Abstract
The aim of this paper is to establish the existence of the principal eigencurve of the p-Laplacian operator with the nonconstant weight subject to Neumann boundary conditions. We then study the nonresonce phenomena under the first eigenvalue and under the principal eigencurve, thus we obtain existence results for some nonautonomous Neumann elliptic problems involving the p-Laplacian operator.
Highlights
Since the 80’s, several works have been devoted to questions of nonresonance for this kind of problem, in the semilinear and autonomous case (p = 2, m1 = 0 and m2 = 1) has been discussed by many authors in connection with various qualitative assumptions on the function g and its potential G
Dakkak considered problem (Pα) in the following particular case α = λ1(m1) and m2 = 1, where λ1 is the first eigenvalue of the p-Laplacian operator with weight and the Neumann boundary condition, they showed the existence of the weak solution of the problem (Pα) with conditions of nonresonance below the first eigenvalue of the −∆p
As (Pα,β) and (Pα) are equivalent, which gives that the problem (Pα) has a solution u ∈ W 1,p(Ω), for every h ∈ L∞(Ω)
Summary
Throughout this paper, Ω will be a smooth bonded domain of IRN , W 1,p(Ω) will denote the usual Sobolev space equipped with the norm. 2) λ is called an eigenvalue of problem (2.1) if there exists u ∈ W 1,p(Ω) \ {0} such that (u, λ) is a solution of problem (2.1) In this case u is called an eigenfunction associated to λ. 0 is a principal eigenvalue of problem (2.1), with the constants as eigenfunctions. 3. Existence of the principal eigencurve of the −∆p with weights in the Neumann case. The set of pairs (α, β) has the structure of a continuous curve called the principal eigencurve of the −∆p with weights in the Neumann case. Therefor, by letting n tends to +∞, we conclude that αm1(x) + τ αm2(x) ≤ 0 a.e. x ∈ Ω It follows from (3.2) and (3.3) that there exist a unique real tα ∈]τ α, 0[ which verifies fα(tα) = 1. The rest of the proof can be carried out in a similar manner to that of the case 2
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