NONLINEAR ELLIPTIC PROBLEMS WITH RESONANCE AT THE TWO FIRST EIGENVALUE : A VARIATIONAL APPROACH

We study the nonlinear elliptic problems with Dirichlet boundary condition { −∆pu = f(x, u) in Ω u = 0 on ∂Ω Resonance conditions at the first or at the second eigenvalue will be considered.


INTRODUCTION
Let us consider the Dirichlet problem (1.1)where −∆ p u = f (x, u) in Ω u = 0 on ∂Ω Ω is a bounded smooth domain in N (N ≥ 1) and the nonlinearity f : Ω × → is assumed to be a Carathéodory function with subcritical growth, that is: for some constants a, b > 0, where 1 ≤ q < p * , if N > p and ∆ p , 1 < p < ∞ is the p-laplacian ∆ p u = div(|∇u| p−2 ∇u).The operator ∆ p with p = 2 arises from a variety of physical phenomena.It is used in non-Newtonian fluids, in some reaction-diffusion problems as well as in flow through porous media.It appears also in nonlinear elasticity, glaceology and petroleum extraction.The linear case when p = 2 has been studied by many authors, see e.g [13], [9], [6] • • • The nonlinear case (p = 2), when the nonlinearity pF (x,s) stays asymptotically between λ 1 and λ 2 , where F (x, s) denotes the primitive F (x, s) = s 0 f (x, t) dt and λ 1 , λ 2 are the first and the second eigenvalues of −∆ p on W 1,p 0 (Ω), has been studied by just a few authors.A contribution in this direction is [12] where the authors use a topological method to study the case N = 1.Another contribution was made by João Marcos B. do Ó in [14] who studied the case when F (x, s) interacts only with the first eigenvalue.In this paper, we will consider three situations.
The first situation is the resonance on the right side of the first eigenvalue, we will prove the following results : Then the problem (1.1) possesses a nonzero solution u ∈ W 1,p 0 (Ω).

Remarks
1. the condition c 3.1 ) and c 3.2 ) can be replaced by 2. In the condition c 1.2 ) of theorem 1.2 we can replace +∞ by −∞, in this case the theorem can be proved without condition c 2.2 ).
The second situation is the resonance between the two first consecutive eigenvalues.To state our result, let us denote by l(x), k(x) and L(x) the corresponding limits These limits are taken uniformly for a.e.x ∈ Ω.
Then the problem (1.1) has a nontrivial solution.
The third situation is the resonance on the left side of the first eigenvalue, we will prove the following : Then the problem (1.1) possesses a nonzero solution.
In the final section, we will give examples to illustrate our results.

PROOF OF THE MAIN RESULTS
We start recalling a compactness condition of the Palais Smale type which was introduced by Cerami and which allows rather general minimax results.
A functional I ∈ 1 (E, ), E is a real Banach space, is said to satisfy the condition () at the level c (() c ) if the following holds : Remark.Using assumption (f 0 ) the functional is well defined and of class 1 on the Sobolev space W 1,p 0 (Ω) with derivative Thus, the critical points of Φ are precisely the weak solutions of (1.1).Moreover, the condition c i ) yields for every c ∈ .Denote the norm in W 1,p 0 (Ω) by .( u p = Ω |∇u| p ) and the norm in L q (Ω) by .q ( u q = ( Ω |u| q ) 1 q ).To obtain a nontrivial critical point of the functional Φ, we will apply the following version of the Mountain-Pass theorem, with condition () Theorem 2.1.Let E be a real Banach space and I ∈ 1 (E, ) satisfying condition () c , for every c > 0. Suppose that I(0) = 0, and for some α, ρ > 0 and e ∈ E with e > ρ, one has α ≤ inf u =ρ I(u) and I(e) < 0, then I has a critical value c ≥ α Remark.It is not difficult to see that the same proof of the standard Mountain-Pass theorem applies to the present context, since the deformation theorem, (theorem 1.3) in [5] is obtained with condition () in Banach space.
To prove the theorems in the first situation, we need the following preliminary lemmas.Lemma 2.1.Assume (f 0 ) and c 1.1 ) if c 3.1 ) holds then, there exists ρ, α > 0 such that Proof.Using (f 0 ) and c 1.1 ) it is easy to show that (1) for some constants A, B > 0.
Choosing ε > 0 such that β +ε < λ 1 , in view of c 3.1 ) and the inequality (1) there exists Ã = Ã(ε) ≥ 0 such that we may assume q > p, with the Poincaré inequality λ 1 u p p ≤ u p and the Sobolev inequality u q q ≤ K u q , we obtain the estimate On the other hand we claim (F (x, and using ( 3) and ( 4) the desired result follows.
Proof.Suppose by negation that there exists a sequence (t n ) such that (5) |t n | → +∞ and Since On the other hand, using lemma 2.2, Fatou's lemma with which contradicts (6), then the proof is complete.
Proof.Let us assume by negation, that Φ does not satisfy ()c for some c ∈ , then there exists a sequence (u n ) such that (7) Φ (u n )u n → 0, Φ(u n ) → c, and u n → +∞.
Proof of theorem 1.1.In view of lemmas 2.1, 2.3, 2.4 we may apply theorem 2.1 taking e = R 1 ϕ 1 , it follows that the functional Φ has a critical value c 0 ≥ α > 0, and, hence that problem (1.1) has a nontrivial solution u 0 ∈ W 1,p 0 (Ω).The proof of theorem 1.2. is similar to that of theorem 1.1 and is omitted.To prove the theorem 1.3 we will use the following lemmas.
Proof.From (f 0 ) and c 1.3 it follows that there exists constants a and b such that (9) |f (x, s)| ≤ a|s| p−1 + b.Now, suppose by negation, that Φ does not satisfy ()c for some c > 0, then there exists a sequence (u n ) such that (7) holds.

Let us define v
. We have the following claim which is inspired from [7].
Letting, m(.) = f By (7) we have |Φ (u n )w| ≤ ε n w for all w ∈ W 1,p 0 (Ω), where ε n → 0, therefore passing to the limit, we obtain Ω f v = 1, so that v = 0 .On the other hand, for any w ∈ W 1,p 0 (Ω) we have passing to the limit, we conclude The result above and claim 1 imply ) on subset of Ω of positive measure), then by the second part of ( 10), the strict monotonicity of λ 1 (cf [11]) and the strict partial monotonicity of λ 2 (cf [4]), we have ), the first part of ( 10) and ( 11) are in contradiction, hence m(.) = λ 1 and v is a λ 1 eigenfunction, so it follows that On the other hand by (8) we have 12) and c 2.3 ), Fatou's lemma yields Via ( 13) we obtain Ω L(x) dx ≤ −pc < 0 which gives a contradiction, then the proof of lemma 2.5 is complete.
Proof of theorem 1.3.In view of lemmas 2.5 and 2.6 we may apply theorem 2.1 letting e = t 0 ϕ 1 .It follows that the functional Φ has a critical value c ≥ α > 0.
Proof.Assume by contradiction that there exists c > 0 and a sequence (u n ) in W 1,p 0 (Ω) such that (7) holds.Then a subsequence of (v n ), still denoted by (v n ), where In view of (7) we have thus by c 1.4 ) and c 2.4 we obtain Passing to the limit in the above inequality, we obtain Since λ 1 v p p ≤ v p ≤ 1, from (18) we conclude that v = 0 and |u n (x)| → +∞ a.e.x ∈ Ω.

SOME EXAMPLES
This final section treats the question of verifying some applications of the hypotheses that are required in the abstract theorems presented earlier.

Example 1
We consider the boundary value problem A simple computation shows that: 1. lim |s|→+∞ pF (x, s) Hence the hypotheses of the theorem 1.1 are satisfied, and (P 1 ) is a resonant problem.where λ 1 < β < λ 2 , so taking A simple calculation shows that the primitive F satisfies : )) > 0.

Example 3
In this example we consider the Dirichlet problem (P 1 ) where the Carathéodory function f is as follows : The primitive F (x, s) = s 0 f (x, t) dt is such that Hence by theorem 1.4 the problem (P 1 ) possesses a nonzero solution in W 1,p 0 (Ω).