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In this short paper we study the Sobolev function property of the Rayleigh 's quotient δ ( q ) = inf u ∈ E 1 , p ( U ) ⁡ ‖ u ‖ ∇ p ‖ u ‖ q , ∂ U p 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 − Δ p u = 0 in U , subject to | ∇ u | p − 2 ∂ u ∂ ν = λ | u | q − 2 u on ∂ U to the first p -harmonic Steklov eigenpair on an exterior region U ⊊ R N when N ≥ 3 . [ABSTRACT FROM AUTHOR]

Abstract: This paper treats some variational principles for solutions of inhomogeneous p-Laplacian boundary value problems on exterior regions with dimension . Existence-uniqueness results when are provided in a space of functions that contains . Functions in are required to decay at infinity in a measure theoretic sense. Various properties of this space are derived, including results about equivalent norms, traces and an -imbedding theorem. Also an existence result for a general variational problem of this type is obtained. [Copyright &y& Elsevier]

Abstract: This paper describes spectral representations and approximations of solutions of second order, self-adjoint, linear elliptic boundary value problems on exterior regions in , for . Inhomogeneous Dirichlet, Robin and Neumann boundary conditions are treated. Some new trace results are proved and used to provide spectral characterizations of the boundary traces of functions in . Orthonormal bases of these Hilbert spaces are derived using a theory of Steklov eigenproblems. The Steklov eigenfunctions of a regularized Laplace operator on the exterior of a ball are explicitly determined when . [Copyright &y& Elsevier]

Abstract: By emplying the main ideas from G. Brosch, two results of R. Nevanlinna''s Four Value Theorem type on the unicity of meromorphic functions in the open complex plane sharing three distinct values, and a fourth value or a pair of values weakly are proved, which extend and generalize previous results of R. Nevanlinna, G.G. Gundersen, H. Ueda and G. Brosch. [Copyright &y& Elsevier]

Abstract: A uniqueness theorem for two distinct non-constant meromorphic functions that share three values of finite weights is proved, which generalizes two previous results by H.X. Yi, and X.M. Li and H.X. Yi. As applications of it, many known results by H.X. Yi and P. Li, etc. could be improved. Furthermore, with the concept of finite-weight sharing, extensions on Osgood–Yang''s conjecture and Mues'' conjecture, and a generalization of some prevenient results by M. Ozawa and H. Ueda, ect. could be obtained. [Copyright &y& Elsevier]

Abstract: In this paper, we deal with the relation between the characteristics of two meromorphic functions that share three weighted values, which improve the results of Q.C. Zhang. And through the results, two conjectures which proposed by Mues and Osgood-Yang are solved in a weaker condition. [Copyright &y& Elsevier]

In this paper, we study the existence and multiplicity results of nontrivial positive solutions to the following quasilinear elliptic equation on R N , when N ≥ 2 , − Δ p u = λ K (x) u r − 1 − V (x) u q − 1. Here, K (x) , V (x) > 0 are suitable potentials, 1 < p < r < q < ∞ , and λ > 0 is a parameter. To study this problem, some compact embedding results regarding M V q , p (R N) ↪ L K r (R N) are proved that unify and extend some recent results of the author [12–14]. [ABSTRACT FROM AUTHOR]