On Singular Problems in Nonreflexive Fractional Orlicz-Sobolev Spaces

In this work, we deal with the existence and uniqueness of a positive solution uₛ u s to the singular quasilinear problem (-Δ Φ)ˢ u=u-γ ( - Δ Φ ) s u = u - γ in the nonreflexive fractional Orlicz–Sobolev space Wˢ₀LΦ(Ω ) W 0 s L Φ ( Ω ) for $$0 0 < s < 1 . Furthermore, we show that $$u_s$$ u s converges in $$LΦ(Ω )$$ L Φ ( Ω ) to the unique positive solution $$u∈ W¹_0LΦ(Ω )$$ u ∈ W 0 1 L Φ ( Ω ) of the problem $$-Δ Ψu=u-γ$$ - Δ Ψ u = u - γ as $$s ↑ 1$$ s ↑ 1 , where $$Ψ $$ Ψ is a suitable N -function equivalent to the N -function $$Φ $$ Φ . The main difficulties in obtaining weak solutions for both singular quasilinear problems are that the associated energy functionals may not be well-defined on the whole of their natural functional frameworks, due to the lack of reflexivity and the presence of the singular term. To overcome these difficulties, we use a minimization method and present a new approach for constructing appropriate test functions, which allows us to prove that the problems admit positive minimizers and that these minimizers are weak solutions.