On Singular Problems in Nonreflexive Fractional Orlicz-Sobolev Spaces

Source abstract

Abstract In this work, we deal with the existence and uniqueness of a positive solution $$u_s$$ u s to the singular quasilinear problem $$(-\Delta _{\Phi })^s u=u^{-\gamma }$$ ( - Δ Φ ) s u = u - γ in the nonreflexive fractional Orlicz–Sobolev space $$W^{s}_0L^{\Phi }(\Omega )$$ W 0 s L Φ ( Ω ) for $$0<s<1$$ 0 &lt; s &lt; 1 . Furthermore, we show that $$u_s$$ u s converges in $$L^{\Phi }(\Omega )$$ L Φ ( Ω ) to the unique positive solution $$u\in W^{1}_0L^{\Phi }(\Omega )$$ u ∈ W 0 1 L Φ ( Ω ) of the problem $$-\Delta _{\Psi }u=u^{-\gamma }$$ - Δ Ψ u = u - γ as $$s \uparrow 1$$ s ↑ 1 , where $$\Psi$$ Ψ is a suitable N -function equivalent to the N -function $$\Phi$$ 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.