EXISTENCE AND MULTIPLICITY OF SOLUTIONS FOR A QUASILINEAR ELLIPTIC SYSTEM ON UNBOUNDED DOMAINS INVOLVING NONLINEAR BOUNDARY CONDITIONS∗

where Ω ⊆ R is an unbounded domain with noncompact smooth boundary ∂Ω, the outward unit normal to which is denoted by n with p > 1 and i = 1, ..., n. The growing attention for the study of the p-Laplacian operator in the last few decades is motivated by the fact that it arises in various applications. The p-Laplacian operator in (1.1) is a special case of the divergence form operator −div(a(x,∇u)), which appears in many nonlinear diffusion problems, in particular in the mathematical modeling of non-Newtonian fluids, for a discussion of some physical background, see [9]. We also refer to Aronsson-Janfalk [1] for the


Introduction and main results
The objective of this paper is to study the nonlinear elliptic boundary value system    −div(a(x)|∇u i | p−2 ∇u i ) = λf (x)u i |u i | p−2 + F ui (x, u 1 , · · · , u n ), x ∈ Ω, where Ω ⊆ R N is an unbounded domain with noncompact smooth boundary ∂Ω, the outward unit normal to which is denoted by n with p > 1 and i = 1, ..., n.
The growing attention for the study of the p-Laplacian operator in the last few decades is motivated by the fact that it arises in various applications. The p-Laplacian operator in (1.1) is a special case of the divergence form operator −div(a(x, ∇u)), which appears in many nonlinear diffusion problems, in particular in the mathematical modeling of non-Newtonian fluids, for a discussion of some physical background, see [9]. We also refer to Aronsson-Janfalk [1] for the mathematical treatment of the Hele-Shaw flow of "power-law fluids". The concept of Hele-Shaw flow refers to the flow between two closely-spaced parallel plates, close in the sense that the gap between the plates is small compared to the dimension of the plates. Quasilinear problems with a variable coefficient also appear in the mathematical model of the torsional creep. This study is based on the observation that a prismatic material rod object to a torsional moment, at sufficiently high temperature and for on extended period of time, exhibits a permanent deformation, called creep. The corresponding equations are derived under the assumptions that the components of strain and stress are linked by a power law referred to as the creep-law [12,15,16].
The boundary condition of the system (1.1) describes a flux through the boundary which depends in a nonlinear manner on the solution itself, for some physical motivation of such boundary conditions, for example see [11,19]. Some related the elliptic type equations and p-Laplacian equations results, we refer the reader to [2, 4-8, 10, 13, 14, 17, 22, 25-40] and the references therein.
Let Ω ⊆ R N be an unbounded domain with smooth boundary ∂Ω. We assume throughout that 1 < p < N , a 0 < a ∈ L ∞ (Ω), for some positive constant a 0 and b : ∂Ω → R is continuous function satisfying (1+|x|) p , and we denote n times product of this space by X = E n with respect to the norm Denote by L p (Ω, w 1 ), L q (Ω, w 2 ) and L m (∂Ω, w 3 ) the weighted Lebesgue spaces with weight functions w i (x) = (1 + |x|) αi for i = 1, 2, 3 and the norms defined by Then we have the following embedding and trace theorem.
then the embedding operator E n → (L q (Ω, w 2 )) n is continuous. If the upper bound for q be strict, then the embedding is compact.
Because the lack of separability for the functions F and h, we need to restrict the problem (1.1) to the following assumptions on f , F and h: The function f is nontrivial measurable satisfying The mapping h : ∂Ω → R is a Caratheodory function which fulfills the assumptions Also we need the following assumptions on F : Moreover, using Homogeneity property in (F 1 ), we have the so-called Euler identity We say that u = (u 1 , ..., u n ) is a weak solution to the system (1. The corresponding energy functional of the problem (1.1) is defined by Note that using Lemmas 1.1 and 1.2 we deduce that J λ is well-defined on X. Now we state our main results: hold. Then the problem (1.1) has a nontrivial weak solution for every Then the problem (1.1) has infinity many solutions for 0 < λ < Λ.

Proof of Theorem 1.1
Let us consider (H 0 ). We need the following proposition
Remark 2.1. Note that λ < Λ implies the existence of some C 0 > 0 such that , J λ is Ferechet differentiable on X and satisfies the Palais-Smale condition.
Proof. We use the notations Then the directional derivative of J λ is Clearly I ′ λ : X → X * is continuous. The operator K ′ H is a composition of the operators As a composition of continuous operators, K ′ H is also continuous. Moreover using (H 1 ), n product of trace operator X → (L m (∂Ω, w 3 )) n is compact and K ′ H is also compact.
In a similar way we obtain that the operator K ′ F is a composition of the operators , l ′ is continuous by Lemma 1.1. Again K ′ ϕ is also continuous. In a similar way K ′ ϕ is also compact. Since the assumptions (F 1 ) and (F 3 ) hold, we get F ui ∈ C(Ω × (R + ) n , R + ) are positively homogeneous of degree p * − 1. Moreover using the above fact, we get the existence of a positive constant M such that (2.1) By the Sobolev embedding theorem, we derive that K ′ F is continuous and compact and the continuous differentiability of J λ follows. Now let U m = (u 1m , ..., u nm ) ∈ X be a Palais-Smale sequence for the functional J λ , i.e., For m large enough we have

This implies
Using a direct calculation we have Also using the property (F 4 ), we have

Relations (2.4) and (2.5) yield
To show that U m contains a Cauchy sequence we use the following inequalities for ξ ∈ R N (see Diaz [9, Lemma 4.10]): In the case p ≥ 2: This concludes that there exists a subsequence of U m which converges in X because of J ′ λ (U m ) → 0 and K ′ γ is compact for γ ∈ {f, H, F }. If 1 < p < 2, modifying the proof of [18, Lemma 3], we can easily deduce that Since ||U m || B is bounded, the same arguments as the case p ≥ 2, lead to a convergent subsequence.
Proof of Theorem 1.1. We shall use the mountain pass lemma to obtain a solution. In what follows, we notice two points to verify the geometric assumptions of the mountain pass theorem. From assumptions (f 2 ) and (H 2 ), for every i > 0 there is a C ϵi > 0 such that Thus using (B 1 ) and Lemma 1.1, we have Additionally, we recall the following result: For all s ∈ (0, ∞) there is a constant C s > 0 such that (x + y) s ≤ C s (x s + y s ) for all x, y ∈ (0, ∞). Now using the estimate (1.2) and Lemma 1.1 we get

Consequently this two facts and Remark 2.1 imply that
For > 0 and R > 0 small enough, we deduce that for every (u 1 , ..., u n ) ∈ X with ||(u 1 , ...u n )|| B = R, the righthand side is strictly greater than 0.

It remains to show that there exists
Since µ > p the righthand side tends to −∞ as t → ∞ and for sufficiently large t 0 , V = (tψ, 0, ..., 0) has the desired property.
Since J λ satisfies the Palais-Smale condition and J λ (0, ..., 0) = 0, the mountain pass lemma shows that there is a nontrivial critical point of J λ in X with critical value

Proof of Theorem 1.2
We recall here a version of the Ljusternik-Schnirelman principle in Banach spaces which was discussed by Browder [3], Zeidler [41], Rabinowitz [23] and Szulkin [24]. We then shall apply the principle to establish the existence of a sequence of solutions for the problem (1.1). Let Y be a real reflexive Banach space and Σ the collection of all symmetric subsets of Y − {0} which are closed in X ( A is symmetric if A = −A). A nonempty set A ∈ Σ is said to be of genus k (denoted by γ(A) = k) if k is the smallest integer with the property that there exists an odd continuous mapping from A to R k − {0}. If there is no such k, γ(A) = ∞, and if A = ∅, γ(A) = 0.
In order to continue the proof we shall need the following proposition. By our hypotheses of f , F and h, Lemma 2.2 and Proposition 3.1, we claim thatJ λ | S F possesses at least γ(S F ) pairs of distinct critical points. Since F : Ω × (R + ) n → R + is a C 1 -function, there exists a nonempty open set O ⊂ Ω such that F (x, t 1 , ..., t n ) > 0 for all (x, t 1 , ..., t n ) ∈ O × (R + ) n . Using the properties of the genus it follows that γ( O) ≥ γ(B O ), where B O is the unit ball of W 1,p 0 ( O) ⊂ X. On the other hand it is well known that the genus of the unit ball of an infinite dimensional Banach space is infinity, so γ(S F ) = ∞. Therefore we conclude that there exists a sequence {(u 1m , ..., u nm )} ⊂ X such that any (u 1m , ..., u nm ) is a constrained critical point ofJ λ on S F . By the Lagrange multipliers rule, there exists a sequence {λ m } ⊂ R such that Since (u 1m , ..., u nm ) ∈ S F and 0 < λ < Λ, so the right hand side of (3.1) is positive and so λ m > 0. Setting we have the following equation Since λ m ̸ = 0, we derive This proves the theorem.