Mini-Workshop: Nonlinear Spectral and Eigenvalue Theory with Applications to the p-Laplace Operator Table of Contents

Asymmetric Eigenvalue Problems with Weights for the p-laplacian with Neumann Boundary Conditions M. Cuesta (Calais) (joint work with M. Arias (Granada), J.-P. Gossez (Bruxelles)) The motivation of this work is the study of (1) −∆pu = f(x, u) in Ω, ∂u ∂n = 0 on ∂Ω, where ∆pu := div(|∇u|p−2∇u), 1 < p < ∞, and Ω is a bounded smooth domain of R and |f(x, s)| ≤ a(x)|s|p−1 + b(x) with a, b belonging to suitable Lebesgue spaces. Our ultimate goal is to find optimal conditions on the limits at +∞ and −∞ of the quotients f(x, s)/|s|p−2s and pF (x, s)/|s|p (where F (x, s) := ∫ s 0 f(x, t) dt) as s → +∞ and s → −∞ to assure solvability of (1). When considering m(x) = lims→+∞ f(x,s) |s|p−2s , n(x) = lims→−∞ f(x,s) |s|p−2s , we are lead to study weighted asymmetric eigenvalue problems of the form (2) −∆pu = λ(m(x)(u+)p−1 − n(x) (u−)p−1) in Ω, ∂u ∂n = 0 on ∂Ω We will always assume that the weights m(x) and n(x) are possibly non constant, different, indefinite and belong to L(Ω) where r > N/p if p ≤ N and r = 1 if p > N . We will also assume that m and n ≡ 0 and we are only interested on positive eigenvalues. Notice that 0 is always an eigenvalue of (2). The case m(x) ≡ n(x) have been studied [5]. When m(x) are n(x) are constant and different, (2) leads to the notion of Fučik spectrum and the so-called problems of Ambrosetti-Prodi type. Analogous problems (1) and (2) have been treated with Dirichlet boundary conditions by [1]. The study of (2) start with the following symmetric eigenvalue problem (3) −∆pu = λm(x)|u|p−2u in Ω, ∂u ∂n = 0 on ∂Ω. The following value introduced by [5] plays a crucial role: λ∗(m) := inf{ ∫