A non resonance under and between the two first eigenvalues in a nonlinear boundary problem

3 Proofs of theorems 61 3.1 Proof of theorem 2.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 3.2 Proof of theorem 2.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 3.3 Proof of theorem 2.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 3.4 Proof of theorem 2.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 3.5 Proof of theorem 2.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . 70


Consider the following nonlinear boundary problem
where Ω is a bounded domain in R N , p > 1, ∆ p u = div(|∇u| p−2 ∇u) is the p-Laplacian and ∂ ∂ν is the outer normal derivative.
In [2], one has proved that, in the case g (x, u) = λV (x) |u| p−2 u + h with V satisfies the same last conditions and h ∈ L s (∂Ω), the solutions are in C 1,α Ω for some α in ]0, 1[.Now we will study the case g (x, u) = f (x, u)+h, with h ∈ L p ′ (∂Ω) where p ′ is the conjugate of p and f : ∂Ω × R → R is a Caratheodory function, we show a non resonance of solutions under and between the two first eigenvalues.

Main results
In the theorems that follow we study a monotonicity of the two first eigenvalues with respect to the weight.One consider two weight's functions V 1 and V 2 satisfying the condition (1.1).Without loss of generality, one can assume that the weights are in the same space L s (∂Ω) .

Remark 2.1
The notation means that one has a large inequality a.e in ∂Ω and a strict inequality in a subset with a positive measure.
In the theorems 2.4 and 2.5 we prove the existence of solutions to the problem where h ∈ L p ′ (∂Ω), p ′ is the conjugate of p and f : ∂Ω × R → R is a Caratheodory function, with conditions on the behavior of the ratios f (x,s) s|s| p−2 and p F (x,s) |s| p under the first eigenvalue and between the two first eigenvalues of the problem Consider the following conditions where |s| p λ 1 and K + 0 a.e in ∂Ω.

One shows the following results
Theorem 2.3 If V is a weight in L ∞ (∂Ω) with λ 1 V (x) ≤ λ 2 a.e in ∂Ω, and if the problem P (V ) admits a non trivial solution u, then and u is an eigenfunction associated to λ 2 .
Remark 2.2 1) The hypothesis (h2), (h3), (h2') and (h3') mean that ∀ε > 0 there exists b ε ∈ L p ′ (∂Ω) and d ε ∈ L 1 (∂Ω) such that a.e in ∂Ω and ∀s ∈ R one has and 2) If (h1) and (h2) or (h1) and (h2') are satisfied then there exists a real a > 0 and a function b ∈ L p ′ (∂Ω) such that a.e in ∂Ω and ∀s ∈ R one has We have to use the theorems Theorem 2.6 (see [1]) Let Φ ∈ C 1 (X, R) be a functional satisfying the palaissmale condition (PS) in a Banach space X, Q 0 ⊂ X \ {0} a symmetric compact and E ⊂ X a nonempty symmetric set .If the following conditions are satisfied Φ is a critical value to the functional Φ, where Theorem 2.7 Let X be a Banach space reflexive and Φ : X −→ R a functional satisfying (i) Φ is weakly lower semi-continuous, (ii) Φ is coercive, then Φ attends his minimum.
Remark 2.3 In theorem 2.6, θ (F ) is defined for a closed and symmetric subset F in X \ {0} by:

Proofs of theorems
3.1.Proof of theorem 2.1.Let u 1 be an eigenfunction associated to λ 1 (V ) then and u 1 do not change sign in ∂Ω.Supposing that u 1 ≥ 0 in ∂Ω, one show that u 1 > 0 on ∂Ω.Indeed, if there exists x ∈ ∂Ω such that u 1 (x) = 0, by the regularity proven in [2], u 1 ∈ C 1,α Ω and by the maximum principle of Vazquez witch is impossible.Let V 1 and V 2 be two weight's functions such that for a.e in ∂Ω one has V 1 (x) V 2 (x) and u 1 be an eigenfunction associated to λ 1 (V 2 ), then and It's clear that γ (C) = 2, and |v| By the Lagrange's theorem about the extremum, this one is attained for α = 0 or β = 0, i.e and it's attained for . Thus we , • Under the condition (i).
Then, if (i) or (ii), we obtain the following inequality .
3.3.Proof of theorem 2.3.V is taken such that λ 1 V (x) a.e in ∂Ω, so, by the theorem 2.1, one has λ 1 (V ) < λ 1 (λ 1 ) = 1, and if u is a non trivial solution to the problem P (V ), then 1 is an eigenvalue, thus u changes sign on ∂Ω, and In addition to this, we have for all w in W 1,p (Ω) i.e u is an eigenfunction associated to λ 2 .
3.4.Proof of theorem 2.4.One introduces the energy's function Φ associated to the problem (2.1) Under the conditions (h1), (h2) and (h3) Φ is well defined, C 1 and for all u and v in W 1,p (Ω) By contradiction, we suppose that there exists a sequence then v n W 1,p (Ω) = 1, so (v n ) n admits a subsequence, noted also, Applying the equality (3.2) at u n and dividing by u n p−1 W 1,p (Ω) , we obtain for all w in W 1,p (Ω) w ∂σ. (3.3) Tending n → +∞ we remark that  Let, now, consider the function m (x) defined as Lemma 3.4.2The operator T : W 1,p (Ω) → W −1,p ′ (Ω) such that for all u and v in W 1,p (Ω): , is monotone of type (S + ).
Proof: For u and v in W 1,p (Ω) one has Proof: From (3.2), one has for all u and v in W 1,p (Ω) Replacing u by u n , v by v n − v in (3.5) and dividing by u n p−1 W 1,p (Ω) , one concluds and since T is (S + ) one has v n → v strongly in W 1,p (Ω) , and Moreover, tending n to +∞ in (3.3) one obtains for all w ∈ W 1,p (Ω) Proof: It's easy to see that λ 1 ≤ m (x) ≤ λ 2 a.e in B, it remains to show that λ 1 ≤ m (x) ≤ λ 2 a.e in ∂Ω\B.For this, we consider the following subsets and we prove that meas σ (D 1 ) = meas σ (D 2 ) = 0. Indeed, from (h2) one has , and one integer on D 1 , it comes ∂σ, and when n tends to +∞ one gets Since ε is arbitrary, one concludes that according to the definition of D 1 , this inequality implies that meas σ (D 1 ) = 0.By the same way one proves that meas σ (D 2 ) = 0. 2 Lemma 3.4.5From (h3) one has λ 1 m (x) < λ 2 a.e in ∂Ω.
Proof: Let us show that λ 1 m (x) a.e in ∂Ω.By contradiction we suppose that m (x) = λ 1 a.e x ∈ ∂Ω, then v is an eigenfunction associated to λ 1 , and one has by (h3), for all ε > 0, Then, one divides by u n p W 1,p (Ω) and one integer, it comes Let us return to the demonstration of the theorem 2.4.The result of theorem 2.2 assures that then 1 is an eigenvalue of the problem P (m) strictly between λ 1 (m) and λ 2 (m), absurd.Finally Φ is (PS).
ii) For applying the theorem 2.6, one constructs two sets E and Q 0 satisfying (P1) and (P2).
3. J.F.Bonder and J.D.Rossi; A nonlinear eigenvalue problem with indefinite weights related to the Sobolev trace embedding, Suported by ANPCyT PICT N. 03-05009.