Electronic Journal of Qualitative Theory of Differential Equations 2003, No. 21, 1-18; http://www.math.u-szeged.hu/ejqtde/

BIFURCATION OF A NONLINEAR ELLIPTIC SYSTEM FROM THE FIRST EIGENVALUE

ABDELOUAHED EL KHALIL, MOHAMMED OUANAN & ABDELFATTAH TOUZANI

Abstract. We study the following bifurcation problem in a bounded domain Ω in IR^{N}:

−Δ_{p}u = λ|u|^{α}|v|^{β}v + f(x, u, v, λ) in Ω
−Δ_{q}v = λ|u|^{α}|v|^{β}u + g(x, u, v, λ) in Ω
(u, v) ∈ W_{0}^{1,p}(Ω) × W_{0}^{1,q}(Ω).

We prove that the principal eigenvalue λ_{1} of the following eigenvalue problem

−Δ_{p}u = λ|u|^{α}|v|^{β}v  in Ω
−Δ_{q}v = λ|u|^{α}|v|^{β}u  in Ω
(u, v) ∈ W_{0}^{1,p}(Ω) × W_{0}^{1,q}(Ω)

is simple and isolated and we prove that (λ_{1}, 0, 0) is a bifurcation point of the system mentioned above.

1. Introduction

The purpose of this paper is to illustrate a global bifurcation phenomenon for the nonlinear elliptic system

(BS)  \{ \begin{matrix}−Δ_{p}u = & λ|u|^{α}|v|^{β}v + f(x, u, v, λ) & in Ω \\ −Δ_{q}v = & λ|u|^{α}|v|^{β}u + g(x, u, v, λ) & in Ω \\ (u, v) ∈ & W_{0}^{1,p}(Ω) × W_{0}^{1,q}(Ω), \end{matrix}

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 Δ_{p}u = ∇.(|∇u|^{p−2}∇u) which reduces to the ordinary Laplacian Δu, when p = 2. The nonlinearities f and g satisfy some hypotheses to be specified later. To

EJQTDE, 2003 No. 21, p. 1 system (BS) we associate the eigenvalue problem system

2000 Mathematics Subject Classification. 35J20, 35J45, 35J50, 35J70.

Key words and phrases. quasilinear system, p-Laplacian operator, principal eigenvalue, Degree theory, Bifurcation point.

(ES)  \{ \begin{matrix}−Δ_{p}u = & λ|u|^{α}|v|^{β}v & in Ω \\ −Δ_{q}v = & λ|u|^{α}|v|^{β}u & in Ω \\ (u, v) ∈ & W_{0}^{1,p}(Ω) × W_{0}^{1,q}(Ω). \end{matrix}

We say that λ is an eigenvalue of (ES) if there exists a nontrivial pair (u, v) ∈ W_{0}^{1,p}(Ω) × W_{0}^{1,q}(Ω) that satisfies (ES) in the following sense

(S)  \{ \begin{matrix}∫_{Ω} |∇u|^{p−2}∇u∇ϕ dx & = & ∫_{Ω}[λ|u|^{α}|v|^{β}vϕ dx \\ ∫_{Ω} |∇v|^{q−2}∇v∇ψ dx & = & ∫_{Ω}[λ|u|^{α}|v|^{β}uψ dx, \end{matrix}

for any (ϕ, ψ) ∈ W_{0}^{1,p}(Ω) × W_{0}^{1,q}(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

−Δ_{p}u = λ|u|^{α−1}|v|^{β+1}u  in Ω
−Δ_{q}v = λ|u|^{α+1}|v|^{β−1}v  in Ω
(u, v) ∈ W_{0}^{1,p}(Ω) × W_{0}^{1,q}(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.

2. Assumptions and Preliminaries

Through this paper Ω will be a bounded domain of IR^{N}. W_{0}^{1,p}(Ω) will denote the usual Sobolev space endowed with norm ∥∇u∥_{p} = (∫_{Ω} |∇u|^{p}dx)^{\frac{1}{p}} . We will write ∥.∥_{p} for the L^{p}−norm. We will also denote for p > 1, p′ = \frac{p}{p−1} the Hölder conjugate exponent of p and p^{∗} the critical exponent, that is p^{∗} = ∞ if p ≥ N and p^{∗} = \frac{Np}{N−p} if 1 < p < N.

Here we use the following norm in product space W_{0}^{1,p}(Ω) × W_{0}^{1,q}(Ω)

∥(u, v)∥ = ∥∇u∥_{p} + ∥∇v∥_{q}.

and ∥.∥_{∗} is the dual norm.

2.1. Assumptions. We assume that p, q > 1, α > 0, β > 0 and

\frac{α + 1}{p} + \frac{β + 1}{q} = 1  (2.1)

(f) f : Ω × IR × IR × IR ⟶ IR satisfies Carathéodory’s conditions in the first three variables and

lim_{|(r,s)|→+∞|} \frac{f(x, r, s, λ)}{|r|^{γ−1}r|s|^{δ+1}} = 0;  (2.2)
f(x, r, s, λ) = o(|r|^{p−1}) as |(r, s)| → 0  (2.3)

EJQTDE, 2003 No. 21, p. 3 uniformly a.e. with respect to x and uniformly with respect to λ in bounded intervals of IR.

(g) g : Ω × IR × IR × IR ⟶ IR satisfies Carathéodory’s conditions in the first three variables and

lim_{|(r,s)|→+∞|} \frac{g(x, r, s, λ)}{|r|^{γ+1}|s|^{δ−1}s} = 0;  (2.4)
g(x, r, s, λ) = o(|s|^{q−1}) as |(r, s)| → 0  (2.5)

uniformly a.e. with respect to x and uniformly with respect to λ in bounded intervals of IR; with

γ + 1 > p or δ + 1 > q

and

δ + γ + 2 < min(p^{∗}, q^{∗}),

for some nonnegative reals γ and δ.

Remark 2.1. The homogeneity assumption (2.1) justifies the notion of eigenvalues.

2.2. Definitions. 1.We say that (λ, (u, v)) in IR × W_{0}^{1,p}(Ω) × W_{0}^{1,q}(Ω) is a solution of (BS) if for any (ϕ, ψ) ∈ W_{0}^{1,p}(Ω) × W_{0}^{1,q}(Ω),

\{ \begin{matrix}∫_{Ω} |∇u|^{p−2}∇u∇ϕ dx & = & ∫_{Ω}[λ|u|^{α}|v|^{β}v + & f(x, u, v, λ)]ϕ dx \\ ∫_{Ω} |∇v|^{q−2}∇v∇ψ dx & = & ∫_{Ω}[λ|u|^{α}|v|^{β}u + & g(x, u, v, λ)]ψ dx, \end{matrix}
(2.6)

for all (ϕ, ψ) ∈ W_{0}^{1,p}(Ω)×W_{0}^{1,q}(Ω). We note that the pair (λ, (0, 0)) is a solution of (BS) for every λ ∈ IR. The pairs of this form will be called the trivial solutions of (BS). We say that P = (λ̄, (0, 0)) is a bifurcation point of (BS) if in any neighbourhood of P in IR × W_{0}^{1,p}(Ω) × W_{0}^{1,q}(Ω) there exists a nontrivial solution of (BS).

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_{n→+∞}⟨T u_{n}, u_{n} − u⟩ ≤ 0, it follows that u_{n} → u strongly inX.

EJQTDE, 2003 No. 21, p. 4 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.[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 ϵ > 0 such that for any u ∈ B_{ϵ}(u_{0}) (open ball in X centered to u_{0} and the radius ϵ) , T u ≠ 0 if u ≠ u_{0}. Then the limit

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

Lemma 2.2. Assume that ⟨T u, u⟩ > 0 for all u ∈ X, ∥u∥_{X} = ρ.Then

Deg[T; B_{ρ}(0); 0] = 1.

2.4. Preliminaries. This subsection establishes an abstract framework used to prove our main results. Let define, for (u, v) ∈ W_{0}^{1,p}(Ω) × W_{0}^{1,q}(Ω) and (ϕ, ψ) ∈ W_{0}^{1,p}(Ω) × W_{0}^{1,q}(Ω), the operators

⟨A_{u}(u, v), (ϕ, ψ)⟩ = ∫_{Ω} |∇u|^{p−2}∇u∇ϕ dx,
⟨A_{v}(u, v), (ϕ, ψ)⟩ = ∫_{Ω} |∇v|^{q−2}∇v∇ψ dx,
⟨B_{u}(u, v), (ϕ, ψ)⟩ = ∫_{Ω} |u|^{α}|v|^{β}vϕ dx,
⟨B_{v}(u, v), (ϕ, ψ)⟩ = ∫_{Ω} |u|^{α}|v|^{β}uψ dx,

EJQTDE, 2003 No. 21, p. 5 with

A(u, v) = \frac{α + 1}{p} ∫_{Ω} |∇u|^{p} dx+\frac{β + 1}{q} ∫_{Ω} |∇v|^{q} dx and B(u, v) = ∫_{Ω} |u|^{α}|v|^{β}uv dx

are functionals of class C^{1} and

A_{u} = \frac{1}{α + 1} \frac{∂A}{∂u}, A_{v} = \frac{1}{β + 1} \frac{∂A}{∂v}, B_{u} = \frac{1}{α + 1} \frac{∂B}{∂u} , B_{v} = \frac{1}{β + 1} \frac{∂B}{∂v} .

Define also

⟨F^{λ}(u, v), (ϕ, ψ)⟩ = ∫_{Ω} f(x, u, v, λ)ϕ dx,
⟨G^{λ}(u, v), (ϕ, ψ)⟩ = ∫_{Ω} g(x, u, v, λ)ψ dx.

Remarks 2.1. (i) Due to (2.6) a pair (u, v) is a weak solution of (BS) if and only if

A_{u}(u, v) − λB_{u}(u, v) = F^{λ}(u, v) in (W_{0}^{1,p}(Ω) × W_{0}^{1,q}(Ω))′

and

A_{v}(u, v) − λB_{v}(u, v) = G^{λ}(u, v) in (W_{0}^{1,p}(Ω) × W_{0}^{1,q}(Ω))′

(ii) 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.

Proof. Step 1. Definition of B_{u} and B_{v}. By Hölder’s inequality, we have

|⟨B_{u}(u, v), (ϕ, ψ)⟩| = |∫_{Ω} |u|^{α}|v|^{β}vϕ dx| ≤ ∥u∥_{p}^{α}∥v∥_{q}^{β+1}∥ϕ∥_{p} < +∞

for any (u, v) ∈ W_{0}^{1,p}(Ω) × W_{0}^{1,q}(Ω) and (ϕ, ψ) ∈ W_{0}^{1,p}(Ω) × W_{0}^{1,q}(Ω). Thus B_{u} is well defined. Similarly, we prove that B_{v} is also well defined.

Step 2. Compactness of B_{u} and B_{v}. Let {(u_{n}, v_{n})}_{n≥0} ⊂ W_{0}^{1,p}(Ω)×W_{0}^{1,q}(Ω) be a sequence such that (u_{n}, v_{n}) ⇀ (u, v) weakly in W_{0}^{1,p}(Ω) × W_{0}^{1,q}(Ω). We claim that B_{u}(u_{n}, v_{n}) → B_{u}(u, v) strongly in W_{0}^{1,p}(Ω) × W_{0}^{1,q}(Ω), i.e. for all (ϕ, ψ) ∈ W_{0}^{1,p}(Ω) × W_{0}^{1,q}(Ω),

sup_{∥∇ϕ∥_{p}+∥∇ψ∥_{q}≤1} |⟨B_{u}(u_{n}, v_{n}) − B_{u}(u, v), (ϕ, ψ)⟨| = o(1) as n → +∞.

Indeed, Hölder’s inequality implies

|⟨B_{u}(u_{n}, v_{n}) − B_{u}(u, v), (ϕ, ψ)⟩| = |∫_{Ω}(|u_{n}|^{α}|v_{n}|^{β}v_{n} − |u|^{α}|v|^{β}v)ϕ dx|
≤ ∫_{Ω} |v_{n}|^{β+1} |(|u_{n}|^{α} − |u|^{α})| |ϕ| dx
+ ∫_{Ω} |u|^{α} ||v_{n}|^{β}v_{n} − |v|^{β}v| |ϕ| dx
≤ ∥|u_{n}|^{α} − |u|^{α}∥_{\frac{p}{α}} ∥ϕ∥_{p}∥v_{n}∥_{q}^{β+1}

+∥u∥_{p}^{α}∥ϕ∥_{p} ∥|v_{n}|^{β}v_{n} − |v|^{β}v|∥_{\frac{q}{β+1}} .

Due to the continuity of Nemytskii operator u ⟶ |u|^{α} ( resp. v ⟶ |v|^{β}v) from L^{p}(Ω) into L^{\frac{p}{α}} (Ω) (resp. fromL^{q}(Ω) into L^{\frac{q}{β+1}} (Ω)) and Rellich’s Theorem, there exists n_{0} ≥ 0 such that for all n ≥ n_{0} we have

∥|u_{n}|^{α} − |u|^{α}∥_{\frac{p}{α}} = o(1),  (2.7)
∥|v_{n}|^{β}v_{n} − |v|^{β}v∥_{\frac{q}{β+1}} = o(1).  (2.8)

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.

Lemma 2.4. F^{λ}(., .) and G^{λ}(., .) are well defined, completely continuous and F^{λ}(0, 0) = G^{λ}(0, 0) = 0 and we have

lim_{\begin{matrix}∥∇u∥_{p} → 0 \\ ∥∇v∥_{q} → 0 \end{matrix}} \frac{F^{λ}(u, v)}{∥∇u∥_{p}^{p−1} + ∥∇v∥_{q}^{q−1}} = 0 in (W_{0}^{1,p}(Ω) × W_{0}^{1,q}(Ω))′, (2.9)

and

lim_{\begin{matrix}∥∇u∥_{p} → 0 \\ ∥∇v∥_{q} → 0 \end{matrix}} \frac{G^{λ}(u, v)}{∥∇u∥p^{p−1} + ∥∇v∥_{q}^{q−1}} = 0 in (W_{0}^{1,p}(Ω) × W_{0}^{1,q}(Ω))′,

(2.10)

uniformly with respect to λ in bounded subsets of IR.

EJQTDE, 2003 No. 21, p. 7

Proof. (2.2) and (2.3) imply that for any ϵ > 0 there are two real ξ = ξ(ϵ) and M = M(ξ) > 0 such that for a.e, x ∈ Ω

|f(x, u, v, λ)| ≤ ϵ|u|^{p−1} + M|u|^{γ}|v|^{δ+1}.

• Compactness: Here, we show the compactness of F^{λ}. If δ + 1 > p, let θ ∈ IR^{+} such that:

\{ \begin{matrix}δ + γ + 2 < θ < min(p^{∗}, q^{∗}) & if & γ + 1 > p \\ max(δ + γ + 2, p) < θ < min(p^{∗}, q^{∗}) & if & γ + 1 ≤ p. \end{matrix}

By using Hölder’s and Young’s inequalities we have

∫_{Ω} |f(x, λ, u, v)|^{θ′} dx ≤ ϵ ∫_{|u(x)|+|v(x)|≤ξ} |u|^{θ′(p−1)}dx + M ∫_{|u(x)|+|v(x)|≥ξ} |u|^{θ′δ}|v|^{θ′(γ+1)}dx
≤ ϵ ∫_{Ω} |u|^{θ′(p−1)}dx + M ∫_{Ω} |u|^{θ′γ}|v|^{θ′(δ+1)}dx

≤ M_{1} ∫_{Ω} |u|^{θ}dx + M_{2} (∫_{Ω} |u|^{θ}dx)^{\frac{γ}{θ−1}} (∫_{Ω} |v|^{θ}dx)^{\frac{δ+1}{θ−1}}

Thus F^{λ} is the Nemytskii continuous operator from L^{θ}(Ω)×L^{θ}(Ω) into L^{θ′}(Ω). On the other hand, let {(u_{n}, v_{n})}_{n} ⊂ W_{0}^{1,p}(Ω) × W_{0}^{1,q}(Ω) such that (u_{n}, v_{n}) ⇀ (u, v) in W_{0}^{1,p}(Ω) × W_{0}^{1,q}(Ω) and strongly in L^{θ}(Ω) × L^{θ}(Ω). So,

∥F^{λ}(u_{n}, v_{n}) − F^{λ}(u, v)∥_{∗} = sup_{|(ϕ,ψ)∥≤1} |∫_{Ω}(f(x, λ, u_{n}, v_{n}) − f(x, λ, u, v))ϕ dx|
≤ ∥f(x, λ, u_{n}, v_{n}) − f(x, λ, u, v∥_{θ′} → 0, as n → ∞.

Hence F^{λ} is compact. By a similar argument, we can show that G^{λ} is compact.

• For the limit (2.9) and (2.10), we argue as follows:

EJQTDE, 2003 No. 21, p. 8 Hölder’s inequality yields for all (ϕ, ψ) ∈ W_{0}^{1,p}(Ω) × W_{0}^{1,q}(Ω),

∥F^{λ}(u, v)∥_{∗} ≤ sup_{∥(ϕ,ψ)∥≤1} |(ϵ ∫_{Ω} |u|^{p−1}ϕ dx + M ∫_{Ω} |u|^{γ}|v|^{δ+1}ϕ dx)|
≤ sup_{∥(ϕ,ψ)∥≤1}(ϵ∥u∥_{p}^{p−1}∥ϕ∥_{p} + M|Ω|^{ω_{0}}∥u∥_{p∗}^{γ}∥v∥_{q∗}^{δ+1}∥ϕ∥_{p∗})
≤ c sup_{∥(ϕ,ψ)∥≤1}(ϵ∥∇u∥_{p}^{p−1} + M∥∇u∥_{p}^{γ}∥∇v∥_{q}^{δ+1})∥∇ϕ∥_{p}

where ω_{0} = \frac{p^{∗}q^{∗}}{p^{∗}q^{∗}−(γ+1)p^{∗}−(δ+1)q^{∗}} and c ∈ IR^{+} is a constant.

So, we distinguish two cases: If γ + 1 > p then,

\frac{∥F^{λ}(u, v)∥_{∗}}{∥∇u∥_{p}^{p−1} + ∥∇v∥_{q}^{q−1}} ≤ c(ϵ + M∥∇u∥_{p}^{γ−(p−1)}∥∇v∥_{q}^{δ+1}).  (2.11)

If δ + 1 > q then,

\frac{∥F^{λ}(u, v)∥_{∗}}{∥∇u∥_{p}^{p−1} + ∥∇v∥_{q}^{q−1}} ≤ cϵ + cM∥∇u∥_{p}^{γ}∥∇v∥_{q}^{(δ+1)−(q−1)}.  (2.12)

Therefore, by passing to the limit in (2.11) or (2.12), (2.9) follows. There is the same proof of (2.10).

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 (ES) if and only if the system

\{ \begin{matrix}A_{u}(u, v) − λB_{u}(u, v) & = & 0 \\ A_{v}(u, v) − λB_{v}(u, v) & = & 0 \end{matrix}

has a nontrivial solution (u, v) ∈ W_{0}^{1,p}(Ω) × W_{0}^{1,q}(Ω).

Proposition 2.1. Under the assumption (2.1), (ES) has a principal eigenvalue λ_{1} characterized variationally as follows

λ_{1} = inf {A(u, v); (u, v) ∈ W_{0}^{1,p}(Ω) × W_{0}^{1,q}(Ω), B(u, v) = 1} .

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

Γ_{p}(u, ϕ) = ∫_{Ω} |∇u|^{p} dx + (p − 1) ∫_{Ω} |∇ϕ|^{p} (\frac{|u|}{ϕ})^{p} dx
− p ∫_{Ω} |∇ϕ|^{p−2}∇ϕ∇u (\frac{|u|^{p−2}u}{ϕ^{p−1}}) dx
= ∫_{Ω} |∇u|^{p} dx + ∫_{Ω} \frac{Δ_{p}ϕ}{ϕ^{p−1}} |u|^{p} dx

for all (u, ϕ) ∈ (W_{0}^{1,p}(Ω) ∩ C^{1}(Ω))^{2} with ϕ > 0 in Ω. The following lemma is the heart on the proof of the simplicity.

Lemma 2.5. For all (u, ϕ) ∈ (W_{0}^{1,p}(Ω) ∩ C^{1,ν}(Ω))^{2} with ϕ > 0 in Ω and ν ∈ (0, 1), we have Γ_{p}(u, ϕ) ≥ 0 i.e

∫_{Ω} |∇u|^{p} dx ≥ ∫_{Ω} \frac{−Δ_{p}ϕ}{ϕ^{p−1}}|u|^{p} dx,  (2.13)

and if Γ_{p}(u, ϕ) = 0 there is c ∈ IR such that u = cϕ.

Proof. By Young’s inequality we have for ϵ > 0,

∫_{Ω} ∇u|∇ϕ|^{p−2}∇ϕ\frac{u|u|^{p−2}}{ϕ^{p−1}} dx ≤ ∫_{Ω} |∇u||∇ϕ|^{p−1}(\frac{|u|}{ϕ})^{p−1} dx
≤ ∫_{Ω}(\frac{ϵ^{p}}{p} |∇u|^{p} + \frac{p − 1}{pϵ^{p}}|\frac{u}{ϕ}|^{p}|∇ϕ|^{p}) dx.  (2.14)

For ϵ = 1 we have by integration over Ω,

p ∫_{Ω} |∇ϕ|^{p−2}∇ϕ∇u (\frac{|u|^{p−2}u}{ϕ^{p−1}}) dx ≤ ∫_{Ω} |∇u|^{p} dx+(p−1) ∫_{Ω} |\frac{u}{ϕ}|^{p}|∇ϕ|^{p} dx.

Thus

Γ_{p}(u, ϕ) ≥ 0.

On the other hand, if Γ_{p}(u, ϕ) = 0 by (2.14), we obtain for ϵ = 1,

p ∫_{Ω} |∇ϕ|^{p−2}∇ϕ∇u (\frac{|u|^{p−2}u}{ϕ^{p−1}}) dx−∫_{Ω} |∇u|^{p} dx−(p−1) ∫_{Ω} |\frac{u}{ϕ}|^{p}|∇ϕ|^{p} dx = 0

(2.15)

and

∫_{Ω} \{∇u∇ϕ|∇ϕ|^{p−2} \frac{u|u|^{p−2}}{ϕ^{p−1}} − |∇u||∇ϕ|^{p−1} (\frac{|u|}{ϕ})^{p−1}\} dx = 0. (2.16)

Thanks to (2.15) we find |∇u| = |\frac{u}{ϕ}∇ϕ|. Thus from (2.16), it follows that ∇u = ϵ\frac{u}{ϕ}∇ϕ, where |ϵ| = 1. Consequently, since Γ_{p}(u, ϕ) = 0, we obtain ϵ = 1 and ∇(\frac{u}{ϕ}) = 0. Therefore, there is c ∈ IR such that u = cϕ.

Remarks 2.2. (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_{loc}^{1}(Ω) × C_{loc}^{1}(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. Main Results

3.1. Simplicity and isolation results.

Theorem 3.1. (i) λ_{1} is simple.

(ii) λ_{1} is the unique eigenvalue of (ES) having an eigenvector (u, v)

not changing its sign, i.e., uv > 0 in Ω. (iii) There is c > 0 such that

|Ω^{−}| ≥ (|λ|c)^{ω} ,

where (λ, (u, v)) is an eigenpair of (ES), Ω^{−} = {x ∈ Ω, u(x) < 0 and v(x) < 0} and ω = \frac{p^{∗}q^{∗}}{p^{∗}q^{∗}−(α+1)q^{∗}−(β+1)p^{∗}}.

(iv) λ_{1} is isolated.

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

A(ϕ, ψ) = λ_{1}B(ϕ, ψ)
≤ λ_{1} ∫_{Ω} u^{α+1}v^{β+1} \frac{|ϕ|^{α+1}|ψ|^{β+1}}{u^{α+1}v^{β+1}} dx
≤ λ_{1} ∫_{Ω} u^{α+1}v^{β+1} [\frac{α + 1}{p} \frac{|ϕ|^{p}}{u^{p}} + \frac{β + 1}{q} \frac{|ψ|^{q}}{v^{q}} ] dx
≤ λ_{1} ∫_{Ω} [\frac{α + 1}{p} \frac{u^{α}v^{β+1}}{u^{p−1}} |ϕ|^{p} + \frac{β + 1}{q} \frac{u^{α+1}v^{β}}{v^{q−1}} |ψ|^{q}] dx
≤ \frac{α + 1}{p} ∫_{Ω} \frac{−Δ_{p}u}{u^{p−1}}|ϕ|^{p} dx + \frac{β + 1}{q} ∫_{Ω} \frac{−Δ_{q}v}{v^{q−1}} |ψ|^{q} dx.

Thanks to lemma 2.5 we have

A(ϕ, ψ) = \frac{α + 1}{p} ∫_{Ω} \frac{−Δ_{p}u}{u^{p−1}}|ϕ|^{p} dx + \frac{β + 1}{q} ∫_{Ω} \frac{−Δ_{q}v}{v^{q−1}} |ψ|^{q} dx.

Thus

∫_{Ω} |∇ϕ|^{p} dx = ∫_{Ω} \frac{−Δ_{p}u}{u^{p−1}}|ϕ|^{p} dx and ∫_{Ω} |∇ϕ|^{p} dx = ∫_{Ω} \frac{−Δ_{q}v}{v^{q−1}} |ϕ|^{q} dx.

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

A(ϕ, ψ) = λ_{1}B(ϕ, ψ)
≤ λ ∫_{Ω} u^{α+1}v^{β+1} \frac{ϕ^{α+1}ψ^{β+1}}{u^{α+1}v^{β+1}} dx
≤ λ ∫_{Ω} u^{α+1}v^{β+1} [\frac{α + 1}{p} \frac{ϕ^{p}}{u^{p}} + \frac{β + 1}{q} \frac{ψ^{q}}{v^{q}} ] dx
≤ λ ∫_{Ω} [\frac{α + 1}{p} \frac{u^{α}v^{β+1}}{u^{p−1}} ϕ^{p} + \frac{β + 1}{q} \frac{u^{α+1}v^{β}}{v^{q−1}} ψ^{q}] dx
≤ \frac{α + 1}{p} ∫_{Ω} \frac{−Δ_{p}u}{u^{p−1}}ϕ^{p} dx + \frac{β + 1}{q} ∫_{Ω} \frac{−Δ_{q}v}{v^{q−1}} ψ^{q} dx.

Therefore, we deduce from Lemma 2.5 that A(ϕ, ψ) = A(u, v) i.e

λ_{1} ∫_{Ω} |ϕ|^{α+1}|ψ|^{β+1} dx = λ ∫_{Ω} |u|^{α+1}|v|^{β+1} dx.

EJQTDE, 2003 No. 21, p. 12 So, by normalization, we conclude that λ_{1} = λ. (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

∫_{Ω} |∇u^{−}|^{p} dx = λ ∫_{Ω} |u|^{α}|v|^{β}vu^{−} dx
= λ [∫_{Ω^{−}} |u^{−}|^{α+1}|v^{−}|^{β+1} + ∫_{Ω} |u|^{α}|v|^{β}u^{−}v^{+}] dx
≤ |λ| ∫_{Ω^{−}} |u^{−}|^{α+1}|v^{−}|^{β+1} dx
≤ |λ|Ω^{−}|^{ω}∥u^{−}∥_{p^{∗}}^{α+1}∥v^{−}∥_{q^{∗}}^{β+1}
≤ c|λ||Ω^{−}|^{ω}∥u^{−}∥_{1,p}^{α+1}∥v^{−}∥_{1,q}^{β+1}
≤ c|λ||Ω^{−}|^{ω} (\frac{α + 1}{p} ∫_{Ω} |∇u^{−}|^{p} dx + \frac{β + 1}{q} ∫_{Ω} |∇v^{−}|^{q} dx) .  (3.1)

Similarly, multiplying the second equation of (ES) by v^{−}, we obtain

∫_{Ω} |∇v^{−}|^{q} dx ≤ c|λ||Ω^{−}|^{ω} (\frac{α + 1}{p} ∫_{Ω} |∇u^{−}|^{p} dx + \frac{β + 1}{q} ∫_{Ω} |∇v^{−}|^{q} dx) .

(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.

3.2. 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 (λ̄; (0, 0)) is a bifurcation point of (BS), then λ̄ is an eigenvalue of (ES).

Proof. (λ̄, (0, 0)) is a bifurcation point of (BS) then there is a sequence {λ_{n}, (u_{n}, v_{n})}_{n} ⊂ IR × W_{0}^{1,p}(Ω) × W_{0}^{1,q}(Ω) of nontrivial solutions of (ES) such that

\begin{matrix}λ_{n} & ⟶ & λ̄ in IR \\ (u_{n}, v_{n}) & ⟶ & (0, 0) in W_{0}^{1,p}(Ω) × W_{0}^{1,q}(Ω). \end{matrix}

EJQTDE, 2003 No. 21, p. 13 Thus

A_{u}(u_{j}, v_{j}) − λ_{j}B_{u}(u_{j}, v_{j}) = F^{λ_{j}}(u_{j}, v_{j})
A_{v}(u_{j}, v_{j}) − λ_{j}B_{v}(u_{j}, v_{j}) = G^{λ_{j}}(u_{j}, v_{j}).  (3.3)

Set ū_{j} = \frac{u_{j}}{∥∇u_{j}∥_{p}^{p−1}+∥∇v_{j}∥_{q}^{q−1}} and v̄_{j} = \frac{v_{j}}{∥∇u_{j}∥_{p}^{p−1}+∥∇v_{j}∥_{q}^{q−1}} . Then ū_{j} and

v̄_{j} are bounded, it follows that there exist a pair (ū, v̄) ∈ W_{0}^{1,p}(Ω) × W_{0}^{1,q}(Ω) such that (ū_{j}, v̄_{j}) ⇀ (ū, v̄) weakly in W_{0}^{1,p}(Ω) × W_{0}^{1,q}(Ω) and a.e in Ω. Thanks to Lemma 2.5 and Lemma 2.4 we deduce the following convergence

λ_{j}B_{u}(ū_{j}, v̄_{j}) + \frac{F^{λ_{j}}(u_{j}, v_{j})}{∥∇u_{j}∥_{p}^{p−1} + ∥∇v_{j}∥_{q}^{q−1}} ⟶ λ̄B_{u}(ū, v̄)

and

λ_{j}B_{v}(u_{j}, v_{j}) + \frac{G^{λ_{j}}(u_{j}, v_{j})}{∥∇u_{j}∥_{p}^{p−1} + ∥∇v_{j}∥_{q}^{q−1}} ⟶ λ̄B_{v}(ū, v̄).

According to (3.3), we have

A_{u}(ū_{j}, v̄_{j}) ⟶ λ̄B_{u}(ū, v̄)

and

A_{v}(ū_{j}, v̄_{j}) ⟶ λ̄B_{v}(ū, v̄),

Since −Δ_{p} and −Δ_{q} are homeomorphisms, it is clear that

(ū_{j}, v̄_{j}) ⟶ (A_{u})^{−1}(λ̄B_{u}(ū, v̄))

and

(ū_{j}, v̄_{j}) ⟶ (A_{v})^{−1}(λ̄B_{v}(ū, v̄)).

Consequently by the convergence a.e in Ω we conclude that

(ū, v̄) = (A_{u})^{−1}(λ̄B_{u}(ū, v̄))

and

(ū, v̄) = (A_{v})^{−1}(λ̄B_{v}(ū, v̄)).

Finally, since ∥∇ū_{j}∥_{p}^{p−1} + ∥∇v̄_{j}∥_{q}^{q−1} = 1 we must have ū ≢ 0 and v̄ ≢ 0. It follows that (ū, v̄) solves (ES). This complete the proof in view of Remark 2.4.

Let X = IR × W_{0}^{1,p}(Ω) × W_{0}^{1,q}(Ω) be equipped with the norm

∥(λ, u, v)∥ = (|λ|^{2} + ∥∇u∥_{p}^{2} + ∥∇v∥_{q}^{2})^{\frac{1}{2}} .

EJQTDE, 2003 No. 21, p. 14 Definition 3.1. We say that

C = {(λ, (u, v)) : (λ, (u, v)) solves (BS), u ≠ 0, v ≠ 0}

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

Now let, L = \frac{α+1}{p}A_{u} + \frac{β+1}{q}A_{v}, G = \frac{1}{2}(B_{u} + B_{v}) and F = F^{λ} + G^{λ}. In virtue of the preceding results Deg[L − λG − F, B_{ϵ}(0, 0), (0, 0)] is well defined. By homotopic we have

Deg[L − λG − F, B_{ϵ}(0, 0), (0, 0)] = Deg[L − λG, B_{ϵ}(0, 0), (0, 0)].

The main bifurcation result in this paper is the following

Theorem 3.2. Under the assumptions (f) and (g), the pair (λ_{1}, (0, 0)) is a bifurcation point of (BS). Moreover, there is a continuum of nontrivial solutions C of (BS) such that (λ_{1}, (0, 0)) ∈ C̄ and C is either unbounded in E or there is λ̄ ≠ λ_{1}, an eigenvalue of (ES), with (λ̄, (0, 0)) ∈ C̄.

Proof. We will give only a sketch of the proof since it follows the lines of the proof of Theorem 14.18 in [9]. The key point in the proof is the fact that the value of

Deg[L − λG; B_{ϵ}(0), 0]  (3.4)

changes when λ crosses λ_{1}. If this fact is proved then the result follows exactly as in the classical bifurcation result of Rabinowitz [15]. Choose a > 0 such that (λ_{1}, λ_{1} + a) does not contain any eigenvalue of (ES). Then the variational characterization of λ_{1} and lemma 2.2 yield

Deg[L − λG; B_{ϵ}(0), 0] = 1,  (3.5)

when λ ∈ (λ_{1} − a, λ_{1}). To evaluate (3.4) for λ ∈ (λ_{1}, λ_{1} + a) we use the following trick. Fix a number K > 0 and define a function ψ : IR ⟶ IR by

ψ(s) = \{ \begin{matrix}0 & for & s ≤ K \\ \frac{2a}{λ_{1}}(s − 2K) & for & s ≥ 3K \end{matrix}

and ψ is positive and strictly convex in (K, 3K). Define a functional

ψ̄_{λ}(u, v) = ⟨L(u, v), (u, v)⟩ − λ⟨G(u, v), (u, v)⟩ + ψ(⟨L(u, v), (u, v)⟩).

Then ψ̄_{λ} is continuously Frèchet differentiable and its critical point (u_{0}, v_{0}) ∈ W_{0}^{1,p}(Ω) × W_{0}^{1,q}(Ω) corresponds to a solution of the following

EJQTDE, 2003 No. 21, p. 15 system

\frac{∂\overline{ψ}_{λ}}{∂u} = \frac{∂\overline{ψ}_{λ}}{∂v} = 0,

where

⟨\frac{∂\overline{ψ}_{λ}}{∂u} (u, v), (u, v)⟩ = (α+1) [ ⟨A_{u}(u, v), (u, v)⟩(1+ψ′(⟨L(u, v), (u, v)⟩)−λ⟨B_{u}(u, v), (u, v)⟩ ] ,

and

⟨\frac{∂\overline{ψ}_{λ}}{∂v} (u, v), (u, v)⟩ = (β+1) [ ⟨A_{v}(u, v), (u, v)⟩(1+ψ′(⟨L(u, v), (u, v)⟩)−λ⟨B_{v}(u, v), (u, v)⟩ ] ,

However, since λ ∈ (λ_{1} − a, λ_{1}), only nontrivial critical points of the derivative of ψ̄ noted Dψ̄_{λ} occur if

ψ′(⟨L(u_{0}, v_{0}), (u_{0}, v_{0})⟩) = \frac{λ}{λ_{1}} − 1.  (3.6)

Due to the definition of ψ we have ⟨L(u_{0}, v_{0}), (u_{0}, v_{0})⟩ ∈ (K, 3K); by (3.6) and the simplicity of λ_{1}, we conclude that (u_{0}, v_{0}) ∈ {(u_{1}, v_{1}), (−u_{1}, −v_{1})}, where (u_{1}, v_{1}) is an eigenvector associated to λ_{1}. Therefore, for λ ∈ (λ_{1}, λ_{1} + a), the derivative Dψ̄_{λ} has precisely three isolated critical point . The existence of such isolated critical points is ensured by the fact ψ̄_{λ} is weakly lower semicontinuous and

lim_{∥(u,v)∥→+∞} ∥ψ̄_{λ}∥_{1,p} = ∞,

due to the definition of ψ. So, ψ̄_{λ} attains local minima at (u_{1}, v_{1}) and (−u_{1}, −v_{1}). Note that (0, 0) is an obvious isolated critical point. It follows from Lemma 2.1 that

ind(Dψ̄_{λ}, (u_{1}, v_{1})) = ind(Dψ̄_{λ}, (−u_{1}, −v_{1})) = 1.  (3.7)

Since also

⟨Dψ̄_{λ}((u, v)), (u, v)⟩ > 0

for ∥u∥_{p} + ∥v∥_{q} = R, with R > 0, sufficiently large, we have according to Lemma 2.2 that

Deg[Dψ̄_{λ}; B_{R}(0), 0] = 1.  (3.8)

Additivity property of the degree, (3.7) and (3.8) yield

Deg[L − λG; B_{ϵ}(0), 0] = −1.  (3.9)

for any λ ∈ (λ_{1}, λ_{1} + a) and ϵ > 0 sufficiently small. Since (3.5) and (3.9) establish the ”jump” of the degree the proof is completed.

EJQTDE, 2003 No. 21, p. 16 Remarks 3.1. (i) For the case of one equation and when ∂Ω of

class C^{2,ν}, a similar bifurcation theorem has proved by Del Pino and Manasevich [11] and Binding and Huang [2]. The first authors used the continuity of λ_{1}(p) with respect to p; on the other hand, the second authors considered the following particular case f ≡ f(x, s) which satisfies f(x, s) ∈ C(Ω̄ × IR) an odd function in s and |f(x, s)| ≤ c|s|^{q−1} uniformly in Ω; where p < q < q̄ = p + \frac{p^{2}}{N}.They use C^{1,ν}(Ω̄) regularity of solutions.

(ii) Fleckinger, Manásevich and De Thelin show a similar bifurcation theorem for the following system

−Δ_{p}u = λ|u|^{p−2}u + b|v|^{p−2}v + f(λ, |u|^{p−2}u) in Ω
−Δ_{p}v = c|u|^{p−2}u + d|v|^{p−2}v in Ω
u_{|∂Ω} = v_{|∂Ω} = 0,

where Ω is sufficiently regular bounded open subset of IR^{N}. The function f : IR × IR → IR is continuous and λ is a parameter; b,c and d are constants satisfying bc > 0, d ≤ 0.

Note that the methods used in all papers are not applicable in our case (any bounded domain). So the method used to evaluate the degree is different.

References

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[9] P. Drábek, Solvability and bifurcation of nonlinear equations, Pitman Research Notes in Math. 264, Longman, Hrlow (1992).

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[14] A. Elkhalil and A. Touzani:On the first eigencurve of p-Laplacian, Partial Differential Equations, Lecture Notes in Pure and Applied Mathematics, 222, Marcel Dekker Inc., 195-205 (2002).

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Abdelouahed El Khalil 3-1390, boul. Decarie saint Laurent Quebec H4L 3N1

E-mail address: lkhlil@hotmail.com

Mohammed Ouanan and Abdelfattah Touzani Department of Mathematics, Faculty of Sciences Dhar-Mahraz P.O. Box. 1796 Atlas, Fez (Morocco)

E-mail address: m_{−}ouanan@hotmail.com
E-mail address: atouzani@iam.net.ma