Comportamento assintótico de soluções de alguns problemas elípticos em espaços de Orlicz-Sobolev

Abstract

Let Ω be a bounded, smooth domain of R N and, for each n ∈ N, let Φn be an Nfunction of the form Φn(t) = Z t 0 sφn(s) ds where φn : R → R is an even function satisfying additional properties. In the first part of this work we study the asymptotic behavior, as n → ∞, of un ∈ W 1,Φn 0 (Ω), solution of a singular problem    −∆Φn u = Λn f(x) u α in Ω, u > 0 in Ω, u = 0 on ∂Ω, (2) where 0 ≤ α ≤ 1, f is a nonnegative, nontrivial function in L 1 (Ω) and Λn is a positive constant. In Chapter 1, we prove that, if Λn = 1, then problem (2) has a unique weak solution un ∈ W 1,Φn 0 (Ω), for any 0 ≤ α ≤ 1. In Chapter 2 we show that un is the global minimizer of the energy functional Jn(u) := Z Ω Φn(|∇u|) dx − Z Ω f u (un) α dx, un ∈ W 1,Φn 0 (Ω), and exploit this fact to prove that limn→∞ un = d uniformly in Ω, where d denotes the distance function to the boundary ∂Ω. In Chapter 3 we consider the modular functional t 7→ Z Ω Φn(|∇u|) dx, under the constraint Z Ω f|u| 1−α dx = 1. In the case 0 ≤ α < 1, we prove that it admits a positive minimizer un ∈ W 1,Φn 0 (Ω) which solves (2) with Λn = Z Ω φn(|∇un|)|∇un| 2 dx. Moreover, we prove that limn→∞ un = ε −1d, uniformly in Ω, where ε = R Ω f d1−α dx. Furthermore, we also show that limn→∞ (Λn) 1 n = limn→∞ Z Ω Φn(|∇un|) dx 1 n = γ1(ε), where the function γ1 : [0, ∞) → [0,∞) is defined by γ1(t) := limn→∞ (φ 0 n (t)) 1 n , if t > 0, and γ1(0) = 0. v In order to prove these convergences, we show that γ1 is continuous, strictly increasing and onto. We also consider the sequences (Λn) 1 n and Z Ω Φn(|∇un|) dx 1 n and prove that they both converge to a positive number Λ∞. Considering the sequence of solutions un, we prove that it converges uniformly to a function u∞ ∈ C0(Ω) ∩ W1,∞(Ω), which solves the equation min{−∆∞u, γ1(|∇u|) − Λ∞} = 0 in the viscosity sense. We conclude that Λ∞ = γ1(ε) and u∞ = ε −1d. In the second part of this work, exposed in Chapter 4, we study the asymptotic behavior of the minimizers of the Rayleigh-type quotient k∇vkΦl kvkΨj , where (Φl) and (Ψj ) are sequences of N-functions. We prove that, up to subsequences, the minimizer of k∇·kΦl k·kΨj converges, as j → ∞, to the minimizer of the quotient k∇·kΦl k·k∞ . On its turn, this quotient converges, as l → ∞, to the minimizer w∞ of the Rayleigh-type quotient k∇·k∞ k·k∞ . We show that w∞ is the viscosity solution of    ∆∞

u k∇uk∞

= 0 em D := Ω \ {x?}, u kuk∞ = d sobre ∂D = Ω ∪ {x?}, where x? ∈ Ω satisfies w∞(x?) = kw∞k∞ = 1 and d(x?) = kdk∞.