂ [Γ]_{ϵ}; where [Γ]_{ϵ} = {x ∈ Ω : d(x, Γ) ≤ ϵ}.

Proof. We distinguish two cases First case: \overline{Γ} ∩ ∂Ω = ∅. Let ε > 0, we consider U = Ω_{1}\ ([∂Ω]_{ϵ} ∪ [Γ]_{ϵ}) and V = Ω_{2}\ ([∂Ω]_{ϵ} ∪ [Γ]_{ϵ}), where [∂Ω]_{ϵ} = \{x ∈ Ω : d(x, Ω) ≤ ϵ\}. Since \overline{Γ} ∩ ∂Ω = ∅, we can choose ϵ enough small, so that [∂Ω]_{ϵ} ∩ [Γ]_{ϵ} = ∅. By theorem 2.1, there exists n ∈ ℕ^{∗} such that Z(u_{2}^{n}) ⊂ [∂Ω]_{ϵ} ∪ [Γ]_{ϵ}. We assume that Z(u_{2}^{n}) ∩ [∂Ω]_{ϵ} ≠ ∅ then there exists a nodal component D_{ϵ} of u_{2}^{n} included in [∂Ω]_{ϵ}. Thus (u_{2_{/Dϵ}}^{n} , λ_{2}^{n}) is a solution of the problem P (D_{ϵ}, ρ_{n_{/Dϵ}}) with λ_{2}^{n} its first eigenvalue [1, 7]. By the remark 2.1, we have

λ_{2}^{n} = λ_{1}(D_{ϵ}, ρ_{n_{/Dϵ}}) ≥ λ_{1}(D_{ϵ}, b)

we have meas(D_{ϵ}) → 0 when ϵ → 0 , consequently λ_{2}^{n} = λ_{1}(D_{ϵ}, ρ_{n}) → +∞ when ϵ → 0 which is absurd with lemma 2.2. So Z(u_{2}^{n}) ⊂ [Γ]_{ϵ}. Second case: \overline{Γ} ∩ ∂Ω ≠ ∅. Let ϵ > 0, there exists a surface Γ′_{ϵ} of C^{1} which subdivide Ω in two connected components such that

Γ′_{ϵ} ⊂ [Γ]_{ϵ} and \overline{Γ′_{ϵ}} ∩ ∂Ω = ∅

Let η > 0 (enough small) so that [Γ′_{ϵ}]_{η} ⊂ [Γ]_{ε} and [∂Ω]_{η} ∩ [Γ′_{ϵ}]_{η} = ∅, finally we conclude the result by applying the proof of the first case with Γ′_{ϵ}. □

Remark 2.2. The result of the corollary 2.5 remains true even if Γ is not of class C^{1}, only it is enough to approach Γ by a surface Γ′ of class C^{1} which located in [Γ]_{ϵ}.

Corollary 2.6. For all L > 0, there exists ρ ∈ L^{∞}(Ω) and an eigenfunction u associated to the second eigenvalue of P (Ω, ρ) such that the length of Z(u) is larger than L.

Proof. Let L > 0, there exists a surface Γ is of class C^{1} in Ω which subdivide Ω in two connected components such that

\overline{Γ} ∩ ∂Ω = ∅ and meas(Γ) > L + 1.

For ϵ > 0 (enough small) we consider [Γ]_{ϵ} and [∂Ω]_{ϵ} two neighborhood of Γ and ∂Ω respectively such that

[∂Ω]_{ϵ} ∩ [Γ]_{ϵ} = ∅

Denote by U = Ω_{1}\([∂Ω]_{ϵ} ∪ [Γ]_{ϵ}) and V = Ω_{2}\([∂Ω]_{ϵ} ∪ [Γ]_{ϵ}) two open. In virtue of the Theorem 2.1 and of the Corollary 2.5, ∃n ∈ ℕ^{∗} such that

U ∩ Z(u_{2}^{n}) = V ∩ Z(u_{2}^{n}) = ∅ and Z(u_{2}^{n}) ⊂ [Γ]_{ϵ}.

Let us suppose that for an infinity of ϵ > 0, u_{2}^{n} admits a nodal component D_{ϵ} included in [Γ]_{ϵ}. So

λ_{2}^{n} = λ_{1}(D_{ϵ}, ρ_{n_{/Dϵ}}) ≥ λ_{1}(D_{ϵ}, b).

Since lim_{ϵ→0} meas(D_{ϵ}) = 0, it follows that λ_{2}^{n} ≥ lim_{ϵ→0} λ_{1}(D_{ϵ}, b) = +∞ which is absurd with the lemma 2.2 . Thus, for ϵ enough small, there exists n ∈ ℕ such that Z(u_{2}^{n}) is a closed surface in [Γ]_{ϵ} with Z(u_{2}^{n}) = ∂W where W is an open containing Ω_{i}^{ϵ} which is an open included in Ω_{i} such that ∂Ω_{i}^{ϵ} ⊂ ∂[Γ]_{ϵ}. So if meas(Γ) > L + 1, then ∃ϵ > 0 (enough small) and ∃n ∈ ℕ^{∗} such that meas(Z(u_{2}^{n}) > L for i = 1, 2. □

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[5] T. Kappeler and B. Ruf, On the nodal line of the second eigenfunction of elliptic operators in two dimension, Journal Rvine und. Ange Maths. (396), 1-13.

[6] M. Otani, A remark on certain nonlinear elliptic equation, Proc. Fac. Sci. Tokai Univ. No. 19 (1984), 23-28.

[7] N. Tsouli, Etude de l’ensemble nodal des fonctions propres et de la non-résonance pour l’opérateur p-Laplacien, Thèse d’état, Université Mohamed I, Oujda (1996).

Abdel R. El Amrouss Département de Mathématiques et Informatique Faculté des Sciences, Université Mohamed 1er, Oujda, Maroc

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

Zakaria El Allali Département de Mathématiques et Informatique Faculté des Sciences, Université Mohamed 1er, Oujda, Maroc

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

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

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