Abstract
Based on properties of vector fields, we prove Hardy inequalities with remainder terms in the Heisenberg group and a compact embedding in weighted Sobolev spaces. The best constants in Hardy inequalities are determined. Then we discuss the existence of solutions for the nonlinear eigenvalue problems in the Heisenberg group with weights for the -sub-Laplacian. The asymptotic behaviour, simplicity, and isolation of the first eigenvalue are also considered.
Highlights
Let Lp,μ u = −ΔH, p u − μψp |u|p−2u dp 0≤μ≤ Q−p p p (1.1)be the Hardy operator on the Heisenberg group
Where 1 < p < Q = 2n + 2, λ ∈ R, f (ξ) ∈ Ᏺp := { f : Ω→R+ | limd(ξ)→0(dp(ξ) f (ξ)/ (ψp(ξ)) = 0, f (ξ) ∈ L∞loc(Ω \ {0})}, Ω is a bounded domain in the Heisenberg group, and the definitions of d(ξ) and ψp(ξ); see below
We investigate the weak solution of (1.2) and the asymptotic behavior of the first eigenvalue for different singular weights as μ increases to ((Q − p)/ p)p
Summary
Be the Hardy operator on the Heisenberg group. Let us recall some elementary facts on the Heisenberg group (e.g., see [15]). Let Ω1 = BH (R2)\ BH (R1) with 0 ≤ R1 < R2 ≤ ∞ and u(ξ) = v(d(ξ)) ∈ C2(Ω1) be a radial function with respect to d(ξ). We denote by c, c1, C, and so forth some positive constants usually except special narrating.
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