Nonresonance between the first two Eigencurves of Laplacian for a Nonautonomous Neumann Problem
Adv. Sci. Technol. Eng. Syst. J. 2(5), 152–159 (2017);
DOI: 10.25046/aj020522
We consider the following Neumann elliptic problem \( \left\{ \begin{array}{rl} -\Delta u =\alpha\,m_{1}(x)\,u+m_{2}(x)\,g(u)+h(x)\quad & in \: \Omega,\\ \quad\\ \frac{\partial u}{\partial\nu} = 0\qquad\qquad\qquad\qquad\qquad\qquad\quad& on\: \partial\Omega. \end{array} \right. \) By means of Leray-Schauder degree and under some assumptions on the asymptotic behavior of the potential of the nonlinearity g, we prove an existence result for our equation for every given \(h\in L^{\infty}(\Omega)\).
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
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