Search

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]

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]