On some critical problems for the fractional Laplacian operator
2012; Elsevier BV; Volume: 252; Issue: 11 Linguagem: Inglês
10.1016/j.jde.2012.02.023
ISSN1090-2732
AutoresBegoña Barrios, Eduardo Colorado, A. de Pablo, Urko Sánchez,
Tópico(s)Nonlinear Differential Equations Analysis
ResumoWe study the effect of lower order perturbations in the existence of positive solutions to the following critical elliptic problem involving the fractional Laplacian:{(−Δ)α/2u=λuq+uN+αN−α,u>0in Ω,u=0on ∂Ω, where Ω⊂RN is a smooth bounded domain, N⩾1, λ>0, 0<q<N+αN−α, 0<α<min{N,2}. For suitable conditions on α depending on q, we prove: In the case q<1, there exist at least two solutions for every 0<λ 0, at least one if λ=Λ, no solution if λ>Λ. For q=1 we show existence of at least one solution for 0<λ 1 the existence is shown for every λ>0. Also we prove that the solutions are bounded and regular.
Referência(s)