Electronic Journal of Qualitative Theory of Differential Equations 2004, No. 9, 1-14; http://www.math.u-szeged.hu/ejqtde/

A GLOBAL BIFURCATION RESULT OF A NEUMANN PROBLEM WITH INDEFINITE WEIGHT

ABDELOUAHED EL KHALIL & MOHAMMED OUANAN

Abstract. This paper is concerned with the bifurcation result of nonlinear Neumann problem

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

We prove that the principal eigenvalue λ_{1} of the corresponding eigenvalue problem with f ≡ 0, is a bifurcation point by using a generalized degree type of Rabinowitz.

1. Introduction

The purpose of this paper is to study a bifurcation phenomenon for the following nonlinear elliptic problem

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

where Ω is a bounded domain of IR^{N}, N ≥ 1, with smooth boundary and ν is the unit outward normal vector on ∂Ω; the weight function m belongs to L^{∞}(Ω) and λ is a parameter. We assume that ∫_{Ω} m(x)dx < 0 and |Ω^{+}| ≠ 0 with Ω^{+} = {x ∈ Ω; m(x) > 0}, where | . | is the Lebesgue measure of IR^{N}. The so-called p-Laplacian is defined by −Δ_{p}u = −∇.(|∇u|^{p−2}∇u) which occurs in many mathematical models of physical processes as glaciology, nonlinear diffusion and filtration problem, see [18], power-low materials [2], the mathematical modelling of non-Newtonian fluids [1]. For a discussion of some physical background, see [10]. In this context and for certain physical motivations, see for example [17]. Observe that in the particular case f ≡ 0 and p = 2, (P) cames linear.

The nonlinearity f is a function satisfying some conditions to be specified later.

Classical Neumann problems involving the p-Laplacian operator have been studied by many authors. Senn and Hess [14, 15] studied an eigenvalue problem with Neumann boundary condition. Bandele, Pizio and Tesei

EJQTDE, 2004, No. 9, p. 1 [4] studied the existence and uniqueness of positive solutions of some nonlinear Neumann problems; we cite also the paper [6] where the authors studied the role played by the indefinite weight on the existence of positive solution. In [16], the author shows that the first positive eigenvalue λ_{1}, of

2000 Mathematics Subject Classification. 35P20, 35J70, 35J20.

Key words and phrases. p-Laplacian operator, Neumann condition, principal eigenvalue, indefinite weight, topological degree, bifurcation point, variational method.

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

is well defined and if ∫_{Ω} m(x) dx < 0, it is simple and isolated. These fundamental properties will be used in proof of our main bifurcation result.

In general, variational method and bifurcation theory have been used in pure and applied mathematics to establish the existence, multiplicity and structure of solutions to Partial Differential Equations. However, the relationship between these two methods have remained largely unrecognized and searchers have tended to use one method or the other. The present paper gives an example of nonlinear partial differential equation with Neumann boundary condition, expanding variational and bifurcation methods to occur the connection between these two distinct ”arguments”.

In recent years, bifurcation problems with a particular with a particular nonlinearity were studied by several authors, with the right hand side of the first equation of the form f and the Direchlet boundary condition. In fact, bifurcation Direchlet boundary condition problems with other conditions on m and f were studied on bounded smooth domains by [5] and [9]. These results were extended for any bounded domain and m is only locally bounded by [11] and [12]. The authors considered the bifurcation phenomena, namely on the interior of domain. The case Ω = IR^{N} was treated by Dràbek and Huang [13] under some appropriate hypotheses.

The purpose of this paper is to study the bifurcation phenomenon from

the first eigenvalue of (E) when ∫_{Ω} m(x) dx < 0, by using a combination

of topological and variational methods. Our main result is formulated by Theorem 3.2, where we investigate the situation improving the conditions of the nonlinearity f for Neumann boundary condition. In Proposition 3.1, we give a characterization of the bifurcation points of (P) related to the spectrum of (E). We establish the existence of a global branch of nonlinear solutions pairs (λ, u), with u ≠ 0, bifurcating from the trivial branch at λ = λ_{1}. Bifurcation here means that there is a sequence of nontrivial solutions (λ, u), with u ≠ 0, going to zero as λ approaches the right eigenvalues.

The rest of the paper is organized as follows: Section 2 is devoted to statement of some assumptions and notations which we use later and prove some technical preliminaries; in Section 3 we verify that the topological degree is well defined for our operators in order to be able to show that this degree has a jump, when λ crosses λ_{1}, which implies the bifurcation result.

EJQTDE, 2004, No. 9, p. 2 In fact, we may employ the global bifurcation result of that of Rabinowitz [19].

2. Assumptions and Preliminaries

We first introduce some basic definitions, assumptions and notations. Here p > 1, Ω is a bounded domain in IR^{N}, (N ≥ 1) with a smooth boundary. W^{1,p}(Ω) is the usual Sobolev space, equipped with the standard norm

∥u∥_{1,p} = (∫_{Ω} |∇u|^{p} dx + ∫_{Ω} |u|^{p} dx)^{\frac{1}{p}} , u ∈ W^{1,p}(Ω).

2.1. Assumptions. We make the following assumptions: f : Ω × IR × IR ⟶ IR is a Carathéodory’s function, satisfying the homogenization condition type

f(λ, x, s) = o(|s|^{p−1}), as s → 0  (2.1)

uniformly a.e. with respect to x ∈ Ω and uniformly with respect to λ in any bounded subset of IR. Moreover f satisfies the asymptotic condition: There is q ∈ (p, p^{∗}) such that

lim_{|s|→+∞} \frac{f(λ, x, s)}{|s|^{q−1}} = 0,  (2.2)

uniformly a.e. with respect to x ∈ Ω and uniformly with respect to λ in any bounded subset of IR. Here p^{∗} is the critical Sobolev exponent defined by

p^{∗} = \{ \begin{matrix}\frac{Np}{N−p} & if & N > p \\ +∞ & if & N ≤ p, \end{matrix}

2.2. Definitions. 1. By a solution of (P), we understand a pair (λ, u) in IR × W^{1,p}(Ω) satisfying (P) in the weak sense, i.e.,

∫_{Ω} |∇u|^{p−2}∇u∇v dx = λ ∫_{Ω} m(x)|u|^{p−2}uv dx + ∫_{Ω} f(λ, x, u)v dx, (2.3)

for all v ∈ W^{1,p}(Ω). This is equivalent to saying that u is a critical point of the energy functional corresponding to (P) defined as

\frac{1}{p} ∫_{Ω} |∇u|^{p} dx − \frac{λ}{p} ∫_{Ω} m(x)|u|^{p}dx − ∫_{Ω} F (λ, x, u)dx,

where F denoted the Nemitskii operator associated to f. In other words, F is the primitive of f with respect to the third variable, i.e., F (λ, x, u) =

∫_{0}^{u} f(λ, x, s) ds. We note that the pair (λ, 0) is a solution of (P) for every

λ ∈ IR. The pairs of this form will be called the trivial solutions of (P). We say that P = (μ, 0) is a bifurcation point of (P), if in any neighborhood of P in IR × W^{1,p}(Ω) there exists a nontrivial solution of (P).

EJQTDE, 2004, No. 9, p. 3 2. Throughout, we shall denote by X a real reflexive Banach space and by X′ stand for its dual with respect to the pairing ⟨., .⟩. We shall deal with mapping T acting from X into X′. T is demicontinuous at u in X, if u_{n} → u strongly in X, implies that T u_{n} ⇀ T u weakly in X′. T is said to belong to the class (S_{+}), if for any sequence u_{n} weakly convergent to u in X and lim sup_{n→+∞}⟨T u_{n}, u_{n} − u⟩ ≤ 0, it follows that u_{n} → u strongly in X.

2.3. Degree theory. If T ∈ (S_{+}) and T is demicontinuous, then it is possible to define the degree Deg[T; D, 0], where D ⊂ X is a bounded open set such that T u ≠ 0 for any u ∈ ∂D. Its properties are analogous to the ones of the Leray-Schauder degree (cf. [7]).

Assume that T is a potential operator, i.e., for some continuously differentiable functional Φ : X → IR, Φ′(u) = T u, u ∈ X. A point u_{0} ∈ X will be called a critical point of Φ if Φ′(u_{0}) = 0. We say that u_{0} is an isolated critical point of Φ if there exists ϵ > 0 such that for any u ∈ B_{ϵ}(u_{0}), Φ′(u) ≠ 0 if u ≠ u_{0}. Then, the limit

Ind(T, u_{0}) = lim_{ϵ→0^{+}} Deg[Φ′; B_{ϵ}(u_{0}), 0]

exists and is called the index of the isolated critical point u_{0}, where B_{r}(w) denotes the open ball of radius r in X centered at w.

Now, we can formulate the following two lemmas which we can find in [20].

Lemma 2.1. Let u_{0} be a local minimum and an isolated critical point of Φ. Then

Ind(Φ′, u_{0}) = 1.

Lemma 2.2. Assume that ⟨Φ′(u), u⟩ > 0 for all u ∈ X, ∥u∥_{X} = r. Then

Deg[Φ′; B_{ρ}(0), 0] = 1.

2.4. Preliminaries. Let us define, for (u, v) ∈ W^{1,p}(Ω) × W^{1,p}(Ω), the operators

A_{p}, G : W^{1,p}(Ω) → (W^{1,p}(Ω))′

and

F : IR × W^{1,p}(Ω) → (W^{1,p}(Ω))′
⟨A_{p}u, v⟩ = ∫_{Ω} |∇u|^{p−2}∇u∇v dx
⟨Gu, v⟩ = ∫_{Ω} m(x)|u|^{p−2}uv dx
⟨F (λ, u), v⟩ = ∫_{Ω} f(λ, x, u)v dx

Remark 2.1. (i) Due to (2.3) a function u is a weak solution of (P) if, and only if,

A_{p}u − λGu − F (λ, u) = 0 in (W^{1,p}(Ω))′  (2.4)

(ii) The operator A_{p} has the following properties:

(a) A_{p} is odd, (p − 1)-homogeneous and strictly monotone, i.e.,

⟨A_{p}u − A_{p}v, u − v⟩ > 0 ∀ u ≠ v.

(b) A_{p} ∈ (S_{+}).

Lemma 2.3. G is well defined, compact, odd and (p − 1)-homogeneous.

Proof. The definition and compactness of G are required by the compactness of Sobolev embedding

W^{1,p}(Ω) ↪ L^{p}(Ω).

The oddness and (p − 1) homogeneity of G are obvious. Thus, the lemma is proved.

Lemma 2.4. For any λ ∈ IR, the Nemitskii operator F (λ, .) is well defined, compact and F (λ, 0) = 0. Moreover, we have

lim_{∥u∥_{1,p}→0} \frac{F (λ, u)}{∥u∥_{1,p}^{p−1}} = 0 in (W^{1,p}(Ω))′,  (2.5)

uniformly for λ in any bounded subset of IR.

Proof. Conditions (2.1) and (2.2) imply that for any ϵ > 0, there are two reals δ = δ(ϵ) and M = M(δ) > 0 such that for a.e., x ∈ Ω, we have

|f(λ, x, s)| ≤ ϵ|s|^{p−1} for |s| ≤ δ  (2.6)

and

|f(λ, x, s)| ≤ M|s|^{q−1} for |s| ≥ δ.  (2.7)

Therefore, for 0 < ϵ ≤ 1, we get by integration on Ω that

∫_{Ω} |f(λ, x, u(x))|^{q′} dx ≤ ∫_{Ω} |u(x)|^{q′(p−1)} dx + M ∫_{Ω} |u(x)|^{q} dx.  (2.8)

We have q′(p − 1) ≤ p′(p − 1) = p < q. Thus L^{q}(Ω) ↪ L^{q′(p−1)}(Ω) and there is a constant c > 0 such that

∫_{Ω} |u(x)|^{q′(p−1)}dx ≤ c ∫_{Ω} |u(x)|^{q}dx.  (2.9)

Inserting (2.9) in (2.7), we deduce the estimate

∫_{Ω} |f(λ, x, u(x))|^{q′}dx ≤ (c + M) ∫_{Ω} |u(x)|^{q} dx.  (2.10)

Thus F (λ, .) maps L^{q}(Ω) into its dual L^{q′}(Ω) continuously ( for more detail on the properties of Nemitskii operator the reader can see [8] ). Moreover,

EJQTDE, 2004, No. 9, p. 5 if u_{n} ⇀ u in W^{1,p}(Ω) then u_{n} → u in L^{q}(Ω), because p < q < p^{∗} and F (λ, u_{n}) → F (λ, u) in L^{q′}(Ω). Since L^{q′}(Ω) ↪ (W^{1,p}(Ω))′, it follows that F (λ, u_{n}) → F (λ, u) in (W^{1,p}(Ω))′. This implies that F (λ, .) is compact. It is not difficult to verify that F (λ, 0) = 0, for all λ ∈ IR.

In virtue of (2.1), we have \frac{F (λ, u)}{∥u∥_{1,p}^{p−1}} → 0 in L^{q′}(Ω), as u → 0 in W^{1,p}(Ω).

Indeed, set \overline{u} = \frac{u}{∥u∥_{1,p}} . Hence

\frac{F (λ, u)}{∥u∥_{1,p}^{p−1}} = \frac{F (λ, u)}{|u|^{p−1}}|\overline{u}|^{p−1}.  (2.11)

From this and Hölder’s inequality, we deduce that

∫_{Ω} |\frac{F (λ, u)}{∥u∥_{1,p}^{p−1}} |^{q′} dx ≤ [∫_{Ω} (\frac{|F (λ, u(x))|}{|u(x)|^{p−1}})^{q′t} dx]^{\frac{1}{t}} [∫_{Ω} |\overline{u}|^{(p−1)q′t′}dx]^{\frac{1}{t′}} ,

(2.12)

for some t > 0 which satisfies

\frac{q′(p − 1)}{p^{∗}} < \frac{1}{t} < \frac{p^{∗} − (p − 1)q′}{p^{∗}} .  (2.13)

This is always possible, since p < q < p^{∗}. By (2.6) and (2.7), we conclude that

∥|\frac{F (λ, u)}{|u|^{p−1}} |^{q′} ∥_{t}^{t} ≤ ϵ|Ω| + M^{q′t} ∫_{Ω} |u|^{q′t(q−p)}dx, ∀ϵ > 0.  (2.14)

From this inequality and the fact that u → 0 in W^{1,p}(Ω), we have the limit

∥|\frac{F (λ, u)}{|u|^{p−1}} |^{q′} ∥_{t}^{t} → 0 as u → 0 in W^{1,p}(Ω).

On the other hand, \overline{u} belongs to L^{p^{∗}}(Ω) ( because ∫_{Ω} |\overline{u}|^{p^{∗}}dx ≤ c). Then,

we find a constant c > 0 such that

∥|\overline{u}|^{(p−1)q′}∥_{t′} ≤ c,

since q′t′(p − 1) < p^{∗} by (2.13). This completes the proof.

Remark 2.2. Note that every continuous map T : X ⟶ X′ is also demicontinuous. Note also, that if T ∈ (S_{+}) then (T +K) ∈ (S_{+}) for any compact operator K : X ⟶ X′.

Remark 2.3. λ is an eigenvalue of (E) if and only if, u ∈ W^{1,p}(Ω)\{0} solves

A_{p}u − λGu = 0.  (2.15)

EJQTDE, 2004, No. 9, p. 6

Now, define an operator T_{λ} : W^{1,p}(Ω) → (W^{1,p}(Ω))′; by T_{λ}u = A_{p}u − λGu − F (λ, u). In view of Lemma 2.3, Lemma 2.4, Remark 2.1 and Remark 2.2, it follows that, for ϵ > 0, sufficiently small, the degree

Deg[T_{λ}; B_{ϵ}(0), 0],  (2.16)

is well defined for any λ ∈ IR such that T_{λ}u ≠ 0 for any ∥u∥_{1,p} = ϵ ( here B_{ϵ}(0) is the open ball of radius ϵ in W^{1,p}(Ω) centered at 0 ). By using the same argument as used in proof of Lemma 2.3, we can state the following proposition which plays a crucial role in our bifurcation result.

Proposition 2.1. If (μ, 0) is a bifurcation point of problem (P), then μ is an eigenvalue of (E).

Proof. Fix μ ∈ IR. Since (μ, 0) ∈ IR × W^{1,p}(Ω) is a bifurcation of (P) there exists a sequence {(λ_{j}, u_{j})}_{j} ⊂ IR × W^{1,p}(Ω) of nontrivial solutions of the problem (P) such that

λ_{j} → μ in IR and u_{j} → 0 in W^{1,p}(Ω),  (2.17)

as j → +∞. (λ_{j}, u_{j}) solve the equation (2.4). Therefore, by (p−1)−homogeneity we have

A_{p}v_{j} − λ_{j}Gv_{j} = \frac{F (λ_{j}, u_{j})}{∥u_{j}∥_{1,p}^{p−1}},

where v_{j} = \frac{u_{j}}{∥u_{j}∥_{1,p}^{p−1}} . The sequence (v_{j})_{j} is bounded in W^{1,p}(Ω). Thus, there

is a function v ∈ W^{1,p}(Ω) such that v_{j} ⇀ v in W^{1,p}(Ω) ( for a subsequence if necessary ). Then, by combining Remark 2.1, Lemma 2.3 and Lemma 2.4, we obtain that v_{j} → v strongly in W^{1,p}(Ω) and

0 = A_{p}v_{j} − λ_{j}Gv_{j} − \frac{F (λ_{j}, u_{j})}{∥u_{j}∥_{1,p}^{p−1}} → A_{p}v − μGv.  (2.19)

Then

A_{p}v = μGv in (W^{1,p}(Ω))′.  (2.20)

It is clear that v ≠ 0, because ∥ v_{j} ∥_{1,p}= 1, ∀j ∈ IN^{∗}. Hence (2.20) proves that μ is an eigenvalue of (E) in view of Remark 2.3. This clearly concludes the proof.

3. Main Results

The goal of this section it to prove our main bifurcation results. In order to do so, we shall introduce further notations and some properties of the principal positive eigenvalue of the eigenvalue problem (E) which will be used in our analysis. For this purpose, consider the variational characterization

EJQTDE, 2004, No. 9, p. 7 of λ_{1}. We recall that λ_{1} can be characterized variationally as follows

(3.1)  λ_{1} = inf {∫_{Ω} |∇u|^{p} dx; u ∈ W^{1,p}(Ω), ∫_{Ω} m(x)|u|^{p}dx = 1} .

In fact, we have the following theorem.

Theorem 3.1. [16] Let us suppose that m ∈ L^{∞}(Ω) such that mes{x ∈ Ω/m(x) > 0} ≠ 0, then we have

(i) λ_{1} is effectively an eigenvalue of (E) with weight m; and 0 < λ_{1} < +∞ if and only if m changes sign and ∫_{Ω} m(x) dx < 0. (ii) λ_{1} is simple, namely, if u and v are two eigenfunctions associated to λ_{1} then u = kv for some k. (iii) If u is an eigenfunction associated with λ_{1}, then min_{\overline{Ω}} |u| > 0.

(iv) λ_{1} is isolated.

Lemma 3.1. Let (u_{n})_{n} be a sequence in W^{1,p}(Ω) such that

∥∇u_{n}∥_{p}^{p} − λ ∫_{Ω} m(x)|u_{n}|^{p} dx ≤ c,  (∗)

for some 0 < λ < λ_{1} and positive constant c independent on n. Then (u_{n})_{n} is bounded in W^{1,p}(Ω).

Proof. From (∗), we deduce that

∥∇u_{n}∥_{p}^{p} − λ∥m∥_{∞}∥u_{n}∥_{p}^{p} ≤ c.

That is,

∥∇u_{n}∥_{p}^{p} ≤ c + λ∥m∥_{∞}∥u_{n}∥_{p}^{p}, ∀n ∈ IN^{∗}.

Thus it suffices to show that (∥u_{n}∥_{p})_{n} is bounded. Suppose by contradiction that ∥u_{n}∥_{p} → ∞ ( for a suitable subsequence if necessary). We distinguish two cases:

• \underline{∥∇u_{n}∥_{p} is bounded}. Set v_{n} = \frac{u_{n}}{∥u_{n}∥_{p}} , ∀n ∈ IN^{∗}. Thus (v_{n})_{n} is bounded in W^{1,p}(Ω). Consequently, by compactness there exists a subsequence (noted also (v_{n})_{n}) such that v_{n} ⇀ v in W^{1,p}(Ω), v_{n} → v in L^{p}(Ω) and v_{n} → v almost everywhere in Ω, for some function v ∈ W^{1,p}(Ω). It is clear that ∥v∥_{p} = 1 ( because ∥v_{n}∥_{p} = 1, ∀n ) and

∥v∥_{1,p} ≤ lim inf_{n→+∞} ∥v_{n}∥_{1,p}

which implies that

∥∇v∥_{p} ≤ lim inf_{n→+∞} \frac{∥∇u_{n}∥_{p}}{∥u_{n}∥_{p}} = 0.

EJQTDE, 2004, No. 9, p. 8 This yields that v is, almost everywhere, in Ω, equal to a nonzero constant d. Now, dividing (∗) by the quantity ∥u_{n}∥_{p}^{p} and letting n → +∞, we obtain

−λ | d | ∫_{Ω} m(x) dx ≤ 0.

So, we get that ∫_{Ω} m(x) dx ≥ 0 which is a contradiction.

• \underline{∥∇u_{n}∥_{p} is unbounded}. We can suppose that, for n large enough,

∥∇u_{n}∥_{p} > c^{\frac{1}{p}} , where c is given by (∗). From (∗), we deduce that

−λ ∫_{Ω} m(x) | u_{n} |^{p} dx < 0,

for all n large enough. That is

∫_{Ω} m(x) | u_{n} |^{p} dx > 0.

As n tends to plus infinity, u_{n} is admissible in the variational characterization of λ_{1} given by (3.1). Thus

λ_{1} ∫_{Ω} m(x) | u_{n} |^{p} dx ≤ ∥∇u_{n}∥_{p}^{p}.

Set v_{n} = \frac{u_{n}}{∥∇u_{n}∥_{1,p}}. Thus (v_{n})_{n} is bounded in W^{1,p}(Ω) and consequently there is a function v ∈ W^{1,p}(Ω) such that v_{n} converges to v weakly in W^{1,p}(Ω) and strongly in L^{p}(Ω) (for a subsequence if necessary). Dividing (∗) by ∥∇u_{n}∥_{1,p}^{p} and combining with last inequality, we arrive at

λ_{1} ∫_{Ω} m(x) | v_{n} |^{p} dx − λ ∫_{Ω} m(x) | v_{n} |^{p} dx ≤ \frac{c}{∥∇u_{n}∥_{1,p}^{p}} .  (∗∗)

Let n goes to +∞ in (∗∗), we conclude that

(λ_{1} − λ) ∫_{Ω} m(x) | v |^{p} dx ≤ 0.

This and the fact that ∫_{Ω} m(x) | v |^{p} dx ≥ 0 imply that

λ_{1} − λ ≤ 0.

Which is a contradiction. Finally, from the both above cases, we conclude that (∥∇u_{n}∥_{p})_{n} is bounded and the proof of the lemma is achieved.

EJQTDE, 2004, No. 9, p. 9 Let E = IR × W^{1,p}(Ω) be equipped with the norm

∥(μ, u)∥ = (|μ|^{2} + ∥u∥_{1,p}^{2})^{\frac{1}{2}}

Definition 3.1. We say that

C = {(λ, u) ∈ E : (λ, u)) solves (P), u ≠ 0}

is a continuum (or branch) of nontrivial solutions of (P), if it is a connected subset in E.

Theorem 3.2. Under the assumptions (2.1) and (2.2), the pair (λ_{1}, 0) is a bifurcation point of (P). Moreover, there is a continuum of nontrivial solutions C of (P) such that (λ_{1}, 0) ∈ \overline{C} and C is either unbounded in E or there is μ ≠ λ_{1}, an eigenvalue of (P), with (μ, 0)) ∈ \overline{C}.

Proof. We shall employ the homotopy invariance principle of the considered degree to deduce that

Deg[A_{p} − λG; B_{ϵ}(0), 0]  (3.2)

jumps from 1 to −1, as λ crosses λ_{1}. If this fact is proved, then Theorem 3.1 follows exactly as in the classical global bifurcation result of Rabinowitz [19]. Choose δ = δ(λ) > 0 small enough, so that λ_{1} − δ > 0 and the interval (λ_{1} − δ, λ_{1} + δ) does not contain any eigenvalue of (E) different of λ_{1}. A such δ exists because λ_{1} is isolated in the spectrum. Then, the variational characterization (3.1) of λ_{1} and Lemma 2.2 yield

Deg[A_{p} − λG; B_{ϵ}(0), 0] = 1,  (3.3)

when λ ∈ (λ_{1} −δ, λ_{1}). To evaluate (3.2) for λ ∈ (λ_{1}, λ_{1} +δ), we use a similar argument developed in [11] (see also [12]). Fix a number t_{0} > 0 and define a continuously differentiable function h : IR ⟶ IR by

h(t) = \{ \begin{matrix}0 & for & t ≤ t_{0} \\ a(t − 2t_{0}) & for & t ≥ 3t_{0}, \end{matrix}

where a > \frac{δ}{λ_{1}} and h is positive and strictly convex in (t_{0}, 3t_{0}). Now, define an auxiliary functional

\begin{aligned}Φ_{λ}(u) & = & \frac{1}{p}∥∇u∥_{p}^{p} − \frac{λ}{p} ∫_{Ω} m(x)|u|^{p}dx + h(\frac{1}{p}∥∇u∥_{p}^{p}) \\ & = & \frac{1}{p}⟨A_{p}u, u⟩ − \frac{λ}{p}⟨Gu, u⟩ + h(\frac{1}{p}∥∇u∥_{p}^{p}). \end{aligned}

Then Φ_{λ} is continuously Frêchet differentiable. It is not difficult to show that any critical point w ∈ W^{1,p}(Ω) of Φ_{λ} solves the equation

A_{p}w − \frac{λ}{1 + h′(\frac{1}{p}∥∇w∥_{p}^{p})}Gw = 0.

However, the only nontrivial critical points of Φ_{λ} occur if

h′ (\frac{1}{p}∥∇u∥_{p}^{p}) = \frac{λ}{λ_{1}} − 1.  (3.4)

Because λ ≠ λ_{1} and by definition of h, we must have \frac{1}{p}∥∇u∥_{p}^{p} ∈ (t_{0}, 3t_{0}). Due to the simplicity of λ_{1}, either u_{0} = ∓αu_{1} for some α ∈ IR_{∗}^{+}, where u_{1} > 0 is the principal eigenfunction normalized by ∥u_{1}∥_{1,p} = 1 corresponding to λ_{1}. Thus, for λ ∈ (λ_{1}, λ_{1} + δ), the derivative Φ′_{λ} possesses precisely three isolated critical points −αu_{1}, 0, αu_{1} . Now, to complete the proof, it suffices to prove that these points are local minimums of Φ_{λ}, so that we can apply Lemma 2.1. For this, we argue by variational method. We claim that: (C1) Φ_{λ} is weakly lower semicontinuous. (C2) Φ_{λ} is coercive, i.e.

lim_{∥u∥_{1,p}→+∞} ∥Φ_{λ}(u)∥_{1,p} = ∞.

Indeed, for (C1) let due to the definition of u_{n} ⇀ u in W^{1,p}(Ω). Thus, Lemma 2.3 implies

⟨Gu_{n}, u_{n}⟩ ⟶ ⟨Gu, u⟩.  (3.5)

Thanks to the weakly lower semicontinuity of the norm, we obtain

∥∇u∥_{p} ≤ lim_{n→+∞} ∥∇u_{n}∥_{p}.  (3.6)

Using (3.5), (3.6) and the fact that h is increasing in (3t_{0}, ∞), we deduce that

lim inf_{n→∞} Φ_{λ}(u_{n}) ≥ Φ_{λ}(u).

We deal now with (C2), Φ_{λ}(u) is coercive otherwise, there exist a sequence (u_{n})_{n} in W^{1,p}(Ω) and a constant c > 0 such that lim_{n→∞} ∥u_{n}∥_{1,p} = ∞ and Φ_{λ}(u_{n}) ≤ c. Therefore

Φ_{λ}(u_{n}) = \frac{1}{p}⟨A_{p}u_{n}, u_{n}⟩ − \frac{λ}{p}⟨Gu_{n}, u_{n}⟩ + h(\frac{1}{p}∥∇u_{n}∥_{p}^{p})

= \frac{1}{p} ∫_{Ω} |∇u_{n}|^{p} dx − \frac{λ}{p} ∫_{Ω} m(x)|u_{n}|^{p} dx + h(\frac{1}{p}∥∇u_{n}∥_{p}^{p})

≥ \frac{1+a}{p} ∫_{Ω} |∇u_{n}|^{p} dx − \frac{λ}{p} ∫_{Ω} m(x)|u_{n}|^{p} dx − 2at_{0}.

It follows that

∥∇u_{n}∥_{p}^{p} − \frac{λ}{1 + a} ∫_{Ω} m(x)|u_{n}|^{p} dx ≤ c + 2at_{0}.

EJQTDE, 2004, No. 9, p. 11 Since λ_{1} > \frac{λ}{1+a} then Lemma 3.1 implies that ∥u_{n}∥_{1,p} is bounded which is contradiction. Consequently, from (C_{1}) and (C_{2}) and the fact that Φ_{λ} is even, there are precisely two points at which the minimum of Φ_{λ} is achieved: −αu_{1} and αu_{1} for some α ∈ IR_{∗}^{+}, in view of [3]. The point origin 0 is obviously an isolated critical point of ”the saddle type”. From Lemma 2.1, we have

Ind(Φ′_{λ}, −αu_{1}) = Ind(Φ′_{λ}, αu_{1}) = 1.  (3.7)

Simultaneously, we have

⟨Φ′_{λ}(u), u⟩ > 0,

for any ∥u∥_{1,p} = R, with R > 0 large enough. Indeed, it is easy to verify that

⟨A_{p}u, u⟩ > λ⟨Gu, u⟩ and ⟨A_{p}u, u⟩ > 3t_{0},

for ∥u∥_{1,p} large enough. Therefore,

⟨Φ′_{λ}(u), u⟩ ≥ (λ_{1} − λ)⟨Gu, u⟩ + a⟨A_{p}u, u⟩
≥ \frac{−δ}{λ_{1}} ⟨A_{p}u, u⟩ + a⟨A_{p}u, u⟩
≥ (a − \frac{δ}{λ_{1}} )∥∇u∥_{p}^{p}.

That is ⟨Φ′_{λ}(u), u⟩ → ∞, as ∥u∥_{1,p} → ∞, due to the choice of a and Lemma 3.1. Lemma 2.2 implies that

Deg[Φ′_{λ}; B_{R}(0), 0] = 1.  (3.8)

We choose R so large such that ∥αu_{1}∥_{1,p} = R, i.e, αu_{1} ∈ ∂B_{R}(0). Now, thanks to additivity property of the degree, (3.7) and (3.8), we deduce that

Deg[L − λG; B_{ϵ}(0), 0] = −1.  (3.9)

On the other hand, it is clear that

⟨A_{p}u, u⟩ − λ⟨Gu, u⟩ → 0,

as ∥u∥_{1,p} → 0. Then, by the definition of h, we obtain

Deg[A_{p}u − λG; B_{ϵ}(0), 0] = Ind(Φ′_{λ}, 0),  (3.10)

for ϵ > 0 small enough. Which implies from (2.3) and the homotopy invariance principle of the degree, that for ϵ > 0 small enough,

Deg[T_{λ}; B_{ϵ}(0), 0] = Deg[A_{p}u − λG; B_{ϵ}(0), 0)],  (3.11)

for λ ∈ (λ_{1} − δ, λ_{1} + δ)\{λ_{1}}. Consequently, we conclude from (3.3), (3.9), (3.10) and (3.11) that

Deg[T_{λ}; B_{ϵ}(0), 0] = 1 for λ ∈ (λ_{1} − δ, λ_{1}),

EJQTDE, 2004, No. 9, p. 12

Deg[T_{λ}; B_{ϵ}(0), 0] = −1 for λ ∈ (λ_{1}, λ_{1} + δ)

for ϵ > 0 sufficiently small. The ”jump” of the degree is established and the proof is completed.

Remark 3.1. We can extend the bifurcation result above to any eigenvalue λ_{n} which is isolated in the spectrum and of odd multiplicity in order to be able to apply the above argument.

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Abdelouahed El Khalil 3-1390 Boulevard Décarie Montéal (QC) H4L 3N1 Canada

   E-mail address: lkhlil@hotmail.com

Mohammed Ouanan Departement of mathematics Faculty of Sciences Dhar-Mahraz P.O. Box 1796 Atlas, Fez 30000, Morocco

   E-mail address: m_{−}ouanan@hotmail.com