ON A NONRESONANCE CONDITION BETWEEN THE FIRST AND THE SECOND EIGENVALUES FOR THE p-LAPLACIAN

We are concerned with the existence of solution for the Dirichlet problem − pu = f(x,u)+h(x) in Ω, u = 0 on ∂Ω, when f(x,u) lies in some sense between the first and the second eigenvalues of the p-Laplacian p . Extensions to more general operators which are (p−1)-homogeneous at infinity are also considered. 2000 Mathematics Subject Classification. 35J65.


Introduction.
In this paper, we are concerned with the existence of solution to the following quasilinear elliptic problem: Here Ω is a smooth bounded domain of R N , N ≥ 1, ∆ p denotes the p-Laplacian ∆ p u = div(|∇u| p−2 ∇u), 1 < p < ∞, h belongs to W −1,p (Ω) with p the Hölder conjugate of p and f is a Caratheodory function from Ω × R to R such that where λ 1 (resp., λ 2 ) is the first (resp., the second) eigenvalue of the problem −∆ p u = λ|u| p−2 u in Ω, u = 0 on ∂Ω. (1.3) Problems of this sort have been extensively studied in the 70s and 80s in the semilinear case p = 2.In the quasilinear case p ≠ 2, (1.1) was investigated for N = 1 in [6] and for N ≥ 1 in [3].In this latter work nonresonance is studied at the left of λ 1 .
One of the difficulties to deal with the partial differential equation case N ≥ 1 is the lack of knowledge of the spectrum of the p-Laplacian in that case.The basic properties of λ 1 were established in [2], while a variational characterization of λ 2 was derived recently in [4].This variational characterization of λ 2 allows the study of its (strict) monotonicity dependence with respect to a weight.This is the property which is used in our approach to (1.1).The asymmetry in our assumption (1.2) between λ 1 and λ 2 also comes from that property.In fact it remains an open question whether the last strict inequality in (1.2) can be replaced by ≤ ≡ .
In Section 3 we extend our existence result to more general operators.We consider where -homogeneity condition at infinity.Such operators were studied by Anane [1] in the variational case.Here we use degree theory for mappings of type (S) + as developed by Browder [7] and Berkowits and Mustonen [5].No variational structure is consequently needed.

2.
A result for the p-Laplacian.We seek a weak solution of (1.1), that is, where •, • denotes the duality product between W −1,p (Ω) and W ) where The first inequality in (2.3) must be understood as "less or equal almost everywhere together with strict inequality on a set of positive measure."We also assume that some uniformity holds in the inequalities in (2.3): where α is some fixed number with λ 1 < α < λ 2 .
To prove Theorem 2.3, we first establish the following estimate: where B(O, R) denotes the ball of center O and radius R in W 1,p 0 (Ω).To prove (2.9) we assume by contradiction that We can also suppose that t n converges to t ∈ [0, 1].To reach a contradiction, we use the following lemmas which give various information on w n and w.
Lemma 2.4.The sequence g n defined by is bounded in L p (Ω), and consequently, for a subsequence, g n converges weakly to some g in L p (Ω).
Proof.This is an immediate consequence of (2.6).
Proof.Since w n verifies, (2.12) we deduce from Lemma 2.4 that   which implies meas(D) = 0, that is, the conclusion of Lemma 2.6.
Lemma 2.8.w is a solution of Proof.We first prove that w is a solution of (2.28) We recall that w n satisfies in Ω, (2.29) (2.30) We also have So w is a solution of (2.27).
We can now conclude by a standard degree argument.Indeed T t is clearly completely continuous, since (2.41) Since T 0 is odd, we have, by Borsuk theorem, that deg(I −T 0 ,B(O,R),O) is an odd integer and so nonzero.It then follows that there exists u ∈ B(O, R) such that T 1 (u) = u, which proves Theorem 2.3.

3.
Generalization.Theorem 2.3 will now be extended to the case of nonhomogeneous operators.We consider the problem where The method used in Section 2 for (−∆ p ) can be adapted under suitable assumptions on A. We basically assume that A is a Leray-Lions operator which is (p − 1)homogeneous at infinity.Our precise assumptions are the following: We will be able to solve (3.1) when f (x,s) lies at infinity between the first and the second eigenvalues of the p-Laplacian (−∆ p ), in the sense of (1.2).
Remark 3.1.Equation (3.5) is a hypothesis which means that A is asymptotically homogeneous to (−∆ p ).An example of an operator which verifies (3.3), (3.4), and (3.5) is the following regularized version of the p-Laplacian: with > 0.
Let (S t ) t∈[0,1] be the family of operators from W 1,p 0 (Ω) to W −1,p (Ω) defined by for some fixed number α with λ 1 < α < λ 2 .Since the operator A is of type (S) + , S t is also of type (S) + .By the degree theory for mappings of type (S) + , as developed in Browder [7] and Berkowits and Mustonen [5], to solve (3.1) it suffices to prove the following estimate:  The rest of the proof of Lemma 3.6 uses the fact that (−∆ p ) is of type (M) and is similar to the proof of Lemma 2.8. 17 ) and consequently by (2.16), D g(x) dx = 0,(2.18) , there exists a subsequence, still denoted by (w n ), and a distribution T ∈ W −1,p (Ω), such that (−∆ p )(w n ) converges weakly to T in W −1,p (Ω); in particular lim n→+∞ − ∆ p w n ,w = T ,w .
We can also suppose that t n converges to t ∈ [0, 1].In the same manner as in the proof of Theorem 2.3, to obtain a contradiction, we use Lemmas 2.4, 2.6, and 2.7 (which do not involve the operator A) together with the following two lemmas.