A RESULT ON THE BIFURCATION FROM THE PRINCIPAL EIGENVALUE OF THE Ap-LAPLACIAN

We study the following bifurcation problem in any bounded domain Ω in IR :   Apu := − N ∑ i,j=1 ∂ ∂xi  ( N ∑ m,k=1 amk(x) ∂u ∂xm ∂u ∂xk ) p−2 2 aij(x) ∂u ∂xj   = λg(x)|u|p−2u + f(x, u, λ), u ∈ W 1,p 0 (Ω). We prove that the principal eigenvalue λ1 of the eigenvalue problem { Apu = λg(x)|u|p−2u, u ∈ W 1,p 0 (Ω), is a bifurcation point of the problem mentioned above.


Introduction
In this paper we study the bifurcation problem where Ω is a bounded domain in IR N , N ≥ 1; g ∈ L ∞ loc (Ω) ∩ L r (Ω) is an indefinite weight function, with r = r(N, p) satisfying the conditions r > Np for 1 < p ≤ N, r = 1 for p > N. (1.2) We assume that |Ω + | = 0 with Ω + = {x ∈ Ω; g(x) > 0}.The so-called A p -Laplacian is defined by = − div(|∇u| p−2 a A(∇u)), where A = (a ij (x)) 1≤i,j≤N is a matrix satisfying the conditions Nonlinearity f is a function satisfying some conditions to be specified later.Problems involving the A p -Laplacian, have been studied in [M, L-T, T, E, E-Li-T].We note that bifurcation problem is not considered there.
Bifurcation problem of the type (1.1), with a ij ≡ δ ij , ∀i, j = 1, . . ., N, and other conditions on g and f , were studied on bounded domains by [B-H], [D1, D2] and [D-M].The later authors consider the regular bounded domain with ∂Ω of class C 2,β for some β ∈]0, 1[ and g ≡ 1.This result was extended for the bounded domain having the segment property and g ∈ L ∞ (Ω) by [E, E-La-T].The case Ω = IR N was studied by [D-H] (cf.also [D-K-N]) under some appropriate conditions on f and g.
In this work we investigate the situation improving the conditions on f and g for any bounded domain.This paper is organized as follows: in Section 2, we introduce 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.We also show that the topological degree has a jump when λ crosses λ 1 , which implies the bifurcation result.

Assumptions, Definitions and Preliminaries
We first introduce some basic definitions, assumptions and notations.For every x fixed in Ω denote The symbol | • | a denotes the norm induced by •, • a .We use W 1,p 0 (Ω)-norm defined by 2.1.Assumptions.We assume that (f 1 ) f : Ω × IR × IR → IR satisfies Caratheodory's conditions in the first two variables and f (x, s, λ) = o(|s| p−1 ) for s → 0 (2.1) uniformly a.e. with respect to x and uniformly with respect to λ in bounded sets of IR; uniformly a.e. with respect to x and uniformly with respect to λ in bounded sets.
2.2.Definitions.1.By a solution of (1.1) we understand a pair (λ, u) in IR × W 1,p 0 (Ω) satisfying (1.1) in the weak sense, i.e., such that (2.3) for all v ∈ W 1,p 0 (Ω).We note that the pair (λ, 0) is a solution of (1.1) for every λ ∈ IR.The pairs of this form will be called the trivial solutions of (1.1).We say that P = (λ, 0) is a bifurcation point of (1.1) if in any neighborhood of P in IR × W 1,p 0 (Ω) there exists a nontrivial solution of (1.1). 2. Let X be a real reflexive Banach space and let X * stand for its dual with respect to the pairing •, • .We shall deal with mappings T acting form X into X * .The strong convergence in X (and in X * ) is denoted by → and the weak convergence by , respectively.
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.[B], [S] or [B-P]).
A point u 0 ∈ X will be called a critical point of T if T u 0 = 0. We say that u 0 is an isolated critical point of T if there exists ε > 0 such that for any exists and is called the index of the isolated critical point u 0 .
Assume, furthermore, that T is a potential operator, i.e. for some continuously differentiable functional Φ : X → IR, Φ (u) = T u, u ∈ X.Then we have the following two lemmas which we can find in [D1], [D2] or [D-H].
Lemma 2.1.Let u 0 be a local minimum of Φ and an isolated critical point of T .Then

3) the function u is a weak solution of (1.1) if and only if
(2.4) (ii) The operator A p has the following properties: A p is odd, (p−1)-homogeneous, strictly monotone, i.e., and A p ∈ (S + ) (cf. [T]).We have also . By Hölder's inequality, we have where s is given by We obtain that By Hölder's inequality, we arrive at Then in this case G is well defined.Third case: if p > N, r = 1.In this case . Then for any u, v ∈ W 1,p 0 (Ω), we have with g ∈ L 1 (Ω), and G is well defined also in this case.
If 1 < p < N, r > Np : Let s be as in (2.7).Then where c is the constant of Sobolev's embedding.We have 1), as n → +∞ due to the continuity of Nemytskii's operator u → |u| p−2 u from L s (Ω) into L s p−1 (Ω).Rellich's theorem yields that u n u weakly in W 1,p 0 (Ω) implies that u n → u strongly in L s (Ω) because max(1, p − 1) < s < p * .The compactness of G then follows. If where q is given by (2.8).By Sobolev's embedding, there is c > 0 such that From the continuity of u → |u| N −2 u from L N (Ω) into L N (Ω), and from the compact embedding of W 1,N 0 (Ω) in L N (Ω), we have the desired result.If p > N, r = 1.By Rellich's embedding theorem of W 1,p 0 (Ω) into C(Ω), we obtain where C is the constant given by embedding of The oddness and (p − 1)-homogeneity of G is obvious.Thus the lemma is proved.
Remark 2.6.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 2.7.λ is an eigenvalue of (P ) By the same argument as used in proof of Lemma 2.4, we can show the following proposition.

and only if the equation
Proposition 2.8.If (λ, 0) is a bifurcation point of problem (1.1), then λ is an eigenvalue of (P).

Bifurcation from λ 1
We recall that λ 1 can be characterized variationally as follows: (3.1) Recall for our problem (P ), (cf., [L-T]), that λ 1 is the principal eigenvalue and it is simple and isolated.Let E = IR × W 1,p 0 (Ω) be equipped with the norm is a continuum of nontrivial solutions of (1.1), if it is a connected set in E.
It is not difficult to prove that Ψ λ is weakly lower semicontinuous and for λ ∈ (λ 1 , λ 1 + δ) and ε > 0 sufficiently small.Since (3.3) and (3.7) establish the "jump" of the degree the proof is complete.