The Dunkl operators are basic tools in multivariate analysis involving certain reflection symmetries. In this paper we are concerned with a fundamental problem on rational Dunkl operators, that is the local boundary behavior of generalized harmonic functions associated with Dunkl operators. We prove a local Fatou-type theorem, characterizing local existence of non-tangential boundary values of the generalized harmonic functions by their local non-tangential boundedness, and we further prove that this can also be characterized by finiteness of a “local” Lusin-type area integral, that is defined by means of Dunkl’s generalized translation and the Dunkl operators.
This paper focuses on the following homogeneous elliptic system with critical growth -Δ_p u + V(x)|u|^p-2u = 1/p^* Q_u(u,v) inℝ^N, -Δ_p v + W(x)|v|^p-2v = 1/p^* Q_v(u,v) inℝ^N, where 1 < p < N, p* = pN/(N − p), and V, W: ℝN → ℝ are two sign-changing functions. By applying a variant of the second concentration-compactness principle, we demonstrate the existence of a mountain-pass solution for the above system. Moreover, we combine a recent global compactness result by Cintra and Correia [7] with Krasnoselskii’s genus theory to demonstrate that the system has at least N distinct pairs of non-trivial solutions in the case of small perturbations of the potentials.
In this paper, we study the characterization of inner uniformity of bounded domains G in ℝn, and prove that the following three conditions are equivalent: (1) G is inner uniform; (2) G is Gromov hyperbolic and its inner metric boundary is naturally quasisymmetrically equivalent to the Gromov boundary; (3) G is Gromov hyperbolic and linearly locally connected with respect to the inner metric. The equivalence between the conditions (1) and (2), and the implication from (2) to (3) affirmatively answer three questions raised by Bonk, Heinonen, and Koskela in 2001.
The paper investigates the well-posedness and the complete regularity of the weak solutions, and the longtime dynamics for the structurally damped wave equation with almost-linear h(x, ut) and supercritical nonlinearity g(x, u) on ℝN (N ⩾ 3): utt − Δu + (− Δ)αut + h(x, ut) + g(x, u) = f, where the perturbed parameter α ∈ (1/2, 1) is a dissipative index determining the dissipative strength. We show that when the growth order p of the nonlinearity g(x, u) is up to the supercritical range: p^∗
We are concerned with a critical Choquard system with prescribed mass -Δ u+λ_1u=(I_μ∗| u|^2_μ^∗)| u|^2_μ^∗-2u+ν p(I_μ∗| v|^q)| u|^p-2u in ℝ^N, -Δ v+λ_2v=(I_μ∗| v|^2_μ^∗)| v|^2_μ^∗-2v+ν q(I_μ∗| u|^q)| v|^p-2v in ℝ^N, ∫_ℝ^Nu^2=a^2, ∫_ℝ^Nv^2=b^2, where N ≥ 3, 0 < μ < N, ν ∈ ℝ, Iμ: ℝN → ℝ is a Riesz potential, 2_μ^∗:=2N-μN-2 and 2N-μN-2
