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
We investigate the existence, asymptotic boundary behavior and uniqueness of viscosity solutions u ∈ C0(Ω) of equations M_𝐚(D^2u)=f(u)+h(x) in Ω ⊂ ℝn such that u(x) → ∞ as x → ∂Ω. Such solutions are referred to as large or boundary blow-up solutions. Here, Ω is a smooth bounded domain, M_𝐚 is a weighted partial trace operator, f is a non-decreasing function that satisfies the Keller–Osserman condition, and h is a continuous function in Ω. The main difficulty in the investigation rests on the possibility that M_𝐚 is very degenerate elliptic, and h is unbounded as well as sign-changing in Ω. To the best of our knowledge, large solutions to equations involving partial trace operators have not been investigated before.
We characterize the sequences nj of integers for which, for every finite continuous Borel measure μ on [0, 1], the Cesàro averages of the sequence {μ̂(n_j)} converge to 0 (where μ̂(n)=∫_0^1exp(-2π inx)dμ(x) stands for the n-th Fourier–Stieltjes coefficient of μ). Some relevant problems on distribution modulo 1 of real sequences are studied. The slow convergence of functions is introduced and used as a tool. The proof of the equivalence of several definitions of slow convergence utilizes H. P. Rosenthal’s combinatorial result.
In this paper, we obtain Calderón–Zygmund estimates for the Laplacian of the following fourth order quasilinear elliptic problem Δ(g(Δ u)Δ u)=Δ(g(f)f), where the primitive of g(t)t, G(t), is an N-function. We prove that if G(f) ∈ Lq, then G(Δu) ∈ Lq for q ≥ 1.
For a smooth, non-degenerate locally integrable structure of hypersurface type on a manifold M, we provide necessary and sufficient conditions for it to be equivalent, near a point, to a real-analytic locally integrable structure the analytic regularizability, generalizing a recent result of Zaitsev and the first author [13]. First, we discover, in our setting, a (previously unknown) invariant CR submanifold Σ in M of hypersurface type, which we call the central submanifold. We prove that the analytic regularizability of M is equivalent to that of the associated CR manifold Σ. Furthermore, as a byproduct of our construction, we show that the central manifold construction reduces the whole (smooth or analytic) equivalence problem for nondegenerate structures with the Levi positivity condition to that of the associated central manifolds, i.e., to CR geometry. Second, we make use of a classical construction due to Marson [15] and show that sufficient for the analytic regularizability of M is the analytic regularizability of the CR manifold M̃ associated with M in the sense of Marson. We show applications of both regularizability conditions to classes of locally integrable structures.
We introduce the class of strongly sofic monoids. This class of monoids strictly contains the class of sofic groups and is a proper subclass of the class of sofic monoids. We define and investigate sofic topological entropy for actions of strongly sofic monoids on compact spaces. We show that sofic topological entropy is a topological conjugacy invariant for such actions and use this fact to prove that every strongly sofic monoid is surjunctive. This means that if M is a strongly sofic monoid and A is a finite alphabet set, then every injective cellular automaton τ: AM → AM is surjective. As an application, we prove that the monoid algebra of a strongly sofic monoid with coefficients in an arbitrary field is always stably finite. Our results are extensions to strongly sofic monoids of two previously known properties of sofic groups. The first one is the celebrated Gromov–Weiss theorem asserting that every sofic group is surjunctive. The second is the Elek–Szabó theorem which says that group algebras of sofic groups satisfy Kaplansky’s stable finiteness conjecture.
We establish global bounds for solutions to stationary and time-dependent Schrödinger equations associated with the sublaplacian L on the Heisenberg group, as well as its pure fractional power L^s and conformally invariant fractional power L_s . The main ingredient is a new abstract uniform weighted resolvent estimate which is proved by using the method of weakly conjugate operators—a variant of Mourre’s commutator method—and Hardy’s type inequalities on the Heisenberg group. As applications, we show Kato-type smoothing effects for the time-dependent Schrödinger equation, and spectral stability of the sublaplacian perturbed by complex-valued decaying potentials satisfying an explicit subordination condition. In the local case s =1, we obtain uniform estimates without any symmetry or derivative loss, which improve previous results.
We prove the arithemtic quantum unique ergodicity (AQUE) conjecture for sequences of Hecke--Maass forms on quotients $\Gamma\backslash (\mathbb{H}^{(2)})^r \times (\mathbb{H}^{(3)})^s$. An argument by induction on dimension of the orbit allows us to rule out the limit measure concentrating on closed orbits of proper subgroups despite many returns of the Hecke correspondence to neighborhoods of the orbit.
We study periodic approximations of aperiodic Schrödinger operators on lattices in Lie groups with dilation structure. The potentials arise through symbolic substitution systems that have been recently introduced in this setting. We characterize convergence of spectra of associated Schrödinger operators in the Hausdorff distance via properties of finite graphs. As a consequence, new examples of periodic approximations are obtained. We further prove that there are substitution systems that do not admit periodic approximations in higher dimensions, in contrast to the one-dimensional case. On the other hand, if the spectra converge, then we show that the rate of convergence is necessarily exponentially fast. These results are new even for substitutions over ℤ^d.
We apply recent circle tangency estimates due to Pramanik–Yang–Zahl to prove sharp weighted Fourier extension estimates for the cone in ℝ^3 and 1-dimensional weights. The idea of using circle tangency estimates to study Fourier extension of the cone is originally due to Tom Wolff, who used it in part to prove the first decoupling estimates. We make an improvement to the best known Mizohata–Takeuchi-type estimates for the cone in ℝ^3 and the 1-dimensional weights as a corollary of our main theorem, where the previously best known bound follows as a corollary of refined decoupling estimates.
We are concerned with positive normalized solutions (u, λ) ∈ H1 (ℝ2) × ℝ to the following semi-linear Schrödinger equations -Δ u+λ u=f(u), in ℝ^2, satisfying the mass constraint ∫_ℝ^2| u|^2dx=c^2 . We are interested in the so-called mass-mixed case in which f has L2-subcritical growth at zero and critical growth at infinity, which in dimension two turns out to be of exponential rate. Under mild conditions, we establish the existence of two positive normalized solutions provided the prescribed mass is sufficiently small: one is a local minimizer and the second one is of mountain-pass type. We also investigate the asymptotic behavior of solutions approaching the zero-mass case, namely when c → 0+.
The momentum formulation of the surface quasi-geostrophic equations consists of two nonlinear terms, besides the pressure term, one of which cannot be written in a divergence form. When the anti-divergence operator is applied to such nonlinear terms, in general, one cannot take advantage of the differentiation operator of order minus one unless the nonlinear terms are compactly supported away from the origin in Fourier frequency. Moreover, the two nonlinear terms of the momentum surface quasi-geostrophic equations are one derivative more singular than that of the Navier-Stokes equations. Upon employing the convex integration technique to the random partial differential equations corresponding to the momentum surface quasi-geostrophic equations forced by linear multiplicative noise, these issues create various difficulties, unseen in the deterministic scenario and even in the case the noise is additive. By making a key observation in case the solution is a shear flow and rewriting the difficult stochastic commutator error in terms of the oscillation error, we prove its non-uniqueness in law.
We solve Boshernitzan's problem of characterization (in terms of so called Furstenberg systems) of bounded sequences that are orthogonal to all uniquely ergodic systems. Some variations of Boshernitzan's problem involving characteristic classes are considered. As an application, we characterize sequences orthogonal to all uniquely ergodic systems whose (unique) invariant measure yields a discrete spectrum automorphism as those satisfying an averaged Chowla property.
We show that every Hardy field extends to an ω-free Hardy field. This result relates to classical oscillation criteria for second-order homogeneous linear differential equations. It is essential in [10], and here we apply it to answer questions of Boshernitzan, and to generalize a theorem of his.
Sobolev mappings exhibiting only pointwise quasiregularity-type bounds have arisen in various applications, leading to a recently developed theory of quasiregular values. In this article, we show that by using rescaling, one obtains a direct bridge between this theory and the classical theory of quasiregular maps. More precisely, we prove that a non-constant mapping f Ω→ℝ^n with a (K, Σ)-quasiregular value at f(x_0) can be rescaled at x_0 to a non-constant K-quasiregular mapping. Our proof of this fact involves establishing a quasiregular values -version of the linear distortion bound of quasiregular mappings. A quasiregular values variant of the small K -theorem is obtained as an immediate corollary of our main result.
We show that the harmonic measure on a product of boundaries satisfies dimension conservation for a random walk with non-elementary marginals on a countable group acting on a product of hyperbolic spaces under the finite first moment condition.
We establish a fourth order sharp Sobolev trace inequality on three-balls, and its equivalence to a third order sharp Sobolev inequality on two-spheres.
