Eugénio Rocha
Departamento de Matemática
Universidade de Aveiro
Abstract
The main issue of the presentation is to discuss the existence and multiplicity of solutions of the (linear) Dirichlet problem
| −Δ u(x) &minus(λ/|x|2) u(x) = |u(x)|2*-2 u(x) + μ|x|α-2 u(x) + f(x) in Ω with u∈H10(Ω), |
where 0∈Ω⊆RN (N≥ 3) is a bounded domain with smooth boundary, 2*:=2N/(N-2) is the Sobolev critical exponent, 0≤λ < ((N-2)/2)² and f&isin L∞(Ω). Briefly, we will also point out the main ideas for the study of the existence of positive solutions of the (nonlinear) Dirichlet problem
| −Δp u(x) = β(x)u(x)−η+f(x,u(x)) in Ω with u∈ W1,p0 (Ω), |
which involves the combined effect of a singular term (η&ge 0) and a (p-1)-linear term f(x,u) near +∞.
( These are joint works with J. Chen and with J. Chen and N. Papageorgiou, respectively.)
Jornada SPM/CIM/CMAT - "Dia das Equações"