Abstract
The objective of this work is to investigate a nonlocal problem involving singular and critical nonlinearities:
{([u](s,p)(p))(sigma-1)(-Delta)(p)(s)u = lambda/(u)gamma + u(ps)*(-1) in Omega,
u > 0, in Omega,
u = 0, in R-N\Omega,
where Omega is a bounded domain in R-N with the smooth boundary partial derivative Omega, 0 < s < 1 < p < infinity, N > sp, 1 < sigma < p(s)*/p, with p(s)* = Np/N-ps , (-Delta)(p)(s) is the nonlocal p-Laplace operator and [u](s,p) is the Gagliardo p-seminorm. We combine some variational techniques with a truncation argument in order to show the existence and the multiplicity of positive solutions to the above problem.