Abstract
We construct new optimal $L^{p}$ Hardy-type inequalities for elliptic Schrödinger-type operators with a potential term.
Highlights
For any ξ ∈ Rn and a positive definite matrix A ∈ Rn×n, let |ξ|A := Aξ, ξ, where ·, · denotes the Euclidean inner product on Rn
Qp,A,V (u) := −div |∇u|Ap−2A∇u + V |u|p−2u defined in a domain Ω ⊂ Rn, n 2, and assume that the equation Qp,A,V (u) = 0 admits a positive solution in Ω
The search for Hardy-type inequalities with optimal weight function W was originally proposed by Agmon, who raised this problem in connection with his theory of exponential decay of Schrodinger eigenfunctions [1, p. 6]
Summary
For any ξ ∈ Rn and a positive definite matrix A ∈ Rn×n, let |ξ|A := Aξ, ξ , where ·, · denotes the Euclidean inner product on Rn. We are interested to find an optimal weight function W 0 (see definition 2.29) such that the equation Qp,A,V −W (u) = 0 admits a positive solution in Ω. [19, theorem 4.3], we are interested to find an optimal weight function W 0 such that the following Hardy-type inequality is satisfied:. The search for Hardy-type inequalities with optimal weight function W was originally proposed by Agmon, who raised this problem in connection with his theory of exponential decay of Schrodinger eigenfunctions [1, p. As a corollary of the proof of theorem 1.1 we obtain the following result. Suppose that Qp,A,V admits a positive minimal Green function G(x) in Ω \ {x0} (see definition 2.22) satisfying lim G(x) = 0, and.
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Similar Papers
More From: Proceedings of the Royal Society of Edinburgh: Section A Mathematics
Disclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.