On the first exterior p-harmonic Steklov eigenvalue

In this short paper we study the Sobolev function property of the Rayleigh's quotient δ(q)=infu∈E1,p(U)⁡‖u‖∇p‖u‖q,∂Up as a function of q∈[1,p⁎] for p⁎=p(N−1)N−p when p∈(1,N), as well as the asymptotic behavior of positive solutions with minimal energy of the following problem−Δpu=0inU,subject to|∇u|p−2∂u∂ν=λ|u|q−2uon∂U to the first p-harmonic Steklov eigenpair on an exterior region U⊊RN when N≥3.