2005-Oujda International Conference on Nonlinear Analysis. Electronic Journal of Differential Equations, Conference 14, 2006, pp. 181–190. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu (login: ftp)

ON THE SPECTRUM OF THE P-LAPLACIAN OPERATOR FOR NEUMANN EIGENVALUE PROBLEMS WITH WEIGHTS

SIHAM EL HABIB, NAJIB TSOULI

Abstract. This paper is devoted to study the spectrum for a Neumann eigenvalue problem involving the p-Laplacian operator with weight in a bounded domain.

1. Introduction

Let Ω be a smooth bounded domain in ℝ^{N}, N ≥ 1, 1 < p < +∞ and m ∈ L^{r}(Ω), with r = r(N, p) satisfying the conditions

\begin{matrix}r > N/p if 1 < p < N \\ r = 1 if p > N \\ r > p if p = N \end{matrix}  (1.1)

We assume that meas(Ω^{+}) ≠ 0, where Ω^{+} = {x ∈ Ω/m(x) > 0}. We consider the nonlinear eigenvalue problem

\begin{matrix}−Δ_{p}u = λm(x)|u|^{p−2}u in Ω, \\ \frac{∂u}{∂ν} = 0 on ∂Ω, \end{matrix}  (1.2)

where Δ_{p}u = div(|∇u|^{p−2}∇u) is the p-Laplacian operator.

The goal of this work is to prove the existence of a sequence of non trivial eigenvalues for the problem (1.2), and we prove the simplicity, isolation and monotonicity with respect to the weight of the first eigenvalue λ_{1} defined by

λ_{1} = inf{∥∇u∥_{p}^{p} : u ∈ W^{1,p}(Ω) and ∫_{Ω} m(x)|u|^{p}dx = 1}.  (1.3)

The semilinear elliptic problems has been treated by many authors; see, e.g. [3, 11, 12, 16] and the references therein. In the case of bounded weight: Anane [1] with Dirichlet boundary conditions and Dakkak [8] with Neumann boundary conditions, proved the existence, simplicity and isolation of the first eigenvalue. Cuesta [7] (for the p-Laplacian) and in Touzani [17] (for the A_{p}-Laplacian: A_{p}u = Σ_{i,j=1}^{N} \frac{∂}{∂x_{i}} (|∇u|_{a}^{p−2}a_{ij}(x)\frac{∂u}{∂x_{j}}), |ξ|_{a}^{2} = Σ_{i,j=1}^{N}a_{ij}(x)ξ_{i}ξ_{j}) studied the above properties in the case of Dirichlet problem with weight in L^{r}(Ω), with r satisfying (1.1). Cuesta [7] showed these results for any r satisfying the conditions (1.1), by using Harnack’s inequality and Picone’s identity. However, in [17], the isolation was establisehd with some appropriate condition on r (r > Np′) in order to use the regularity results established by Di Benedetto [9]. We will try to adapt these results to the case of Neumann boundary conditions.

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

Key words and phrases. Nonlinear eigenvalue problem; p-Laplacian; indefinite weight;

Neumann boundary value problem.

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

Published September 20, 2006.

This paper is organized as follows. In section 2, we recall some definitions and results that we will use later. In section 3, we prove that the problem (1.2) has a sequence of eigenvalues, by using a perturbation of the initial problem ([7]), and then by applying the Ljusternik-Schnirelmann theory ([4, 5]) to the perturbed problem. In section 4, we show simplicity, isolation and monotonicity of the first eigenvalue λ_{1}.

2. Preliminaries

Note by W^{1,p}(Ω) the Sobolev space with norm ∥.∥_{1,p} = (∥.∥_{p}^{p} + ∥∇(.)∥_{p}^{p})^{1/p}, where ∥.∥_{p} is the L^{p}-norm.

We say that λ ∈ ℝ is an eigenvalue of problem (1.2) if there exists u ∈ W^{1,p}(Ω)\ {0} such that

∫_{Ω} |∇u|^{p−2}∇u∇φ dx = λ ∫_{Ω} m(x)|u|^{p−2}uφ dx ∀φ ∈ W^{1,p}(Ω).  (2.1)

Theorem 2.1 ([1]). Let v > 0 and u ≥ 0 be two continuous functions in Ω, differentiable a.e, and

L(u, v) = |∇u|^{p} + (p − 1)\frac{u^{p}}{v^{p}} |∇v|^{p} − p\frac{u^{p−1}}{v^{p−1}} |∇v|^{p−2}∇v∇u,

R(u, v) = |∇u|^{p} − |∇v|^{p−2}∇(\frac{u^{p}}{v^{p−1}} )∇v.

Then we have

(i) L(u,v) =R(u,v) (ii) L(u, v) ≥ 0 a.e. in Ω (iii) L(u, v) = 0 a.e. in Ω if and only if there exists k ∈ ℝ such that u = kv .

Proposition 2.2. Let u ∈ W^{1,p}(Ω) be an eigenfunction associated to λ then

(i) u ∈ L^{∞}(Ω)

(ii) u is locally Hölder continuous; i.e., there exists α = α(p, N, ∥λm∥_{r}) in ]0, 1[ such that for each Ω′ ⊂ Ω, there exists C = C(p, N, ∥λm∥_{s}, dist(Ω′, ∂Ω)) such that

|u(x) − u(y)| ≤ C∥u∥_{∞}|x − y|^{α} ∀x, y ∈ Ω′.

The proof of (i) can be found in [10, propositions 1.2 and 1.3], and of (ii) in [14, theorem 8].

Proposition 2.3. Let u ∈ W^{1,p}(Ω) be a nonnegative weak solution of (1.2), then either u ≡ 0 or u(x) > 0 for all x ∈ Ω.

The proof is a direct consequence of Harnack’s inequality (see [15, theorem 5, 6, 9]).

3. Existence of solutions

In this section, we establish the existence of solutions by using a perturbation of problem (1.2) in order to use the general theory of nonlinear eigenvalue problems. So, let us consider the perturbed problem

\begin{matrix}−Δ_{p}u + ε|u|^{p−2}u = λm(x)|u|^{p−2}u in Ω, \\ \frac{∂u}{∂ν} = 0 on ∂Ω, \end{matrix}  (3.1)

where ε is enough small (0 < ε < 1), λ > 0. We first show that problem (3.1) has at least one sequence of eigenvalues, and deduce the solutions of problem (1.2) when ε tends to 0.

Let X = W^{1,p}(Ω) , G_{ε}(u) = \frac{1}{p}∥∇u∥_{p}^{p} + \frac{ε}{p}∥u∥_{p}^{p}, and F (u) = \frac{1}{p} ∫_{Ω} m(x)|u|^{p}dx. It is well known that F and G_{ε} are differentiable [8]. The problem (3.1) is equivalent to the problem G′_{ε}(u) = λF′(u). Let us consider the functional Φ_{ε} : W^{1,p}(Ω) → ℝ defined by Φ_{ε}(v) = (G_{ε}(v))^{2} − F (v).

Lemma 3.1. The eigenvalues and eigenfunctions associated to the problem (3.1) are entirely determined by a non trivial critical values of Φ_{ε}.

Proof. Let u ≢ 0 be a critical point of Φ_{ε} associated with a critical value c_{ε} then Φ_{ε}(u) = c_{ε} and Φ′_{ε}(u) = 0, i.e c_{ε} = −(G_{ε}(u))^{2} < 0 and ⟨Φ′_{ε}(u), v⟩ = \frac{1}{2\sqrt{−c_{ε}}} ⟨F′(u), v⟩ for any v ∈ C_{c}^{∞}(Ω). Thus we deduce that λ = \frac{1}{2\sqrt{−c_{ε}}} is a positive eigenvalue of (3.1) and u is its associated eigenfunction.

Conversely, let (u ≢ 0, λ) be a solution of (3.1), then for every β ∈ ℝ^{∗}, βu is also an eigenfunction associated to λ. In particular for β = (\frac{1}{2λG_{ε}(u)})^{\frac{1}{p}}, v =

(2λG_{ε}(u))^{− \frac{1}{p}} u is an eigenfunction associated to λ = \frac{1}{2\sqrt{−c_{ε}}}, thus v is a critical point associated to the critical value c_{ε} = −\frac{1}{4λ^{2}} . □

Let us now consider the sequence

c_{n,ε} = inf_{K∈A_{n}} sup_{v∈K} Φ_{ε}(v),  (3.2)

where A_{n} = {K ⊂ W^{1,p}(Ω) : K is compact symmetric and γ(K) ≥ n}, n ≥ 1.

Theorem 3.2. The values c_{n,ε} defined by (3.2) are the critical values of Φ_{ε}, moreover c_{n,ε} < 0 for n ≥ 1 and lim_{n→∞} c_{n,ε} = 0.

Proof. The proof of this theorem is based on the fundamental theorem of multiplicity [6] and the approximation of Sobolev imbedding by operators of finite rank. We first show that for all n ≥ 1, c_{n,ε} is a critical value of Φ_{ε} and c_{n,ε} < 0.

Since ϕ_{ε} is even and is C^{1} on W^{1,p}(Ω), then the result follows from the fundamental theorem of multiplicity if Φ_{ε} satisfies the following conditions:

(i) Φ_{ε} is bounded below (ii) Φ_{ε} verify the Palais Smale condition (PS). (iii) for all n ≥ 1, there exists a compact symmetric subset K such that γ(K) =

n and sup_{v∈K}{Φ_{ε}(v)} < 0.

Let us verify assertion (i): Let us take ε fixed (0 < ε < 1) and let u ∈ W^{1,p}(Ω), then by Hölder’s inequality and the Sobolev imbeddings; there exist k_{1} > 0 , k_{2} > 0 and k_{3} > 0 such that:

∫_{Ω} m|u|^{p}dx ≤ \begin{cases}k_{1}∥m∥_{r}∥u∥_{1,p}^{p} & if 1 < p < N \\ k_{2}∥m∥_{r}∥u∥_{1,N}^{N} & if p = N \\ k_{3}∥m∥_{r}∥u∥_{1,p}^{p} & if p > N \end{cases}

In addition, note that ∥u∥_{1,p,ε} = (∥∇u∥_{p}^{p} + ε∥u∥_{p}^{p})^{\frac{1}{p}} defines a norm on W^{1,p}(Ω) equivalent to the usual norm on W^{1,p}(Ω). It is easy to see that there exists c_{1} > 0, c_{2} > 0 and c_{3} > 0 such that for Φ_{ε}, we have the following inequalities:

Φ_{ε}(u) ≥ \begin{cases}\frac{c_{1}}{p^{2}} ∥u∥_{1,p}^{p}(ε^{2}∥u∥_{1,p}^{p} − p∥m∥_{r}) & if 1 < p < N \\ \frac{c_{2}}{N^{2}} ∥u∥_{1,N}^{N}(ε^{2}∥u∥_{1,N}^{N} − p∥m∥_{r}) & if p = N \\ \frac{c_{3}}{p^{2}} ∥u∥_{1,p}^{p}(ε^{2}∥u∥_{1,p}^{p} − p∥m∥_{r}) & if p > N \end{cases}

From where by treating each case one has: Φ_{ε} is bounded below and Φ_{ε}(u) → +∞ as ∥u∥_{1,p} → +∞.

(ii) Let us now show that Φ_{ε} verify the palais Smale condition: Let (u_{n})_{n} be a sequence in W^{1,p}(Ω) such that (Φ(u_{n}))_{n} is bounded and Φ′(u_{n}) → 0 in (W^{1,p}(Ω))′. Since Φ_{ε} is coercive then (u_{n})_{n} is bounded in W^{1,p}(Ω), thus there exists a subsequence still denoted (u_{n}) such that u_{n} converges to u strongly in L^{p}(Ω) and weakly in W^{1,p}(Ω). Suppose that ∥u_{n}∥_{1,p} converges to α ≥ 0. We distinguish two cases: Case 1: α = 0. Since u_{n} ⇀ u in W^{1,p}(Ω) and ∥u_{n}∥_{1,p} → 0 then u_{n} → 0 in W^{1,p}(Ω), consequently, the condition (PS) is satisfied. Case 2: α > 0. For n ≥ 1 we have:

Φ′_{ε}(u_{n}) = 2G_{ε}(u_{n})G′_{ε}(u_{n}) − F′(u_{n})

then

G′_{ε}(u_{n}) = \frac{1}{2G_{ε}(u_{n})}(Φ′_{ε}(u_{n}) + F′(u_{n}))

i.e.,

[\frac{p}{2}(Φ′_{ε}(u_{n}) + m|u_{n}|^{p−2}u_{n})]/(∥∇u_{n}∥_{p}^{p} + ε∥u_{n}∥_{p}^{p}) = G′_{ε}(u_{n}).

Since u ↦ m|u_{n}|^{p−2}u is strongly continuous, ∥u_{n}∥_{1,p} → α > 0 and Φ′_{ε}(u_{n}) → 0, then the expression

û_{n} = [\frac{2}{p}(Φ′_{ε}(u_{n}) + m|u_{n}|^{p−2}u_{n})]/(∥∇u_{n}∥_{p}^{p} + ε∥u_{n}∥_{p}^{p})

converges strongly in (W^{1,p}(Ω))′. However, G′_{ε} is continuous, thus u_{n} = (G′_{ε})^{−1}û_{n} converges strongly in W^{1,p}(Ω), from where the (PS) condition holds.

(iii) For all n ≥ 1, there exists a compact symmetric subset K such that

γ(K) = n and sup_{u∈K} Φ_{ε}(u) < 0.

Indeed, since meas(Ω^{+}) ≠ 0, then there exists a family of balls (B_{i})_{1≤i≤n} in Ω such that B_{i} ∩ B_{j} = ∅ if i ≠ j and meas(Ω^{+} ∩ B_{i}) ≠ 0. By approximating the characteristic function χ_{Ω^{+}∩B_{i}} by C_{c}^{∞}(Ω) functions in L^{p}, there exists a sequence (u_{i})_{1≤i≤n} ⊂ C_{c}^{∞}(B_{i}) such that ∫_{Ω} m|u_{i}|^{p}dx > 0 for all i = 1, …, n. We normalize u_{i} in order to have F (u_{i}) = 1. Let X_{n} be the subset generated by (u_{i})_{i}. For all u ∈ X_{n}, we have u = Σ_{i=1}^{n}α_{i}u_{i}, and F (u) = Σ_{i=1}^{n}|α_{i}|^{p}. Then u ↦ (F (u))^{\frac{1}{p}} define one norm on X_{n}; moreover, since ε is fixed it follows that (∥∇u∥_{p}^{p} + ε∥u∥_{p}^{p})^{1/p} is also a norm on W^{1,p}(Ω). However, the dimension of X_{n} is finite, then there exists c > 0 such that

cF (u) ≤ G_{ε}(u) ≤ \frac{1}{c}F (u) ∀ u ∈ X_{n}.

Let K be defined as

K = \{u ∈ W^{1,p}(Ω) such that \frac{c^{2}}{3} ≤ F (u) ≤ \frac{c^{2}}{2} \}

It is clear that K_{1} = K ∩ X_{n} ≠ ∅ and sup_{u∈K_{1}} Φ_{ε}(u) ≤ −\frac{c^{2}}{12} < 0. Since X_{n} is isomorphous to ℝ^{n}, one can identify K_{1} to a crown K′_{1} of ℝ^{n} such that S^{n−1} ⊂ K′_{1} ⊂ ℝ^{n} \ {0} where S^{n−1} is the unit sphere of ℝ^{n}. then γ(K_{1}) = n and the result follows.

For the proof of lim_{n→∞} c_{n,ε} = 0, we use an approximation of Sobolev imbeddings by operators of finite rank; see for example [17]. □

Remark 3.3. It is clear that the sequence (λ_{n,ε})_{n} defined by the formula λ_{n,ε} = \frac{1}{2\sqrt{−c_{n,ε}}} for all n ≥ 1 is a sequence of positive eigenvalues of (3.1) which satisfy

lim_{n→+∞} λ_{n,ε} = +∞.

Next, we show that problem (1.2) has a sequence of eigenvalues (λ_{n})_{n} which is the limit of the sequence (λ_{n,ε})_{n}, as ε → 0 .

Remark 3.4. We write λ_{n,ε} as λ_{n,ε} = inf_{K∈Γ_{n}} sup_{u∈K} G_{ε}(u), where

Γ_{n} = {K ⊂ W^{1,p}(Ω) \ {0} : K is compact, symmetric γ(K) ≥ n, ∫ m|u|^{p} = 1}.

For more details see [8]. Put

G(u) = \frac{1}{p}∥∇u∥_{p}^{p} and λ_{n} = inf_{K∈A_{n}} sup_{u∈K} G(u).

Lemma 3.5. The following assertions hold:

(i) lim_{ε→0} λ_{n,ε} = λ_{n}

(ii) λ_{n} → +∞ as n → +∞.

Proof. (i) Let ε > 0, from Remark 3.4, we have λ_{n,ε} ≥ λ_{n}. Let α > 0 such that λ_{n} < α. Then there exists K = K(α) ∈ A_{n} such that λ_{n} ≤ sup_{u∈K} G(u) < α. Set δ = sup_{u∈K} ∥u∥_{p}. Then

λ_{n} ≤ λ_{n,ε} ≤ sup_{u∈K} G_{ε}(u) ≤ sup_{u∈K} G(u) + \frac{εδ}{p}

For ε = \frac{1}{k} → 0, there exists k(α) such that for every k ≥ k(α), we obtain sup_{u∈K} G(u) + \frac{δ}{kp} ≤ α. Thus λ_{n} ≤ λ_{n,ε} ≤ α for all k ≥ k(α). From where we obtain the desired result.

(ii) From (3.2), we have for all m, m′ ∈ L^{r}(Ω) such that m′ ≥ m; c_{n,ε}(m) ≤ c_{n,ε}(m′), thus λ_{n,ε}(m) ≥ λ_{n,ε}(m′). When ε → 0, we obtain λ_{n}(m′) ≤ λ_{n}(m). Let δ > 0, and set

m′(x) = \begin{cases}m(x) & if m(x) ≥ δ \\ δ & if m(x) < δ \end{cases}

It is clear that λ_{n,ε}(m′) ≤ λ_{n}(m′)+ε/δ. Since (λ_{n,ε}(m′))_{n} ↗ ∞, lim_{n→∞} λ_{n}(m′) = +∞. The result follows. □

Corollary 3.6. (λ_{n})_{n} is a sequence of positives eigenvalues associated to the Neumann problem (1.2).

Proof. Let n ∈ ℕ^{∗} fixed and ε = \frac{1}{k}, k ∈ ℕ^{∗}. There exists a sequence (u_{k})_{k∈ℕ^{∗}} of eigenfunctions associated with (λ_{n,k})_{k∈ℕ^{∗}} which verify G_{k}(u_{k}) + ∥u_{k}∥_{p}^{p} = 1 then (u_{k})_{k} is bounded in W^{1,p}(Ω). Hence for a subsequence (u_{k})_{k}, (u_{k})_{k} converges strongly in L^{p}(Ω), and weakly in W^{1,p}(Ω) towards a limit u. u_{k} → u. Indeed, set J(u) = |u|^{p−2}u. The operator G′ + J is of S^{+} type and G′ + J : W^{1,p}(Ω) → (W^{1,p}(Ω))′ is an homeomorphism (see [13]), then (u_{k})_{k} converges strongly to u. However, G′(u_{k}) + \frac{1}{k} |u_{k}|^{p−2}u_{k} = λ_{n,k}F′(u_{k}) and F′ is strongly continuous on W^{1,p}(Ω), thus G′(u)= λ_{n}F′ and G(u) + ∥u∥_{p}^{p} = 1. Consequently, (u, λ_{n}) is a solution of (1.2). □

4. On the first eigenvalue

In this section, we are going to prove some properties of the first eigenvalue λ_{1} of problem (1.2) defined by (1.3). We refer in this section to the work of Cuesta [7]; so we prove that λ_{1} is simple, isolated and strictly monotone with respect to the weight.

Proposition 4.1. λ_{1} is an eigenvalue of problem (1.2). Moreover λ_{1} > 0 if and only if m changes its sign on Ω and ∫_{Ω} m(x)dx < 0.

The proof of the above proposition is a straight application of [15, Theorem 1.2].

Proposition 4.2. The eigenfunctions associated with λ_{1} > 0 are either positive or negative in Ω.

Proof. Let u be an eigenfunction associated to λ_{1}. Without loss of generality, we can suppose that u ∈ M = {u ∈ W^{1,p}(Ω) : ∫_{Ω} m|u|^{p} = 1}. Then, the infimum in (1.3) is achieved at one u ∈ W^{1,p}(Ω). It’s easy to prove that ∥∇|u|∥_{p} = ∥∇u∥_{p} and |u| ∈ M, it follows then that |u| is an eigenfunction associated to λ_{1}, and therefore, from Proposition 2.3, the result holds. □

Proposition 4.3. Any eigenfunction associated with a positive eigenvalue λ ≠ λ_{1} changes its sign.

Proof. For the proof, we use the Picone’s identity (see Theorem 2.1). Indeed, let (v, λ), (u, λ_{1}) be two positive solutions of (1.2) and suppose that u > 0. Suppose also that v ≥ 0 (the case v ≤ 0 is proved similarly). By Theorem 2.1, we have

∫_{Ω} L(u, v + ε)dx = ∫_{Ω} R(u, v + ε)dx ≥ 0 ∀ε > 0.

However,

∫_{Ω} R(u, v + ε)dx = ∫_{Ω}[|∇u|^{p} − |∇v|^{p−2}∇(\frac{u^{p}}{(v + ε)^{p−1}} )∇v]dx,

By taking \frac{u^{p}}{(v+ε)^{p−1}} as a test function in the formula ⟨−Δ_{p}v, φ⟩ = λ⟨m|v|^{p−2}v, φ⟩, we obtain

∫_{Ω}[λ_{1}mu^{p} − λmv^{p−1} \frac{u^{p}}{(v + ε)^{p−1}} ]dx ≥ 0

i.e.,

∫_{Ω} mu^{p}(λ_{1} − λ\frac{v^{p−1}}{(v + ε)^{p−1}} )dx ≥ 0.

Let ε goes to 0, we obtain ∫_{Ω} mu^{p}(λ_{1}−λ)dx ≥ 0 which is impossible because λ > λ_{1} and ∫_{Ω} mu^{p}dx > 0. Hence, v changes its sign on Ω. □

Proposition 4.4 (Simplicity). λ_{1} is simple in the sense that if (u, λ_{1}) and (v, λ_{1}) are two solutions associated to the problem (1.2), then there exists α ∈ ℝ such that u = αv.

The proof of the above proposition is the same as in the preceding proposition. We use the assertion (iii) of Theorem 2.1 (see preliminaries section).

Now, we establish the isolation of the first eigenvalue. We use for this the same method as in [2, 7, 17]. We will show first the following estimates:

Proposition 4.5. For m ∈ L^{r}(Ω) with r satisfying the conditions (1.1), the following estimates hold:

min(meas(Ω^{+}), meas(Ω^{−})) ≥ (λC^{p}∥m∥_{r})^{σ}

with

σ = \begin{cases}\frac{rN}{N−rp} & if 1 < p < N \\ \frac{r}{1−N} & if p = N \\ \frac{N}{N−p} & if p > N \end{cases}  (4.1)

Proof. Let u be an eigenfunction associated with λ, then

∫_{Ω} |∇u|^{p−2}∇u∇v dx = λ ∫_{Ω} m(x)|u|^{p−2}uv dx ∀v ∈ W^{1,p}(Ω).  (4.2)

For λ ≠ λ_{1}, u changes its sign i.e. u^{+} ≠ 0 and u^{−} ≠ 0, so, since u^{+} ∈ W^{1,p}(Ω) we have

∫_{Ω} |∇u^{+}|^{p}dx = λ ∫_{Ω} m(x)|u^{+}|^{p}.  (4.3)

Case 1: 1 < p < N. By Hölder inequality, we have

λ ∫_{Ω} m|u^{+}|^{p}dx ≤ ∥m∥_{r}∥u^{+}∥_{p^{∗}}^{p}(meas(Ω^{+}))^{1− \frac{1}{r} − \frac{p}{p^{∗}}}

where

p^{∗} = \begin{cases}\frac{pN}{N−p} & if p < N \\ +∞ & if p ≥ N \end{cases}

On the other hand, using Sobolev imbeddings, there exists a constant C depending on p and N such that ∥u^{+}∥_{p^{∗}} ≤ C(∫_{Ω} |∇u^{+}|^{p})^{\frac{1}{p}} dx. Hence from (4.3) it follows that

∥∇u^{+}∥_{p}^{p} ≤ λC^{p}∥m∥_{r}∥u^{+}∥_{1,p}^{p}(meas(Ω^{+}))^{\frac{rp−N}{rN}}

and meas(Ω^{+}) ≥ (λC^{p}∥m∥_{r})^{\frac{rN}{N−rp}}. Case 2: p = N. In this case, we proceed in the same way as previously. On the one hand, we have

λ ∫_{Ω} m|u^{+}|^{N}dx ≤ λ∥m∥_{r}∥u^{+}∥_{N}^{N−1}∥u^{+}∥_{s}

where s = \frac{Nr}{r−N}. From Hölder inequality, we have

∥u^{+}∥_{N}^{N−1} ≤ ∥u^{+}∥_{s}^{N−1}(meas(Ω^{+}))^{(1− \frac{N}{s})( \frac{N−1}{N})}

On the other hand, while s > 1, by the Sobolev imbedding W^{1,N} ↪ L^{s}(Ω), there exists C > 0 such that ∥u^{+}∥_{s} ≤ C∥∇u^{+}∥_{N}. However, it results

∥∇u^{+}∥_{N}^{N} ≤ λC^{N}∥m∥_{r}∥∇u^{+}∥_{N}^{N}(meas(Ω^{+}))^{(1− \frac{N}{s})( \frac{N−1}{N})}.

Finally, we obtain meas(Ω^{+}) ≥ (λC^{N}∥m∥_{r})^{\frac{r}{1−N}}. Case 3: p > N. In this case r = 1. From (4.3), we have

∫_{Ω} |∇u^{+}|^{p}dx ≤ λ∥m∥_{1}∥u^{+}∥_{∞}^{p},

and by Morrey’s lemma, there exists C > 0 depending on p and N such that

∥u^{+}∥_{∞} ≤ C(meas(Ω^{+}))^{\frac{1}{N} − \frac{1}{p}} ∥∇u^{+}∥_{p}.

Thus, meas(Ω^{+}) ≥ (λC^{p}∥m∥_{1})^{\frac{N}{N−p}}.

In conclusion , in the three cases we obtain the desired estimate for u^{+}. By proceeding in the same way for u^{−}, from (4.3) (for u^{−}), we deduce the same estimates for u^{−}. From where the proposition 4.5 holds. □

The following proposition is a consequence of the previous proposition.

Proposition 4.6 (Isolation). λ_{1}(m) is isolated, that is, there exists β > λ_{1} such that if λ ∈]0, β[ then λ = λ_{1} or λ is not an eigenvalue associated to (1.2).

Proof. Let u be an eigenfunction associated with λ ∈]0, β[. Then ∫_{Ω} |∇u|^{p}dx = λ ∫_{Ω} m(x)|u|^{p}. Since ∫_{Ω} |∇u|^{p}dx ≥ λ_{1} ∫_{Ω} m(x)|u|^{p}, it follows that λ_{1} ≤ λ; thus λ_{1} is isolated on the left. Assume now by contradiction that there exists a sequence of eigenvalues of (1.2) (λ_{n})_{n} decreasing to λ_{1}, thus one has a sequence of eigenfunctions u_{n} associated with λ_{n}. We can suppose that ∥∇u_{n}∥ = 1, for all n ∈ ℕ. (u_{n}) is bounded in W^{1,p}(Ω), so there exists a subsequence (still denoted u_{n}) such that u_{n} converges weakly in W^{1,p}(Ω) and strongly in L^{p}(Ω) to a limit u ∈ W^{1,p}(Ω). On the other hand, we have

∫_{Ω} |∇u|^{p}dx ≥ lim inf_{n→∞} ∫_{Ω} |∇u_{n}|^{p}dx = λ_{1}

then by the definition of λ_{1}, we conclude that ∫_{Ω} |∇u_{n}|^{p}dx = λ_{1} and u is an eigenfunction associated to λ_{1}. It follows then by Proposition 4.4 that either u > 0 or u < 0. Suppose that u > 0 (the other case is analogous). By Egorov and Lusin theorem, we obtain for every ε > 0, there exists N_{ε} > 0 such that

meas({x ∈ Ω, u_{n}(x) > 0}) ≥ meas(Ω) − ε, ∀n ≥ N_{ε}.

Then for ε suitably chosen by the estimates (4.1) (for more details see [2, p. 24]), we obtain a contradiction with the estimates (4.1) related to meas(Ω^{−}). □

The last proposition in this paper deals with the monotonicity of λ_{1} with respect to the weight.

Proposition 4.7. λ_{1} verify the monotonicity and the monotonicity strict with respect to the weight, i.e. if m′ ≤ m then λ_{1}(m) ≤ λ_{1}(m′), moreover if m′ ≤ m and meas({x ∈ Ω : m′(x) < m(x)}) ≠ 0 then λ_{1}(m) < λ_{1}(m′).

Proof. Let u > 0 be an eigenfunction associated to λ_{1}(m′). By λ_{1}(m′) definition we have

0 < \frac{1}{(λ_{1}(m′))} ∫_{Ω} |∇u|^{p}dx = ∫_{Ω} m′u^{p}dx ≤ ∫_{Ω} mu^{p}dx.

Since

λ_{1}(m) = inf{∥∇u∥_{p}^{p}/u ∈ W^{1,p}(Ω) and ∫_{Ω} mu^{p}dx = 1}

and

\frac{u}{(∫_{Ω} mu^{p}dx)^{1/p}} ∈ W^{1,p}(Ω) verifies ∫_{Ω} m(\frac{u^{p}}{∫_{Ω} mu^{p}dx})dx = 1

it follows that

λ_{1}(m) ≤ \frac{∫_{Ω} |∇u|^{p}dx}{∫_{Ω} mu^{p}dx} ≤ \frac{∫_{Ω} |∇u|^{p}dx}{∫_{Ω} m′u^{p}dx} = λ_{1}(m′)

and from where it follows that λ_{1}(m) ≤ λ_{1}(m′). The equality holds if and only if

∫_{Ω} mu^{p}dx = ∫_{Ω} m′u^{p}dx.

However, u > 0 thus m ≡ m′, which is a contradiction with meas({x ∈ Ω/m′(x) < m(x)}) ≠ 0. □

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Siham El Habib University Mohamed 1er, Faculty of sciences, Department of Mathematics, Oujda, Morocco

E-mail address: s-elhabib@hotmail.com

Najib Tsouli University Mohamed 1er, Faculty of sciences, Department of Mathematics, Oujda, Morocco

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