EXTENSION OF DÍAZ-SAÁ’S INEQUALITY IN R AND APPLICATION TO A SYSTEM OF p-LAPLACIAN

The proof is based on the Picone’s identity which is very useful in problems involving p-Laplacian. In a second part, we study some properties of the first eigenvalue for a system of p-Laplacian. We use Dı́az-Saá’s inequality to prove uniqueness and Egorov’s theorem for the isolation. These results generalize J. Fleckinger, R. F. Manásevich, N. M. Stavrakakis and F. de Thélin’s work [9] for the first property and A. Anane’s one for the isolation.

In a well-known paper [7] H. Brézis and L. Oswald obtain some necessary and sufficient conditions for the existence and uniqueness of a positive solution of the equation: Later J. I. Díaz and J. E. Saá [8] extend these results to the case of an equation involving the p-Laplacian, ∆ p u = div |∇u| p−2 ∇u (p > 1).In their proof a fundamental ingredient is the so called Díaz-Saá inequality: (1) where ∆ p u = div(|∇u| p−2 ∇u).
In the case of bounded domains, the Hopf's Maximum Principle gives them the hypothesis zi zj ∈ L ∞ (Ω) which is necessary in the original proof of (1).But when we consider unbounded domains such as R N this condition is not verified in general.
Here, under some adapted hypotheses for solutions of elliptic problems, we establish an inequality of Díaz-Saá type which remains true in whole R N .When we say the hypotheses are well adapted, it means that they naturally appear in weighted problems involving the p-Laplacian on R N .
In the second part, we apply this Díaz-Saá type inequality to obtain an uniqueness result of the first eigenvalue for a system of p-Laplacian in R N (S λ ) This work follows an A. Anane's paper [3] where he studies the case of one equation and J. Fleckinger, R. F. Manásevich, N. M. Stavrakakis and F. de Thélin [9] where a system quite different is considered.In [9], a local method is presented to control a possible blowing-up of the quotient zi zj in R N .But, in the integration by parts, some integrals on the boundary appear which make the demonstration quite boring.
Finally, following A. Anane's article [3], we will conclude this paper by showing the isolation of the first eigenvalue of the system (S λ ).As in this article, we will use the Egorov's theorem.This property was not proved in [9]; besides it seems that the method used here cannot be applied for their system because intuitively the sign of u must depend on the sign of v and vice versa.

Díaz-Saá inequality
Denote by D 1,p The following results are the consequences of the principal theorems which will be presented later.
Then we have

If we have equality then there exists a constant
) such that z i ≥ 0 ( ≡ 0) and differentiable.Then the assertions above remain true if we assume that for N = p, there exists some s > 1 such that The proof of the theorems that we will present here needs Picone's identity for the p-Laplacian described for example in [2] by W. Allegretto and Y. X. Huang.Picone's result is an equality true almost everywhere which avoids some integration problems when we work in unbounded open sets.Now, we present a Picone's identity for the p-Laplacian quite more general than the W. Allegretto and Y. X. Huang's one because we only assume a sign condition for one of the two equations.Proposition 3 (Generalized Picone's identity for the p-Laplacian).Let u, v differentiable and v > 0 a. e. on Ω a subset of R N .Note a. e. on Ω.
To prove positivity, we observe that and by Young's inequality Hence by ( 2) and (3) Moreover, if L(u, v) = 0 then by ( 2) and ( 3) On N , by the equation ( 4 Indeed ∇v cannot be perpendicular to u v ∇v−∇u because it would signify that ∇u is the hypotenuse of a right triangle with edges u v ∇v and u v ∇v − ∇u, that is not possible because u v |∇v| = |∇u|.By ( 5) and ( 7) we have u v ∇v = ∇u a. e. on Ω and finally, We can easily remark that in Picone's identity, we can take any set Ω, for example non connected and unbounded.Now, we can establish the following theorem whose principal argument of the proof is the above identity.
Then we have , we can apply Vázquez's Strong Maximum Principle [16] to prove that z > 0 on R N .Moreover, we use P. Tolksdorf's regularity theorem [15] to show that for all r > 0, there exists α(r) > 0 such that z ∈ C 1,α (B r ).In particular for Ω 0 a bounded domain of R N , there exists We apply Picone's identity to the functions Φ n and z And the result follows: We now consider the equality case Let Ω 0 a bounded domain in R N and (Φ n ) n∈N defined as before. 0 But Ω0 L(Φ n , z) dx converges to Ω0 L(Φ, z) dx because Ω 0 is bounded and z ∈ C 1,α0 (Ω 0 ).So L(Φ, z) = 0 a. e. on Ω 0 , the set Ω 0 being taken arbitrary in R N we can conclude that L(Φ, z) = 0 a. e. on R N .And, by Picone's identity (Proposition 3), there exists C > 0 such that Φ = Cz.
Then we have if we assume that for p = N , there exists some s > 1 such that Moreover, if the above integral is zero then there exists C such that z = CΦ on R N .
The proof of this theorem is quite similar to Theorem 1.We solve the problem of convergence in (8) by the particular embeddings of the space W 1,p (R N ) in the case p ≥ N (see [6]): where q ∈ [p, +∞); The existence result of a solution in W 1,p (R N ) for the p-Laplacian where p ≥ N has been established by W. Allegretto and Y. X. Huang in [1].

Proofs of Proposition 1 and Proposition 2 (Díaz-Saá inequality):
We only have to apply the preceding results for the couple of the functions (z 1 , z 2 ) and we have Doing the same work for the couple of the functions (z 2 , z 1 ) and adding the two inequalities obtained we arrive to the expected result.We note that in Díaz-Saá inequality, we assume functions are positives so we needn't the absolute values as in the theorems.
Following a discussion with P. Takáč, it appears that Díaz-Saá inequality in R N and both theorems can be proved using very carefully J. I. Díaz and J. E. Saá's method.For this, we have to remark first that if this inequality is true for positive functions Φ it remains true for functions Φ changing sign [11, "Chain rule", Lemma 7.6].After that, we just have to prove the convexity of the application w −→ |∇w To have more details on this method we can see J. Fleckinger, J. Hernández, P. Takáč and F. de Thélin's article [10].But that proof needs a good attention in the computation because some terms, in particular the derivative of J, have to be defined correctly, and we think that our proof using Picone's identity is easier.

Property of first eigenvalue for a system of p-Laplacian
In this second part, we study some properties of the first eigenvalue for a potential system of p-Laplacian: And we assume: ) and b ≥ 0 ( ≡ 0).For a start, we establish existence of a first eigenvalue for (S λ ) and the regularity of the associated eigenfunctions.Theorem 3. We suppose that Hypotheses (H1) and (H2) are satisfied.
(i) System (S λ ) admits a first eigenvalue λ 1 which is positive and defined by ) is a couple of eigenfunctions solution of (S λ1 ) then for all r > 0, u ∈ C 1,ρ (B r ) and v ∈ C 1,γ (B r ) where ρ = ρ(r) > 0 and γ = γ(r) > 0; (iii) There exists a couple of eigenfunctions solution of (S λ1 ) which are positive on R N .
The proof of this theorem is more or less the same as J. Fleckinger, R. F. Manásevich, N. M. Stavrakakis and F. de Thélin's one in [9] for the system: The approach done in [9] is standard because their problem as (S λ ) is variational.
We present now an uniqueness and isolation result for the first eigenvalue.The proof is an interesting application of Theorem 1 and is simpler than in [9].Theorem 4. We suppose that Hypotheses (H1) and (H2) are satisfied.
(i) In the set of continuous functions, the dimension of eigenspace corresponding to principal eigenvalue λ 1 is 1; (ii) λ 1 is the only one eigenvalue of (S λ1 ) which corresponds to a constant sign eigenvector; (iii) λ 1 is isolated i. e. there exists > 0 such that for all λ ∈ (λ 1 , λ 1 + ] the system (S λ ) has no solution.
We apply Theorem 1 to the functions (φ, u) and (ψ, v), On the other hand, by the definition of (φ, ψ) we have By Young's inequality, we obtain So, by (10) and (11) we arrive to Again with (10) and (11) we conclude that Then applying the second result of Theorem 1 we conclude the existence of two constants c and ĉ such that u = cφ and v = ĉψ.By the particular form of the problem and Maximum Principle we can say that if c > 0 then ĉ > 0 and c p = ĉq .That means that the solutions u and v have the same sign.
To prove the uniqueness of the eigenvalue which corresponds to a constant sign eigenvector, we suppose φ > 0, ψ > 0. (u, v) is a solution of (S λ ) and satisfies u = cφ and v = ĉψ so we can write Finally, we obtain λ = λ 1 because φ > 0, ψ > 0 and b ≡ 0.
(iii) Let (u 0 , v 0 ) an eigenvector corresponding to λ 0 (λ 0 > λ 1 > 0) solution of (S λ0 ) such that b|u 0 | α |v 0 | β u 0 v 0 = 1.We have seen above that the solutions u 0 and v 0 change sign because λ 0 = λ 1 , so we note: Multiplying the first equation by u − 0 = max(0, −u 0 ), the second by v − 0 = max(0, −v 0 ) and integrating over R N we obtain Using Hölder and Sobolev's inequalities, we have: because 1 q = 1 β+1 − α+1 p(β+1) .And in a same way: Combining Inequalities (12) and ( 13), we deduce that there exists a constant C > 0 (independent of If the claim is not true there exists a sequence (λ n , u n , v n ) n∈N where (u n , v n ) is the couple of eigenfunctions associated to the eigenvalue λ n , such that (λ n ) n∈N converges to λ 1 .We can assume that By (14), there exists a constant C independent of n such that Let R be such that ||b|| .
And therefore there is some C 0 independent of n such that: In the same way we can prove that In the remainder of the proof, we choose R such that ( 16) is satisfied and intentionally we will forget to extract sub-sequences.
First, we observe that for all (Φ, Ψ) Doing the same work for (v n ) n∈N , we arrive to conclude: (∇u n , ∇v n ) n∈N converges to some limit for the norm of L p (R N ) N × L q (R N ) N ; and because (∇u n , ∇v n ) n∈N converges weakly to (∇ũ, ∇ṽ) in L p (R N ) N × L q (R N ) N , we obtain that (∇u n , ∇v n ) n∈N (or a subsequence) converges strongly to (∇ũ, ∇ṽ) in L p (R N ) N × L q (R N ) N .
We can take the limit in the system (S λn ) and we obtain that (ũ, ṽ) is a solution of (S λ1 ).
Consequently, considering the convergence of (u n , v n ) n∈N to (u 1 , v 1 ) in D 1,p (R N ) × D 1,q (R N ), we have also (u n , v n ) n∈N (or a sub-sequence) which converges strongly to (u 1 , v 1 ) in L p (B R ) × L q (B R ) where R is fixed such that ( 16) is satisfied.And by Egorov's theorem (u n , v n ) n∈N converges uniformly to (u 1 , v 1 ) on B R except on a set with arbitrary small measure.It signifies that for n great enough, u n and v n are positive on B R except on a set with arbitrary small measure: this fact contradicts the assertion (16).Identically, if we put (ũ, ṽ) = (−u 1 , −v 1 ), we follow the same way, replacing the sets U − n and V − n by U + n and V + n .Finally, we conclude that the first principal eigenvalue λ 1 is isolated and the claim is proved.

1 p
| p and compute the directional derivative of J(w) = Ω |∇w 1 p | p dx which is formally: