In this paper, we consider a Hénon-type equation for the Grushin operator. After proving a radial lemma, we establish the existence of a solution for a superlinear and supercritical problem. Additionally, we derive a symmetry-breaking result for ground-state solutions in the subcritical case.

Let $M$ denote the centered Hardy–Littlewood operator on $\mathbb{R}$. We prove that

\begin{equation*}{\rm Var} (Mf)\le {\rm Var} (f) - \frac12\big| |f(\infty)|-|f(-\infty)|\big|,\end{equation*}

for piecewise constant functions $f$ with nonzero and zero values alternating. The above inequality strengthens a recent result of Bilz and Weigt [3] proved for indicator functions of bounded variation vanishing at $\pm\infty$. We conjecture that the inequality holds for all functions of bounded variation, representing a stronger version of the existing conjecture ${\rm Var} (Mf)\le {\rm Var} (f)$. We also obtain the discrete counterpart of our theorem, moreover proving a transference result on equivalency between both settings that is of independent interest.

Commutator blueprints can be seen as blueprints for constructing RGD systems over $\mathbb {F}_2$ with prescribed commutation relations. In this paper, we construct several families of Weyl-invariant commutator blueprints, mostly of universal type. Also applying another result of the author, we obtain new examples of exotic RGD systems of universal type over $\mathbb {F}_2$. In particular, we generalize Tits’ construction of uncountably many trivalent Moufang twin trees to higher rank, we obtain an example of an RGD system of rank $3$ such that the nilpotency degree of the groups $U_w$ is unbounded, and we construct a commutator blueprint of type $(4, 4, 4)$ that is used to answer a question of Tits from the late $1980$s about twin buildings.

This paper is concerned with the following problem involving almost critical growth

\begin{equation*}\begin{cases}(-\Delta) ^su=|u|^{2_{s}^{*}-2-\varepsilon}u,& \text{in } \varOmega ,\\u=0,& \text{on } \Sigma ,\\\end{cases}\end{equation*}

where $2_{s}^{*}=\frac{2N}{N-2s}$, $s\in(\frac{1}{2},1)$, $N \gt 2s$, Ω is a bounded domain in $\mathbb{R}^N$, ɛ is a small parameter, and the boundary Σ is given in different ways according to the different definitions of the fractional Laplacian operator $(-\Delta)^{s}$. The operator $(-\Delta)^{s}$ is defined in two types: the spectral fractional Laplacian and the restricted fractional Laplacian. For the spectral case, Σ stands for $\partial \Omega$; for the restricted case, Σ is $\mathbb{R}^{N}\setminus \Omega$. Firstly, we provide a positive confirmation of the fractional Brezis–Peletier conjecture, that is, the above almost critical problem has a single bubbling solution concentrating around the non-degenerate critical point of the Robin function. Furthermore, the non-degeneracy andlocal uniqueness of this bubbling solution are established.

Let M be an open Riemann surface and $n\ge 3$ be an integer. In this paper, we establish some generic properties (in Baire category sense) in the space of all conformal minimal immersions $M\to{\mathbb{R}}^n$ endowed with the compact-open topology, pointing out that a generic such immersion is chaotic in many ways. For instance, we show that a generic conformal minimal immersion $u\colon M\to {\mathbb{R}}^n$ is non-proper, almost proper, and ${\mathfrak{g}}$-complete with respect to any given Riemannian metric ${\mathfrak{g}}$ in ${\mathbb{R}}^n$. Further, its image u(M) is dense in ${\mathbb{R}}^n$ and disjoint from ${\mathbb{Q}}^3\times {\mathbb{R}}^{n-3}$, and has infinite area, infinite total curvature, and unbounded curvature on every open set in ${\mathbb{R}}^n$. In case n = 3, we also prove that a generic conformal minimal immersion $M\to {\mathbb{R}}^3$ has infinite index of stability on every open set in ${\mathbb{R}}^3$.

We derive faithful inclusions of C*-algebras from a coend-type construction in unitary tensor categories. This gives rise to different potential notions of discreteness for an inclusion in the non-irreducible case and provides a unified framework that encloses the theory of compact quantum group actions. We also provide examples coming from semi-circular systems and from factorization homology. In the irreducible case, we establish conditions under which the C*-discrete and W*-discrete conditions are equivalent.