1. Introduction

Let be a bounded domain of IRN (N  1), with C1;1 boundary and let  be the outward unit normal vector on @ D. Del Santo and P. Omari, have studied in [1] the Dirichlet problem

They have proved the existence of nontrivial weak solutions for this problem for every given h 2 Lp( ) under some assumptions on the function g. In the case of Neumann elliptic problem J.-P. Gossez and P. Omari, have considered in [2] the following problem

They have shown the existence of weak solutions for this problem for every given h 2 L1 under some conditions on function g. A.Dakkak and A. Anane studied in [3] the existence of weak solutions for the problem

where 2 = 2(m) is the second eigenvalue of ?
with weight m, with m 2 M+( ) = fm 2 L1( ) : meas(fx 2 : m(x) > 0g) , 0g. We investigate in the present work the following Neumann elliptic problem

where ?
is the Laplacian operator. The functions m1;m2 2 M+( ), h 2 L1( ), g : IR ! IR is a continuous function and  is a real parameter such that  m1) or 2(m1), with 2(m1) =1).
By a solution of (P) we mean a function u 2 H1\ L1, such that

This paper is organized as follows. In section 2, we recall some results that we will use later. Section 3 is concerned with the existence of principal eigencurve of the Laplacian operator with Neumann boundary conditions. In section 4, we show a theorem of nonresonance between the first and second eigenvalue (see theorem 2). In section 5, we prove the nonresonance between the first two eigencurves for problem (P).

2. Preliminary

Let us briefly recall some properties of the spectrum of  with weight and with Neumann boundary condition to be used later. Let be a smooth bounded domain in IRN (N  1) and let m 2 M+. the eigenvalue problem is

3. Existence of the second eigen curve of the with weighs in the Neumann case

The second eigencurve of the with weighs is defined as a set C2 of those 2 IR2 such that the following Neumann problem

4. Nonresonance between the first and second eigenvalue

In this section we are interesting to the study of the existence results for the following Neumann problem

For the proof of theorem 2, we observe that the main trick introduce in [7] can be adapted in our situation. Furthermore the proof needs some technical lemmas, the two next lemmas concern an a-priori estimates on the possible solutions of the following homotopic problem.

5. Proof of Theorem 2

Our purpose now, consists in building in C an open bounded set O such that, there exist [ such that, no solution of (14) with  2 [0; 1[ occurs on the boundary @O. Homotopy invariance of the degree then yields the conclusion. The set O will have the following form

6. Nonresonance between the first two Eigencurves

1. Del Santo et P. Omari, Nonresonance conditions on the potential for a semilinear elliptic problem. J. Diff. Equ. pp. 120-138, (1994).
2. -P. Gossez and P. Omari, A necessary and sufficient condition of nonresonance for a semilinear Neumann problem, Proc. Amer. Math. Soc. Vol. 114, No. 2 (Feb., 1992), pp. 433-442.
3. Dakkak, A. Anane, Nonresonance Conditions on the Potential for a Neumann Problem, proceedings of the international conference held in Fez, partial differential equations pp. 85-102.
4. Dakkak, Etude sur le spectre et la r´esonance pour des probl`emes elliptiques de Neumann, Th`ese de 3eme cycle, Facult´e des sciences, Oujda, Maroc, (1995).
5. Dakkak , M. Hadda, Eigencurves of the p-Laplacian with weights and their asymptotic behavior EJDE, Vol 2007(2007), No. 35, pp. 1–7.
6. Dakkak and M. Moussaoui On the second eigencurve for the p-laplacian operator with weight, Bol. Soc. Paran. Mat., (3s.) v. 35 1 (2017): 281289.
7. -P. Gossez and P. Omari, On a semilinear elliptic Neumann problem with asymmetric nonlinearities. Transactions of Amer. Math. Soc. Volume 347, Number 7, pp. 2553-2562, July 1995.
8. Anane, Etude des valeurs propres et de la r´esonance pour l’operateur p-Laplacien , these de Doctorat, Universite Libre de Bruxelles, (1988).
9. Tolksdorf, Regularity for more general class of quasilinear elliptic equation J.Diff. Eq. 8, (1983), pp. 773–817.

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