We consider the existence, nonexistence and multiplicity of normalized solutions to the $p$-Laplacian equation on a bounded domain(0.1)

\begin{equation}\left\{\begin{array}{ll}-\Delta_p u+\lambda |u|^{p-2}u=g(u),&\text{in }\Omega,\\\int_\Omega |u|^p=\rho, u=0&\text{on }\partial \Omega,\end{array}\right.\end{equation}

where $\Omega$ is a bounded domain, $p\geq 2$. Firstly, under suitable assumptions on $\rho$, if $g$ is at most mass-critical at infinity, we prove the existence of infinitely many solutions. Secondly, for $\rho$ large, if $g$ is mass-supercritical, we perform a blow-up analysis to show the nonexistence of finite Morse index solutions. At last, for $\rho$ suitably small, combining with the monotonicity argument, we obtain a multiplicity result. In particular, when $p=2$, we obtain the existence of infinitely many normalized solutions.

In this paper, we study the validity of the sub-supersolution method for the equation

\begin{equation*}\begin{cases}-\mbox{div}(K(x)\nabla u)=K(x)|x|^{\alpha-2}f(x,u) \,\mbox{in } {\mathbb{R}}^{N},\\u \gt 0 \,\mbox{in } {\mathbb{R}}^{N},\end{cases}\end{equation*}

$N \geq 3$

$K(x)=exp(|x|^{\alpha}/4)$

$\alpha\geq 2$

$f$

$f$

$f$

We study the existence and multiplicity of positive bounded solutions for a class of nonlocal, non-variational elliptic problems governed by a nonhomogeneous operator with unbalanced growth, specifically the double phase operator. To tackle these challenges, we employ a combination of analytical techniques, including the sub-super solution method, variational and truncation approaches, and set-valued analysis. Furthermore, we examine a one-dimensional fixed-point problem.To the best of our knowledge, this is the first workaddressing nonlocal double phase problems using these methods.

We prove that a partially hyperbolic attracting set for a $C^2$ vector field, having slow recurrence to equilibria, supports an ergodic physical/SRB measure if, and only if, the trapping region admits non-uniform sectional expansion on a positive Lebesgue measure subset. Moreover, in this case, the attracting set supports at most finitely many ergodic physical/SRB measures, which are also Gibbs states along the central-unstable direction. This extends to continuous time systems a similar well-known result obtained for diffeomorphisms, encompassing the presence of equilibria accumulated by regular orbits within the attracting set. In codimension two the same result holds, assuming only the trajectories on the trapping region admit a sequence of times with asymptotical sectional expansion, on a positive volume subset. We present several examples of application, including the existence of physical measures for asymptotically sectional hyperbolic attracting sets, and obtain physical measures in an alternative unified way for many known examples: Lorenz-like and Rovella attractors, and sectional-hyperbolic attracting sets (including the multidimensional Lorenz attractor).

We study necessary and sufficient conditions for a 4-dimensional Lefschetz fibration over the 2-disk to admit a ${\text{Pin}}^{\pm}$-structure, extending the work of A. Stipsicz in the orientable setting. As a corollary, we get existence results of ${\text{Pin}}^{+}$ and ${\text{Pin}}^-$-structures on closed non-orientable 4-manifolds and on Lefschetz fibrations over the 2-sphere. In particular, we show via three explicit examples how to read-off ${\text{Pin}}^{\pm}$-structures from the Kirby diagram of a 4-manifold. We also provide a proof of the well-known fact that any closed 3-manifold M admits a ${\text{Pin}}^-$-structure and we find a criterion to check whether or not it admits a ${\text{Pin}}^+$-structure in terms of a handlebody decomposition. We conclude the paper with a characterization of ${\text{Pin}}^+$-structures on vector bundles.

A meta-conjecture of Coulson, Keevash, Perarnau, and Yepremyan [12] states that above the extremal threshold for a given spanning structure in a (hyper-)graph, one can find a rainbow version of that spanning structure in any suitably bounded colouring of the host (hyper-)graph. We solve one of the most pertinent outstanding cases of this conjecture by showing that for any $1\leq j\leq k-1$, if $G$ is a $k$-uniform hypergraph above the $j$-degree threshold for a loose Hamilton cycle, then any globally bounded colouring of $G$ contains a rainbow loose Hamilton cycle.

Let $\mathcal{G}$ be the class of all connected simple graphs. The Hoffman program of graphs with respect to a spectral invariant $\lambda(G)$ consists of determining all the limit points of the set $\{\lambda(G)\,\vert\, G\in\mathcal{G}\}$ and characterising all $G$’s such that $\lambda(G)$ does not exceed a fixed limit point. In this paper, we study the Hoffman program for Laplacian matching polynomials of graphs in regard to their largest Laplacian matching roots. Precisely, we determine all the limit points of the largest Laplacian matching roots of graphs less than $\tau = 2+\omega^{\frac{1}{2}}+\omega^{-\frac{1}{2}}(=4.38+)$, and then characterise the connected graphs with the largest Laplacian matching roots less than $2+\sqrt{5}$, where $\omega=\frac{1}{3}(\sqrt[3]{19+3\sqrt{33}}+\sqrt[3]{19-3\sqrt{33}}+1)$.

Over an algebraically closed field $\mathbb F$ of characteristic $p \gt 0$, the restricted twisted Heisenberg Lie algebras are studied. We use the Hochschild–Serre spectral sequence relative to its Heisenberg ideal to compute the trivial cohomology. The ordinary 1- and 2-cohomology spaces are used to compute the restricted 1- and 2-cohomology spaces and describe the restricted one-dimensional central extensions, including explicit formulas for the Lie brackets and $-^{[p]}$-operators.

The Cahn–Hilliard model with reaction terms can lead to situations in which no coarsening is taking place and, in contrast, growth and division of droplets occur which all do not grow larger than a certain size. This phenomenon has been suggested as a model for protocells, and a model based on the modified Cahn–Hilliard equation has been formulated. We introduce this equation and show the existence and uniqueness of solutions. Then, formally matched asymptotic expansions are used to identify a sharp interface limit using a scaling of the reaction term, which becomes singular when the interfacial thickness tends to zero. We compute planar solutions and study their stability under non-planar perturbations. Numerical computations for the suggested model are used to validate the sharp interface asymptotics. In addition, the numerical simulations show that the reaction terms lead to diverse phenomena such as growth and division of droplets in the obtained solutions, as well as the formation of shell-like structures.

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.

A finite point set in $\mathbb{R}^d$ is in general position if no $d + 1$ points lie on a common hyperplane. Let $\alpha _d(N)$ be the largest integer such that any set of $N$ points in $\mathbb{R}^d$, with no $d + 2$ members on a common hyperplane, contains a subset of size $\alpha _d(N)$ in general position. Using the method of hypergraph containers, Balogh and Solymosi showed that $\alpha _2(N) \lt N^{5/6 + o(1)}$. In this paper, we also use the container method to obtain new upper bounds for $\alpha _d(N)$ when $d \geq 3$. More precisely, we show that if $d$ is odd, then $\alpha _d(N) \lt N^{\frac {1}{2} + \frac {1}{2d} + o(1)}$, and if $d$ is even, we have $\alpha _d(N) \lt N^{\frac {1}{2} + \frac {1}{d-1} + o(1)}$. We also study the classical problem of determining $a(d,k,n)$, the maximum number of points selected from the grid $[n]^d$ such that no $k + 2$ members lie on a $k$-flat, and improve the previously best known bound for $a(d,k,n)$, due to Lefmann in 2008, by a polynomial factor when $k$ = 2 or 3 (mod 4).

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.

In present paper, we study the limit behavior of normalized ground states for the following mass critical Kirchhoff equation

\begin{align*}\left\{\begin{array}{ll}-(a+b\int_{\Omega}|\nabla u|^2\mathrm{d}x)\Delta u+V(x)u=\mu u+\beta^*|u|^{\frac{8}{3}}u &\text{in}\ {\Omega}, \\[0.1cm]u=0&\text{on}\ {\partial\Omega}, \\[0.1cm]\int_{\Omega}|u|^2\mathrm{d}x=1, \\[0.1cm]\end{array}\right.\end{align*}

$a\geq 0$

$\Omega\subset\mathbb R^3$

$\beta^*:=\frac{b}{2}|Q|_2^{\frac{8}{3}}$

$-2\Delta u+\frac{1}{3}u-|u|^{\frac{8}{3}}u=0$

$\mathbb R^3.$

$a\searrow 0$

We compute the Fukaya category of the symplectic blowup of a compact rational symplectic manifold at a point in the following sense: suppose a collection of Lagrangian branes satisfy Abouzaid’s criterion [Abo10] for split-generation of a bulk-deformed Fukaya category of cleanly intersecting Lagrangian branes. We show (Theorem 1.1) that for a small blowup parameter, their inverse images in the blowup together with a collection of branes near the exceptional locus split-generate the Fukaya category of the blowup. This categorifies a result on quantum cohomology by Bayer [Bay04] and is an example of a more general conjectural description of the behaviour of the Fukaya category under transitions occurring in the minimal model program, namely that minimal model program transitions generate additional summands.

The Pósa–Seymour conjecture determines the minimum degree threshold for forcing the $k$th power of a Hamilton cycle in a graph. After numerous partial results, Komlós, Sárközy, and Szemerédi proved the conjecture for sufficiently large graphs. In this paper, we focus on the analogous problem for digraphs and for oriented graphs. We asymptotically determine the minimum total degree threshold for forcing the square of a Hamilton cycle in a digraph. We also give a conjecture on the corresponding threshold for $k$th powers of a Hamilton cycle more generally. For oriented graphs, we provide a minimum semi-degree condition that forces the $k$th power of a Hamilton cycle; although this minimum semi-degree condition is not tight, it does provide the correct order of magnitude of the threshold. Turán-type problems for oriented graphs are also discussed.

We develop a high-order asymptotic expansion for the mean first passage time (MFPT) of the capture of Brownian particles by a small elliptical trap in a bounded two-dimensional region. This new result describes the effect that trap orientation plays on the capture rate and extends existing results that give information only on the role of trap position on the capture rate. Our results are validated against numerical simulations that confirm the accuracy of the asymptotic approximation. In the case of the unit disk domain, we identify a bifurcation such that the high-order correction to the global MFPT (GMFPT) is minimized when the trap is orientated in the radial direction for traps centred at $0\lt r\lt r_c :=\sqrt {2-\sqrt {2}}$. When centred at position $r_c\lt r\lt 1$, the GMFPT correction is minimized by orientating the trap in the angular direction. In the scenario of a general two-dimensional geometry, we identify the orientation that minimizes the GMFPT in terms of the regular part of the Neumann Green’s function. This theory is demonstrated on several regular domains such as disks, ellipses and rectangles.