Bifurcation of nonlinear elliptic system from the first eigenvalue, Electron

We study the following bifurcation problem in a bounded domain in IR N : 8 : pu = juj jvj v + f(x; u; v; ) in qv = juj jvj u + g(x; u; v; ) in (u; v) 2 W 1;p 0 () W 1;q 0 () : We prove that the principal eigenvalue 1 of the following eigen- value problem 8


Introduction
The purpose of this paper is to illustrate a global bifurcation phenomenon for the nonlinear elliptic system (BS) where Ω is a bounded domain not necessary regular in IR N , N ≥ 1, α, β, p and q are real numbers satisfying suitable conditions which ensure the results.The system (BS) is weakly coupled in the sense that the interaction is present only in the " source terms", while the differential terms have only one dependent variable each.The differential operator involved is the well-known p-Laplacian ∆ p u = ∇.(|∇u|p−2 ∇u) which reduces to the ordinary Laplacian ∆u, when p = 2.The nonlinearities f and g satisfy some hypotheses to be specified later.To system (BS) we associate the eigenvalue problem system (ES) in Ω (u, v) ∈ W 1,p 0 (Ω) × W 1,q 0 (Ω).We say that λ is an eigenvalue of (ES) if there exists a nontrivial pair (u, v) ∈ W 1,p 0 (Ω) × W 1,q 0 (Ω) that satisfies (ES) in the following sense (S) for any (φ, ψ) ∈ W 1,p 0 (Ω) × W 1,q 0 (Ω).The goal of this work is the study of the main properties (simplicity, isolation) of the least positive eigenvalue denote λ 1 of (ES) in theorem 3.1.These properties are well-known in the scalar case of one equation, we refer the reader to the bibliography contained in [14] where several results are cited.
We prove that the eigenvectors (u, v) associated to λ 1 have definite sign in Ω.So, we show that this property implies that λ 1 is simple and isolated in the spectrum.Note that our result of isolation follows by adaptation a technique used by Anane [1] for scalar p-Laplacian in a smooth bounded domain.Concerning the bifurcation problem, we prove the existence of bifurcation branch of nontrivial solutions of (BS) from λ 1 , ( see Theorem 3.2.).Here we use abstract methods of nonlinear functional analysis based on the generalized topological degree in order to get a new result.System (BS) has been studied by Chaïb [6] in the case Ω = IR N .The author extended Diaz-Saás inequality in IR N and claimed that λ 1 is simple.In [12], the authors obtained similar result by considering for a quite different system.In the case of a bounded domain sufficiently regular the following eigenvalue system q 0 (Ω) was studied in [5], where the author proved only the simplicity result of the first eigenvalue.
Here, we address to (ES) and (BS) without any regularity assumption on the boundary ∂Ω.
EJQTDE, 2003 No. 21, p. 2 Bifurcation problems in the case of one equation in a smooth bounded domain involving p-Laplacian operator were studied by [2], [9] and [10] under some restrictive conditions on the nonlinearity sources.In any bounded domain, we cite recent results of [7] and [8].
A global bifurcation result from the first eigenvalue of a particular system is considered in [13] under some restrictive hypotheses on the nonlinearities f and g to be specified at the end of this paper.
In this work, we consider a bifurcation system for which we investigate the system improving the conditions on the nonlinearities f and g in any bounded domain with some condition of homogeneity type connecting p, q, α and β.For this end we use a generalized degree of Leray-Shauder.
This paper is organized as follows: in Section 2 we introduce some assumptions, definitions and we prove some auxiliary results that are the key point for proving our results; Section 3 contains a results and proofs of simplicity and isolation of λ 1 , finally we verify that the topological degree has a jump when λ crosses λ 1 , which implies the bifurcation results.

Assumptions and Preliminaries
Through this paper Ω will be a bounded domain of IR N .W 1,p 0 (Ω) will denote the usual Sobolev space endowed with norm ∇u p = ( Ω |∇u| p dx) 1 p .We will write .p for the L p −norm.We will also denote for p > 1, p = p p−1 the Hölder conjugate exponent of p and p * the critical exponent, that is p * = ∞ if p ≥ N and p * = N p N −p if 1 < p < N.Here we use the following norm in product space W 1,p 0 (Ω) × W 1,q 0 (Ω) (u, v) = ∇u p + ∇v q .and .* is the dual norm.
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 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 If T ∈ (S + ) and T is demicontinuous, then it is possible to define the degree 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.[3]).
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 > 0 such that for any u ∈ B (u 0 ) (open ball in X centered to u 0 and the radius ) , T u = 0 if u = u 0 .Then the limit Ind(T, u 0 ) = lim →0 + Deg[T ; B (u 0 ); 0] 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 [9] or [10].These lemmas are crucial results in our bifurcation argument.
Lemma 2.1.Let u 0 be a local minimum of Φ and an isolated critical point of T. Then Ind(T, u 0 ) = 1.

Preliminaries.
This subsection establishes an abstract framework used to prove our main results.Let define, for (u, v) ∈ W 1,p 0 (Ω) × W 1,q 0 (Ω) and (φ, ψ) ∈ W 1,p 0 (Ω) × W 1,q 0 (Ω), the operators are functionals of class C 1 and Define also Remarks 2.1.(i) Due to (2.6) a pair (u, v) is a weak solution of (BS) if and only if The operators A u and A v are odd and satisfy (S + ).
Lemma 2.3.B u and B v are well defined, completely continuous and odd functionals.
Step 2. Compactness of B u and Indeed, Hölder's inequality implies Due to the continuity of Nemytskii operator (Ω)) and Rellich's Theorem, there exists n 0 ≥ 0 such that for all n ≥ n 0 we have ) Thus (v n ) is bounded in L p (Ω). Finally from (2.7) and (2.8) we have the claim.
The compactness of B v can be proved by the same argument.The oddness of B u and B v is obvious.The proof is completed.
So, we distinguish two cases: q .
There is the same proof of (2.10).
Remark 2.3.λ is an eigenvalue of (ES) if and only if the system Under the assumption (2.1), (ES) has a principal eigenvalue λ 1 characterized variationally as follows The proof of this proposition is more or less the same as F. De Thélin [5] for the system case (ES) modulo a suitable modification.EJQTDE, 2003 No. 21, p. 9 Now, let The following lemma is the heart on the proof of the simplicity. ) Proof.By Young's inequality we have for > 0, For = 1 we have by integration over Ω, On the other hand, if Γ p (u, φ) = 0 by (2.14), we obtain for = 1, (i) By adapting the argument of [5] we can show that the eigenvectors associated to λ 1 without any additional smoothness assumption on ∂Ω, are in L ∞ × L ∞ .(ii) Thanks to an advanced result in regularity theory of [4], we deduce with (i) that the positive solution of (ES) associated to λ 1 is in C 1 loc (Ω) × C 1 loc (Ω).(iii) According to the Maximum principle of [17] applied to each equation we conclude that (ES) has a priori a positive eigenvector associated to λ 1 .

3.1.
Simplicity and isolation results.

Proof.
(i) Let (u, v) and (φ, ψ) be two eigenvectors of (ES) associated to λ 1 with (u, v) is positive (u ≥ 0, v ≥ 0).Thanks to definition of λ 1 and EJQTDE, 2003 No. 21, p. 11 Hölder's inequality we have in view of remark 2.4 that Thanks to lemma 2.5 we have Thus Hence by Lemma 3.2 again, there exist k 1 and k 2 in IR such that u = k 1 φ and v = k 2 ψ.
(ii) Let (u, v) be a positive eigenvector of (ES) associated to λ and (φ, ψ) a positive solution of (ES) associated to λ 1 .It is clear that λ 1 ≤ λ and by Hölder's inequality we have Therefore, we deduce from Lemma 2.5 that A(φ, ψ) = A(u, v) i.e (iii) Let (λ, (u, v)) be an eigenpair of (ES), u − (x) = min(u(x), 0) and v − (x) = min(v(x), 0).Thus, by multiplying the first equation of (ES) by u − we have Similarly, multiplying the second equation of (ES) by v − , we obtain (3.2) Hence by (3.1) and (3.2) we complete the proof of (iii).(iv) The proof is a rather a simple adaptation of argument of [1] modulo suitable modification.

Bifurcation Result.
Once we have proved in the previous subsection that λ 1 is simple and isolated, we can study the bifurcation when λ is near λ 1 .Proposition 3.1.If ( λ; (0, 0)) is a bifurcation point of (BS), then λ is an eigenvalue of (ES).