On existence of minimizers for weighted Lᵖ-Hardy inequalities on C1,γ-domains with compact boundary

Let p ∈ (1,∞) , α∈ R , and Ω RN be a C1,γ -domain with a compact boundary ∂ Ω , where γ∈ (0,1] . Denote by δΩ(x) the distance of a point x∈ Ω to ∂ Ω . Let {W}1,p;α₀(Ω) be the closure of Cc∞(Ω) in {W}1,p;α(Ω) , where {W}1,p;α(Ω):= \{φ ∈{W}1,ploc(Ω) : ( \|{}|∇ φ{}|\|_{L^p(Ω;δΩ-α)}ᵖ + \|φ\|_{L^p(Ω;δΩ-(α+p))}ᵖ)<∞ \}{}. We study the following two variational constants: the weighted Hardy constant {align*} Hα,p(Ω): ={}inf \{∫Ω |∇ φ|^p δΩ-α dx : ∫Ω |φ|^p δΩ-(α+p) dx{}={}1, φ ∈ {W}1,p;α_0(Ω) \} , {align*} and the weighted Hardy constant at infinity {align*} λα,p∞(Ω) :=K Ω{} inf_{W1,pc(Ω K)} \{∫Ω K |∇ φ|^p δΩ-α dx : ∫Ω K |φ|^p δΩ-(α+p) dx=1 \}. {align*} We show that Hα,p(Ω) is attained if and only if the spectral gap Γα,p(Ω):= λα,p∞(Ω)-Hα,p(Ω) is strictly positive. Moreover, we obtain tight decay estimates for the corresponding minimizers. Furthermore, when Ω is bounded and α+p=1 , then λ1-p,p∞(Ω)=0 (no spectral gap) and the associated operator -Δ1-p,p is null-critical in Ω with respect to the weight δΩ⁻¹ , whereas, if α +p < 1 , then λα,p∞(Ω)=|α+p-1/p|ᵖ>0=Hα,p(Ω) (positive spectral gap) and -Δα,p is positive-critical in Ω with respect to the weight δΩ-(α+p) .