Abstract
The objective of this work is to investigate a nonlocal problem involving singular and critical nonlinearities: ( [ u ] s , p p ) σ − 1 ( − Δ ) p s u = λ u γ + u p s ∗ − 1 in Ω , u > 0 , in Ω , u = 0 , in R N ∖ Ω , where Ω is a bounded domain in R N with the smooth boundary ∂ Ω, 0 < s < 1 < p < ∞, N > s p, 1 < σ < p s ∗ / p, with p s ∗ = N p N − p s , ( − Δ ) 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.
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