A NOTE ON THE VARIATIONAL STRUCTURE OF AN ELLIPTIC SYSTEM INVOLVING CRITICAL SOBOLEV EXPONENT

We consider an elliptic system involving critical growth conditions. We develop a technique of variational methods for elliptic systems. Using the well-known results of maximum principle for systems developed by Fleckinger et al. (1995), we can find positive solutions. Also, we generalize the systems results obtained (for the scalar case) by Brézis and Nirenberg (1983). Also, we give applications to biharmonic equations.


Introduction
In this paper, we are concerned with the existence of solutions of the elliptic system −∆u = λu + δv + g 1 (u, v), on Ω and u = v = 0 on ∂Ω, where Ω ⊂ R n , n > 2, is a bounded domain with the smooth boundary ∂Ω, λ, δ, θ, and γ are real numbers, and g 1 , g 2 are real-valued functions with critical growth.The purpose of this paper is to extend the results, obtained in [4], of elliptic equations for the case of only one equation (the scalar case) to the case of elliptic systems as (1.1).Our main tools are a variational approach developed for functionals with values on R 2 (we want to remark that it is an important innovation in this paper), a maximum principle for systems developed in [6], and a minimax approach as in [1].
we can write (1.1) as on Ω and U = Θ on ∂Ω.
It is not common to find in the literature a variational approach of problems like (1.1).Our starting point is [5] where resonance cases were considered.In that paper, the authors considered the functional where ∇F = (g 1 , g 2 ), for the study problem (1.1) in the cases The first case is known as cooperative problem and the second one as noncooperative problem.It is important to remark that J ± are real-valued functionals and thus it is no clear how critical points of J ± , called weak solutions in that paper, became classical solutions of (1.3).So, it is necessary to maintain the classical concept of weak solutions extended now to systems like (1.1) and then to develop a critical point theory for functionals with values on R 2 .

Weak solutions of (1.1)
It is natural to define weak solutions of (1.1) as follows: The novelty here is that we can choose a functional whose critical points are weak solutions of (1.1) in the sense of (1.6).In Section 3, we present such a functional.Also, we can use the regularity theory to show that weak solutions of (1.1) are classical solutions as well.
A superlinear case was considered in [9] where g 1 (u, v) = |u| r and g 2 (u, v) = |v| r , r < (n + 2)/(n − 2); sufficient conditions were given in that paper for the existence of positive solutions.The techniques used there was Leray-Schauder degree theory and measure theory.

Pohozaev's identity
Consider the general elliptic system on Ω and U = Θ on ∂Ω, where G(U) = (g 1 (u, v), g 2 (u, v)).Then we define As in the scalar case, if U is a smooth function satisfying (2.1), then it is easy to check that where ∂ η denotes the normal outer vector on ∂Ω.
Particular cases 2) , n > 2, we see that ) 3), we conclude that there is no positive solution of (2.1).This result is well known from 1965, see [8]. (b) where . Identity (2.5) gives us the following negative result.
Now, we will borrow the ideas of a maximum principle developed in [6] to prove the following theorem. (2.9) on Ω and u = v = 0 on ∂Ω are no negative solutions.
Proof.In this proof, we use the arguments of maximum principle for systems developed in [6], which plays a crucial role in the proof of Theorem 3.1.Suppose that (2.10) 2) by Φ, we get and then From (2.13) and Cauchy-Schwarz inequality, we conclude that Inequality (2.15) implies that φ = 0 or ψ = 0 and then φ = 0 and ψ = 0.By regularity, we conclude that U ≥ Θ.
Also, for weakly coupled cooperative elliptic systems, a maximum and strong maximum principle and its characterization have been developed by López-Gómez and Molina-Meyer in [7, Theorems 2.1 and 2.6] which can be used in Theorem 2.2 in the cooperative case.

Main results
For all U = (u, v), Φ = (ϕ, ψ) ∈ L 2 (Ω).We define an R 2 inner product with the following bracket: It is easy to check that The weak solutions of our problem (1.1), like we have defined in (1.6), can be represented as critical points of functional J : where

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In fact, a calculation shows that for U = (u, v) and Φ = (φ, ψ), Now, in the case Φ = (φ, φ), the previous equality is transformed in then critical points of J become weak solutions of (1.1).
For our main theorem, we use a Lagrange multiplier method which has been adapted to our purpose.Let where |U| p+1 = 1 means that u p+1 = 1 and v p+1 = 1, and whether Proof.Here we follow similar arguments to the one used in [4] for the scalar case.Let {U n } ⊂ H 1 0 (Ω) be a minimizing sequence of S C .Then as n → ∞.
A direct calculation shows that S C ≤ S D , where A well-known result, due to Brézis and Lieb [3], tells us that and therefore (3.17) A direct calculation shows that for all V, Φ ∈ H 1 0 (Ω), (3.18) Let U ∈ H 1 0 (Ω) be the function for which S C is achieved, then there exists a Lagrange multiplier µ ∈ R 2 such that, for all Φ ∈ H 1 0 (Ω),

.19)
In particular, for Φ = U, we get, from (3.18) and (3.19), that Remark 3.2.It is important to note that, for the case δ = θ = 0, hypothesis (d) of Theorem 3.1 is superfluous.In this case, our system is uncoupled and each equation can be handled separately, so we are in the context of [4, Theorem (1.1)].

Regularity of solutions
As in [2]
It is clear that weak solutions of (4.1) are the critical points of the functional Our main tool is the following theorem.
, n ≥ 4, the functional J satisfies the Palais-Smale condition on c.
The foregoing inequality, united with our hypothesis we deduce from (4.15) that Proof.First, we conclude, from hypothesis (e), that S C > Θ, then for all U 0 = (u 0 , v 0 ) ∈ H 1 0 (Ω), U 0 Now, we show that, for all U 0 = (u 0 , v 0 ) ∈ H where inf and sup are taken on suitable sets and J −1 ( c) = Φ.Therefore, U ∈ J −1 ( c) is a nonzero weak solution of (4.1).
.21) as i → ∞.Now, from (4.19) and since n ≥ 4, we deduce that S 2 * /2 − V i is greater than a positive constant for i large enough.Then from (4.21), we conclude that V i → Θ.