A GLOBAL BIFURCATION RESULT OF A NEUMANN PROBLEM WITH INDEFINITE WEIGHT

This paper is concerned with the bifurcation result of nonlinear Neumann problem pu = m (x)juj p 2 u + f(; x; u) in @u @ = 0 on @ : We prove that the principal eigenvalue 1 of the corresponding eigenvalue problem with f 0; is a bifurcation point by using a generalized degree type of Rabinowitz.


Introduction
The purpose of this paper is to study a bifurcation phenomenon for the following nonlinear elliptic problem (P) 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 −∆ 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 [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 (E) −∆ p u = λm(x)|u| p−2 u in Ω ∂u ∂ν = 0 on ∂Ω 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.

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 2.1.Assumptions.We make the following assumptions: 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 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 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., 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 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) = u 0 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 > 0 such that for any u ∈ B (u 0 ), Φ (u) = 0 if u = u 0 .Then, the limit 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.
Proof.The definition and compactness of G are required by the compactness of Sobolev embedding The oddness and (p − 1) homogeneity of G are obvious.Thus, the lemma is proved.
In virtue of (2.1), we have F (λ, u) (2.11) From this and Hölder's inequality, we deduce that ) for some t > 0 which satisfies q (p − 1) This is always possible, since p < q < p * .By (2.6) and (2.7), we conclude that From this inequality and the fact that u → 0 in W 1,p (Ω), we have the limit On the other hand, u belongs to L p * (Ω) ( because we find a constant c > 0 such that |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 > 0, sufficiently small, the degree is well defined for any λ ∈ IR such that T λ u = 0 for any u 1,p = ( here B (0) is the open ball of radius 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.

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) In fact, we have the following theorem.
(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 for some 0 < λ < λ 1 and positive constant c independent on n.
Proof.From ( * ), we deduce that 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: . Consequently, by compactness there exists a subsequence (noted also 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 So, we get that Ω m(x) dx ≥ 0 which is a contradiction.
• ∇u n p is unbounded.We can suppose that, for n large enough, , where c is given by ( * ).From ( * ), we deduce that As n tends to plus infinity, u n is admissible in the variational characterization of λ 1 given by (3.1).Thus Set v n = un ∇un 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 p 1,p and combining with last inequality, we arrive at Let n goes to +∞ in ( * * ), we conclude that This and the fact that 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 Definition 3.1.We say that is a continuum (or branch) of nontrivial solutions of (P), if it is a connected subset in E.
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 .
Thanks to the weakly lower semicontinuity of the norm, we obtain Using (3.5), (3.6) and the fact that h is increasing in (3t 0 , ∞), we deduce that lim inf 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 It follows that EJQTDE, 2004, No. 9, p. 11 Since λ 1 > 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 Deg[T λ ; B (0), 0] = −1 for λ ∈ (λ 1 , λ 1 + δ) for > 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.