In this work, we consider a class of uniformly elliptic operators with a nonlocal term and mixed boundary conditions in bounded domains. We establish the existence of a principal eigenvalue and provide a result that offers both sufficient and necessary conditions for the validity of the maximum principle. As a consequence of these findings, we conduct a detailed study of an eigenvalue problem with an indefinite weight, as well as establish existence and uniqueness results for a logistic-type equation and prove some blow-up results.

This article concerns the existence and multiplicity of solutions for the following class of non-linear elliptic equations with variable exponents

\begin{equation*}\left\{\begin{array}{l}-\Delta u+\lambda u=f(x,u), \quad \text{in} \quad \mathbb{R}^N, \\\,u\in H^{1}(\mathbb{R}^N),\\\end{array}\right.\end{equation*}

where λ > 0, $N\geq 2$ and $f:\mathbb{R}^N \times \mathbb{R} \to \mathbb{R}$ is a function of the following types:

Type 1: The subcritical case:

\begin{equation*}f(x,t)=|t|^{p(\varepsilon x)-2}t, \quad \forall (x,t) \in \mathbb{R}^N\times \mathbb{R}, N\geq3, \end{equation*}

Type 2: The critical case:

\begin{equation*}f(x,t)=\mu|t|^{p(\varepsilon x)-2}t+|t|^{2^*-2}t, \quad \forall (x,t) \in \mathbb{R}^N\times \mathbb{R}, N\geq3, \end{equation*}

Type 3: The exponential subcritical growth case:

\begin{equation*} f(x,t)=\mu|t|^{p(\varepsilon x)-2}te^{\alpha|t|^\beta}, \quad \forall (x,t) \in \mathbb{R}^2\times \mathbb{R}, \end{equation*}

where parameterɛ > 0, α > 0, $\beta \in (0,2)$, $2^*=\frac{2N}{N-2}$ if $N \geq 3$ and $2^*=+\infty$ if N = 2. Related to the function $p:\mathbb{R}^{N}\rightarrow \mathbb{R}$, we assume that it is a continuous function with $p_{\max}, p_{\min} \in (2,2^*)$, where $p_{\max}=\displaystyle \max_{x \in \mathbb{R}^N}p(x)$ and $p_{\min}=\displaystyle \min_{x \in \mathbb{R}^N}p(x)$.

We show that for each λ > 0 the number of solutions is associated with the number of global maximum or global minimum points of p when ɛ is small enough. The proof of is based on the variational methods, Ekeland’s variational principle, Trundiger–Moser inequality, and the monotonicity of the ground state energy with respect to p. Our results extend those of Cao and Noussair (Multiplicity of positive and nodal solutions for nonlinear elliptic problem in $\mathbb{R}^{N}$. Ann. Inst. Henri Poincaré, Anal. Non Linéaire. 13 (1996), 567–588) and Ji, Wang and Wu (A monotone property of the ground state energy to the scalar field equation and applications. J. Lond. Math. Soc., II. Ser. 100 (2019), 804–824).

This paper deals with the exponential separation of type II, an important concept for random systems of differential equations with delay, introduced in Mierczyński et al. [18]. Two different approaches to its existence are presented. The state space X will be a separable ordered Banach space with $\dim X\geq 2$, dual space $X^{*}$, and positive cone $X^+$ normal and reproducing. In both cases, appropriate cooperativity and irreducibility conditions are assumed to provide a family of generalized Floquet subspaces. If in addition $X^*$ is also separable, one obtains an exponential separation of type II. When this is not the case, but there is an Oseledets decomposition for the continuous semiflow, the same result holds. Detailed examples are given for all the situations, including also a case where the cone is not normal.

The aim of this article is to extend the scope of the theory of regularity structures in order to deal with a large class of singular stochastic partial differential equations of the form

\begin{equation*}\partial_t u = \mathfrak{L} u+ F(u, \xi),\end{equation*}

where the differential operator $\mathfrak{L}$ fails to be elliptic. This is achieved by interpreting the base space $\mathbb{R}^{d}$ as a non-trivial homogeneous Lie group $\mathbb{G}$ such that the differential operator $\partial_t -\mathfrak{L}$ becomes a translation invariant hypoelliptic operator on $\mathbb{G}$. Prime examples are the kinetic Fokker-Planck operator $\partial_t -\Delta_v - v\cdot \nabla_x$ and heat-type operators associated with sub-Laplacians. As an application of the developed framework, we solve a class of parabolic Anderson type equations

\begin{equation*}\partial_t u = \sum_{i} X^2_i u + u (\xi-c)\end{equation*}

on the compact quotient of an arbitrary Carnot group.

In this paper, we analyse the possible homotopy types of the total space of a principal SU(2)-bundle over a 3-connected 8-dimensional Poincaré duality complex. Along the way, we also classify the 3-connected 11-dimensional complexes E formed from a wedge of S4’s and S7’s by attaching a 11-cell.

In the article, we investigate Trudinger–Moser type inequalities in presence of logarithmic kernels in dimension N. A sharp threshold, depending on N, is detected for the existence of extremal functions or blow-up, where the domain is the ball or the entire space $\mathbb{R}^N$. We also show that the extremal functions satisfy suitable Euler–Lagrange equations. When the domain is the entire space, such equations can be derived by a N-Laplacian Schrödinger equation strongly coupled with a higher order fractional Poisson’s equation. The results extends [16] to any dimension $N \geq 2$.

We give a simple way to study the isotypical components of the homology of simplicial complexes with actions of finite groups and use it for Milnor fibres of isolated complete intersection singularity (icis). We study the homology of images of mappings ft that arise as deformations of complex map germs $f:(\mathbb{C}^n,S)\to(\mathbb{C}^p,0)$, with n < p, and the behaviour of singularities (instabilities) in this context. We study two generalizations of the notion of image Milnor number µI given by Mond and give a workable way of compute them, in corank one, with Milnor numbers of icis. We also study two unexpected traits when $p \gt n+1$: stable perturbations with contractible image and homology of $\text{im} f_t$ in unexpected dimensions. We show that Houston’s conjecture, µI constant in a family implies excellency in Gaffney’s sense, is false, but we give a correct modification of the statement of the conjecture which we also prove.

We prove two sharp anisotropic weighted geometric inequalities that hold for star-shaped and F-mean convex hypersurfaces in $\mathbb{R}^{n+1}$, which involve the anisotropic p-momentum, the anisotropic perimeter, and the volume of the region enclosed by the hypersurface. We also consider their quantitative versions characterized by asymmetry index and the Hausdorff distance between the hypersurface and a rescaled Wulff shape. As a corollary, we obtain the stability of the Weinstock inequality for the first non-zero Steklov eigenvalue for star-shaped and strictly mean convex domains.

We discuss, in a non-Archimedean setting, the distribution of the coefficients of L-polynomials of curves of genus g over $\mathbb{F}_q$. Among other results, this allows us to prove that the $\mathbb{Q}$-vector space spanned by such characteristic polynomials has dimension g + 1. We also state a conjecture about the Archimedean distribution of the number of rational points of curves over finite fields.

Let Λ be an artin algebra and $\mathcal{M}$ be an n-cluster tilting subcategory of Λ-mod with $n \geq 2$. From the viewpoint of higher homological algebra, a question that naturally arose in Ebrahimi and Nasr-Isfahani (The completion of d-abelian categories. J. Algebra 645 (2024), 143–163) is when $\mathcal{M}$ induces an n-cluster tilting subcategory of Λ-Mod. In this article, we answer this question and explore its connection to Iyama’s question on the finiteness of n-cluster tilting subcategories of Λ-mod. In fact, our theorem reformulates Iyama’s question in terms of the vanishing of Ext and highlights its relation with the rigidity of filtered colimits of $\mathcal{M}$. Also, we show that ${\rm Add}(\mathcal{M})$ is an n-cluster tilting subcategory of Λ-Mod if and only if ${\rm Add}(\mathcal{M})$ is a maximal n-rigid subcategory of Λ-Mod if and only if $\lbrace X\in \Lambda-{\rm Mod}~|~ {\rm Ext}^i_{\Lambda}(\mathcal{M},X)=0 ~~~ {\rm for ~all}~ 0 \lt i \lt n \rbrace \subseteq {\rm Add}(\mathcal{M})$ if and only if $\mathcal{M}$ is of finite type if and only if ${\rm Ext}_{\Lambda}^1({\underrightarrow{\lim}}\mathcal{M}, {\underrightarrow{\lim}}\mathcal{M})=0$. Moreover, we present several equivalent conditions for Iyama’s question which shows the relation of Iyama’s question with different subjects in representation theory such as purity and covering theory.

We prove a large deviation principle for the slow-fast rough differential equations (RDEs) under the controlled rough path (RP) framework. The driver RPs are lifted from the mixed fractional Brownian motion (FBM) with Hurst parameter $H\in (1/3,1/2)$. Our approach is based on the continuity of the solution mapping and the variational framework for mixed FBM. By utilizing the variational representation, our problem is transformed into a qualitative property of the controlled system. In particular, the fast RDE coincides with Itô stochastic differential equation (SDE) almost surely, which possesses a unique invariant probability measure with frozen slow component. We then demonstrate the weak convergence of the controlled slow component by averaging with respect to the invariant measure of the fast equation and exploiting the continuity of the solution mapping.

Recently it has been shown that the unique local perimeter minimizing partitioning of the plane into three regions, where one region has finite area and the other two have infinite measure, is given by the so-called standard lens partition. Here we prove a sharp stability inequality for the standard lens, hence strengthening the local minimality of the lens partition in a quantitative form. As an application of this stability result we consider a nonlocal perturbation of an isoperimetric problem.

The notion of effective topological complexity, introduced by Błaszczyk and Kaluba, deals with using group actions in the configuration space in order to reduce the complexity of the motion planning algorithm. In this article, we focus on studying several properties of this notion of topological complexity. We introduce a notion of effective LS category which mimics the behaviour the usual LS category has in the non-effective setting. We use it to investigate the relationship between these effective invariants and the orbit map with respect to the group action, and we give numerous examples. Additionally, we investigate non-vanishing criteria based on a cohomological dimension bound of the saturated diagonal.

We establish a weak local boundedness to Lane–Emden systems in two-dimensional domains involving general second-order elliptic operators in divergence form and arbitrary positive powers whose product equals 1. Our result is complete in the sense that it reduces to that of Trudinger for single equations. As a counterpart, we derive a new Harnack estimate for such systems and, as a by-product, for biharmonic equations.

We investigate the existence of 4-torsion in the integral cohomology of oriented Grassmannians. We establish bounds on the characteristic rank of oriented Grassmannians and prove some cases of our previous conjecture on the characteristic rank. We also discuss the relation between the characteristic rank and a result of Stong on the height of w1 in the cohomology of Grassmannians. The existence of 4-torsion classes follows from the results on the characteristic rank via Steenrod square considerations. We thus exhibit infinitely many examples of 4-torsion classes for oriented Grassmannians. We also prove bounds on torsion exponents of oriented flag manifolds. The article also discusses consequences of our results for a more general perspective on the relation between the torsion exponent and deficiency for homogeneous spaces.

We prove a synthetic Bonnet–Myers rigidity theorem for globally hyperbolic Lorentzian length spaces with global curvature bounded below by K < 0 and an open distance realizer of length $L=\frac{\pi}{\sqrt{|K|}}$: It states that the space necessarily is a warped product with warping function $\cos: (-\frac{\pi}{2},\frac{\pi}{2})\to\mathbb{R}_+$. From this, one also sees that a globally hyperbolic spacetime with curvature bounded above by K < 0 and an open distance realizer of length $L=\frac{\pi}{\sqrt{|K|}}$ is a warped product with warping function cos.

Let X be a complex Banach space and B be a closed linear operator with domain $\mathcal{D}(B) \subset X,\,\, a,b,c,d\in\mathbb{R},$ and $0 \lt \beta \lt \alpha.$ We prove that the problem

\begin{equation*}u(t) -(aB+bI)(g_{\alpha-\beta}\ast u)(t) -(cB+dI)(g_{\alpha}\ast u)(t) = h(t), \quad t\geq 0,\end{equation*}

where $g_{\alpha}(t)=t^{\alpha-1}/\Gamma(\alpha)$ and $h:\mathbb{R}_+\to X$ is given, has a unique solution for any initial condition on $\mathcal{D}(B)\times X$ as long as the operator B generates an ad-hoc Laplace transformable and strongly continuous solution family $\{R_{\alpha,\beta}(t)\}_{t\geq 0} \subset \mathcal{L}(X).$ It is shown that such a solution family exists whenever the pair $(\alpha,\beta)$ belongs to a subset of the set $(1,2]\times(0,1]$ and B is the generator of a cosine family or a C0-semigroup in $X.$ In any case, it also depends on certain compatibility conditions on the real parameters $a,b,c,d$ that must be satisfied.

The topological structure of ‘mean dichotomy spectrum’ is shown in this article, as an extension of Sacker–Sell spectrum and non-uniform dichotomy spectrum. With regard to mean hyperbolic systems, the coexistence of expansion and contraction behaviours can lead to non-hyperbolic phenomena during evolution process. To be precise, distinct from uniform and non-uniform hyperbolic cases, error hyperbolic degree $\varepsilon(t,\tau)$ is vital to depict the spectral manifolds. As application, the reducibility theorem for mean hyperbolic systems is provided to deduce block diagonalization.

We characterize the functions with ‘small’ bounded mean oscillation (BMO) norm by establishing the precise connection between the space BMO and class $A_\infty$ of Muckenhoupt weights. We prove that there exists a universal constant $c^*_2$ such that $\Vert f \Vert_{BMO} \lt c^*_2$ if and only if $\exp f \in A_2$, where $c^*_2$ is the sharp constant in the John and Nirenberg inequality. Similarly, in dimension one, we prove that $\Vert f \Vert_{BLO} \lt 1$ if and only if $\exp f \in A_1$. As application we introduce a structure of metric space in $A_\infty$ and prove that the closed unit ball of $A_\infty$ is a Banach space.

In this article, we study the two membranes problem for operators given in terms of a mean value formula on a regular tree. We show existence of solutions under adequate conditions on the boundary data and the involved source terms. We also show that, when the boundary data are strictly separated, the coincidence set is separated from the boundary and thus it contains only a finite number of nodes.