Search

The main result of this paper states that if $${\mathcal {F}}$$ is a subgroup-closed saturated formation of full characteristic, then the $${\mathcal {F}}$$ -residual of a K- $${\mathcal {F}}$$ -subnormal subgroup S of a finite group G is a large subgroup of G provided that the $${\mathcal {F}}$$ -hypercentre of every subgroup X of G containing S is contained in the $${\mathcal {F}}$$ -residual of X. This extends a recent result of Aivazidis, Safonova and Skiba.

The aim of this paper is to study the existence of solutions for critical Schrodinger–Kirchhoff-type problems involving a nonlocal integro-differential operator with potential vanishing at infinity. As a particular case, we consider the following fractional problem: $$\begin{aligned}&M\left( \iint _{{\mathbb {R}}^{2N}}\frac{|u(x)-u(y)|^{p}}{|x-y|^{N+sp}}\mathrm{dxdy}+\int _{{\mathbb {R}}^N}V(x)|u(x)|^{p}\mathrm{{d}}x\right) ((-\Delta )_p^{s}u(x)+V(x)|u|^{p-2}u)\\&\quad =K(x)(\lambda f(x,u)+|u|^{p_s^{*}-2}u), \end{aligned}$$ where $$M:[0, \infty )\rightarrow [0, \infty )$$ is a continuous function, $$(-\Delta )_p^{s}$$ is the fractional p-Laplacian, $$00$$ is a real parameter. Using the mountain pass theorem, we obtain the existence of the above problem in suitable space W. For this, we first study the properties of the embedding from W into $$L_{K}^{\alpha }({\mathbb {R}}^N), \alpha \in [p, p_s^{*}].$$ Then, we obtain the differentiability of energy functional with some suitable conditions on f. To the best of our knowledge, this is the first existence results for degenerate Kirchhoff-type problems involving the fractional p-Laplacian with potential vanishing at infinity. Finally, we fill some gaps of papers of do O et al. (Commun Contemp Math 18: 150063, 2016) and Li et al. (Mediterr J Math 14: 80, 2017).

This paper is concerned with a Delsarte-type extremal problem. Denote by$${\mathcal {P}}(G)$$P(G)the set of positive definite continuous functions on a locally compact abelian groupG. We consider the function class, which was originally introduced by Gorbachev,$$\begin{aligned}&{\mathcal {G}}(W, Q)_G = \left\{ f \in {\mathcal {P}}(G) \cap L^1(G)~:\right. \\&\qquad \qquad \qquad \qquad \qquad \left. f(0) = 1, ~ {\text {supp}}{f_+} \subseteq W,~ {\text {supp}}{\widehat{f}} \subseteq Q \right\} \end{aligned}$$G(W,Q)G=f∈P(G)∩L1(G):f(0)=1,suppf+⊆W,suppf^⊆Qwhere$$W\subseteq G$$W⊆Gis closed and of finite Haar measure and$$Q\subseteq {\widehat{G}}$$Q⊆G^is compact. We also consider the related Delsarte-type problem of finding the extremal quantity$$\begin{aligned} {\mathcal {D}}(W,Q)_G = \sup \left\{ \int _{G} f(g) \mathrm{d}\lambda _G(g) ~ : ~ f \in {\mathcal {G}}(W,Q)_G\right\} . \end{aligned}$$D(W,Q)G=sup∫Gf(g)dλG(g):f∈G(W,Q)G.The main objective of the current paper is to prove the existence of an extremal function for the Delsarte-type extremal problem$${\mathcal {D}}(W,Q)_G$$D(W,Q)G. The existence of the extremal function has recently been established by Berdysheva and Révész in the most immediate case where$$G={\mathbb {R}}^d$$G=Rd. So, the novelty here is that we consider the problem in the general setting of locally compact abelian groups. In this way, our result provides a far reaching generalization of the former work of Berdysheva and Révész.

This paper continues the ongoing effort to study the structure of the set of nilpotent elements in noncommutative ring constructions. Let R be any ring, $${(S,\leq)}$$ a strictly (partially) ordered monoid and also $${\omega:S\rightarrow}$$ End(R) a monoid homomorphism. A skew generalized power series ring $${R[[{S,\omega\leq}}]]$$ consists of all functions from a monoid S to a coefficient ring R, whose support contains neither infinite descending chains nor infinite antichains, with pointwise addition, and with multiplication given by convolution twisted by an action $${\omega}$$ of the monoid S on the ring R. Our studies in this paper is strongly connected to the question of whether or not a skew generalized power series ring $${R[[{S,\omega\leq}}]]$$ over a nil coefficient ring R is nil, which is related to the famous question of Amitsur. However, we show that under mild “Armendariz-like” hypothesis on a coefficient ring R, we obtain stronger conditions on the coefficients of elements of a skew generalized power series ring $${R[[{S,\omega\leq}}]]$$ . We will also explore some annihilator conditions in the skew generalized power series ring setting, unifying and generalizing a number of known Armendariz-like and McCoy-like conditions in the special cases.

In this paper, we are interested in the following damped vibration system: where B is an antisymmetric $$N\times N$$ constant matrix, $$q:{\mathbb {R}}\longrightarrow {\mathbb {R}}$$ is a continuous function, $$L(t)\in C({\mathbb {R}},{\mathbb {R}}^{N^{2}})$$ is a symmetric matrix, and $$W(t,x)\in C^{1}({\mathbb {R}}\times {\mathbb {R}}^{N},{\mathbb {R}})$$ are neither autonomous nor periodic in t. The novelty of this paper is that, supposing that $$Q(t)=\int ^{t}_{0}q(s)\mathrm{d}s$$ is bounded from below and L(t) is coercive unnecessarily uniformly positively definite for all $$t\in {\mathbb {R}}$$ , we establish the existence of ground-state homoclinic solutions for (1) when the potential W(t, x) satisfies a kind of superquadratic conditions due to Ding and Luan for Schr $${\ddot{o}}$$ dinger equation. The main idea here lies in an application of a variant generalized weak linking theorem for strongly indefinite problem developed by Schechter and Zou. Some recent results in the literature are generalized and significantly improved.

Let N be a normal subgroup of a finite group G, and x an element of N. Objective that $$|x^G|=|G:C_G(x)|$$ , so $$|x^G|$$ is called “minimal” when $$C_G(x)$$ is a maximal subgroup of G. In this paper, we characterize the structure of N when $$|x^G|$$ is minimal for every non-G-central element x of N.

Let h be a non-vanishing convex univalent function and p be an analytic function in $$\mathbb {D}$$ . We consider the differential subordination $$\psi _i(p(z), z p'(z)) \prec h(z)$$ with the admissible functions $$ \psi _1:=(\beta p(z)+\gamma )^{-\alpha }\left( \tfrac{(\beta p(z)+\gamma )}{\beta (1-\alpha )}+ z p'(z)\right) $$ and $$\psi _2:=\tfrac{1}{\sqrt{\gamma \beta }}\arctan \left( \sqrt{\tfrac{\beta }{\gamma }}p^{1-\alpha }(z)\right) +\left( \tfrac{1-\alpha }{\beta p^{2 (1-\alpha )}(z)+\gamma }\right) \tfrac{z p'(z)}{p^{\alpha }(z)}$$ . The objective of this paper is to find the dominants, preferably the best dominant (say q) of the solution of the above differential subordination satisfying $$\psi _i(q(z), n zq'(z))= h(z)$$ . Furthermore, we show that $$\psi _i(q(z),zq'(z))= h(z)$$ is an exact differential equation and q is a convex univalent function in $$\mathbb {D}$$ . In addition, we estimate the sharp lower bound of $${{\,\mathrm{Re}\,}}p$$ for different choices of h and derive a univalence criterion for functions in $$\mathcal {H}$$ (class of analytic normalized functions) as an application to our results.

The interest of this paper is to deal with long-time behavior of the solutions to the following Von Karman equation involving variable sources and clamped boundary conditions: $$\begin{aligned} \quad u_{tt}+\Delta ^{2} u+a|u_t|^{m(x)-2}u_{t}=[u,F(u)]+b|u|^{p(x)-2}u,\quad \Delta ^{2}F(u)=-[u,u]. \end{aligned}$$ First of all, the authors construct a new control function and apply the Sobolev embedding inequality to establish some qualitative relationships among initial energy value, the term $$\int _{\Omega }\frac{1}{p(x)}|u|^{p(x)}\mathrm{d}x$$ and the Airy stress functions, which ensure that the energy functional are nonnegative with respect to time variable. And then, some energy estimates and Komornik inequality is used to prove a uniform estimate of decay rates of the solution which provides an estimation of long-time behavior of solutions. As we know, such results are seldom seen for the variable exponent case.

The aim of this paper is to study the following non-homogeneous Neumann-type problem 0.1 $$\begin{aligned} \left\{ \begin{array} {ll} - \mathrm{{div}}(\alpha (\vert \nabla u\vert )\nabla u) + u = u \vert u\vert ^{p - 2}, &{} \quad \mathrm{in } \, B_{1}, \\ \dfrac{\partial u}{\partial \nu } = 0, &{} \quad \mathrm{on } \, \partial B_{1} , \end{array}\right. \end{aligned}$$ where $$B_{1}$$ is the unit ball in $$\mathbb {R}^{n}$$ and $$p > 2$$ . We establish the existence of a non-constant, positive, radially non-decreasing weak solution for (0.1), under certain assumptions on $$\alpha $$ . Our approach relies on the theory of Orlicz spaces combined with a new variational method that allows one to deal with problems beyond the usual locally compactness structure and a variant of Mountain Pass Theorem.

In the last decade, many authors essentially contributed to the attractive theory of multiple zeta values. Nevertheless, since their introduction in 1992, there are still many hypotheses and open problems waiting to be solved. The aim of this paper is to develop a method for transforming the multiple zeta-star values $$\zeta ^\star (\{2\}_K,c)$$ leading to a new sum formula for alternating multiple zeta-star values. Its most simple case has the intelligible form $$\begin{aligned} \sum _{t=0}^{c-2}(-2)^{t+1} \sum _{\begin{array}{c} i\ge 2,\,\varvec{s}\in \mathbb {N}^t\\ i+|\varvec{s}\!|=c \end{array}} \zeta ^\star ({\overline{i}},\varvec{s}) =(-1)^c\cdot \zeta (c). \end{aligned}$$ As a by-product, we also establish a closed form for a new harmonic-like finite summation containing binomial coefficients.

We study the product weighted Hardy–Littlewood average $$\mathcal {H}_{\varphi }$$ in this paper. More precisely, we first give the sufficient and necessary condition for the boundedness of $$\mathcal {H}_{\varphi }$$ on the product central Morrey space $$\vec {\dot{B}}^{p,\lambda }(\mathbb {R}^n\times \mathbb {R}^m)$$ , and obtain the sharp constant at the same time. Then we obtain a characterization of the boundedness for the commutator formed by $$\mathcal {H}_{\varphi }$$ and a product central bounded mean oscillation function b. As a consequence, we give a complete answer to a question posed by Fu et al. (Forum Math 27(5):2825–2851, 2015).

The purpose of this paper is the presentation of an identity which is closely related to the sum relation involving multiple zeta star values with even arguments. Let $$E^{\star }(m,n,k)$$ be the sum of all multiple zeta star values of depth k and weight mn with arguments multiples of $$m\ge 2$$ . In this paper, we give two formulas for $$E^{\star }(2s,n,k)$$ for $$s=1,2,3$$ and in particular, by comparing the two we obtain a Bernoulli numbers identity. There are corresponding results included in a special kind of alternating multiple zeta values.

Given a graph G and a subset of vertices $$D\subseteq V(G)$$ D ⊆ V ( G ) , the external neighbourhood of D is defined as $$N_e(D)=\{u\in V(G){\setminus } D:\, N(u)\cap D\ne \varnothing \}$$ N e ( D ) = { u ∈ V ( G ) \ D : N ( u ) ∩ D ≠ ∅ } , where N(u) denotes the open neighbourhood of u. Now, given a subset $$D\subseteq V(G)$$ D ⊆ V ( G ) and a vertex $$v\in D$$ v ∈ D , the external private neighbourhood of v with respect to D is defined to be $$\mathrm{epn}(v,D)=\{u\in V(G){\setminus } D: \, N(u)\cap D=\{v\}\}.$$ epn ( v , D ) = { u ∈ V ( G ) \ D : N ( u ) ∩ D = { v } } . The strong differential of a set $$D\subseteq V(G)$$ D ⊆ V ( G ) is defined as $$\partial _s(D)=|N_e(D)|-|D_w|,$$ ∂ s ( D ) = | N e ( D ) | - | D w | , where $$D_w=\{v\in D:\, \mathrm{epn}(v,D)\ne \varnothing \}$$ D w = { v ∈ D : epn ( v , D ) ≠ ∅ } . In this paper, we focus on the study of the strong differential of a graph, which is defined as $$\begin{aligned} \partial _s(G)=\max \{\partial _s(D):\, D\subseteq V(G)\}. \end{aligned}$$ ∂ s ( G ) = max { ∂ s ( D ) : D ⊆ V ( G ) } . Among other results, we obtain general bounds on $$\partial _s(G)$$ ∂ s ( G ) and we prove a Gallai-type theorem, which states that $$\partial _s(G)+\gamma _{_I}(G)=\mathrm{n}(G)$$ ∂ s ( G ) + γ I ( G ) = n ( G ) , where $$\gamma _{_I}G)$$ γ I G ) denotes the Italian domination number of G. Therefore, we can see the theory of strong differential in graphs as a new approach to the theory of Italian domination. One of the advantages of this approach is that it allows us to study the Italian domination number without the use of functions. As we can expect, we derive new results on the Italian domination number of a graph.

In this paper, we study the constrained minimization problem (1.1) of the Kirchhoff–Schrodinger energy functional under an $$L^{2}$$ -subcritical perturbation. The existence and nonexistence of constraint minimizers are completely classified in terms of the $$L^{2}$$ -subcritical exponent q. Especially for $$q\in (\frac{4}{3},\frac{8}{3})$$ , we prove that there exists a critical value $$\beta ^{*}$$ such that (1.1) has no minimizer if the coefficient $$\beta $$ of $$L^{2}$$ -critical term satisfies $$\beta =\beta ^{*}$$ . For $$q\in (\frac{4}{3},\frac{8}{3})$$ , the blow-up behavior of minimizers as $$\beta \nearrow \beta ^{*}$$ are also analyzed rigorously if the coefficient $$\lambda $$ of $$L^{2}$$ -subcritical term satisfies $$\lambda >\lambda _{0}$$ , where $$\lambda _{0}$$ is a positive constant.

In this paper, we exhibit that non-trivial concircular vector fields play an important role in characterizing spheres, as well as Euclidean spaces. Given a non-trivial concircular vector field $$\xi $$ on a connected Riemannian manifold (M, g), two smooth functions $$\sigma $$ and $$\rho $$ called potential function and connecting function are naturally associated to $$\xi $$ . We use non-trivial concircular vector fields on n-dimensional compact Riemannian manifolds to find four different characterizations of spheres $$ {\mathbf {S}}^{n}(c)$$ . In particular, we prove an interesting result namely an n-dimensional compact Riemannian manifold (M, g) that admits a non-trivial concircular vector field $$\xi $$ such that the Ricci operator is invariant under the flow of $$\xi $$ , if and only if, (M, g) is isometric to a sphere $$ {\mathbf {S}}^{n}(c)$$ . Similarly, we find two characterizations of Euclidean spaces $${\mathbf {E}}^{n}$$ . In particular, we show that an n-dimensional complete and connected Riemannian manifold (M, g) admits a non-trivial concircular vector field $$\xi $$ that annihilates the Ricci operator, if and only if, (M, g) is isometric to the Euclidean space $${\mathbf {E}}^{n}$$ .

In this paper, we define a dual $$\phi $$ -combination $${\widetilde{Q}}_{\phi , \xi }$$ . Using the log-convexity of strictly decreasing convex function $$\phi ^{-1}$$ , we give the dual $$\phi $$ -Brunn–Minkowski inequality. Moreover, the equivalence between the dual $$\phi $$ -Brunn–Minkowski inequality and the dual $$\phi $$ -Minkowski mixed volume inequality is demonstrated.

Let $$(M^n, g(t))$$ be a compact Riemannian manifold. In this paper, we derive the evolution formula for the geometric constant $$\lambda _{a}^{b} (g)$$ as an infimum of a certain energy function when the following partial differential equation: $$\begin{aligned} -\Delta _{\phi } u + a u \log u + b S u = \lambda _{a}^{b}(g) u \end{aligned}$$ with $$\int _M u^2 d\mu = 1$$ , has positive solutions, where a and b are real constants along the extended Ricci flow and the normalized extended Ricci flow. In addition, we derive some monotonicity formulas by imposing some conditions along both the extended Ricci flow and the normalized extended Ricci flow.

Let $$(F_n)_{n\ge 0}$$ be the Fibonacci sequence given by $$F_{n+2}=F_{n+1}+F_n$$ for $$n\ge 0$$ , where $$F_0=0$$ and $$F_1=1$$ . In this paper, we explicitly find all solutions of the title Diophantine equation using lower bounds for linear forms in logarithms and properties of continued fractions. Further, we use a version of the Baker–Davenport reduction method in Diophantine approximation due to Dujella and Peth $$\ddot{\text {o}}$$ .

Let $${\mathcal {H}}$$ be a separable Hilbert space. It is known that the finite sum of Bessel sequences in $${\mathcal {H}}$$ is still a Bessel sequence. But the finite sum of generalized notions of frames does not necessarily remain stable in its initial form. In this paper, for a prescribed Bessel sequence $$F=\{f_n\}_{n=1}^\infty $$ , we introduce and study $${\mathcal {KF}}$$ , the set consisting of all operators $$K\in {\mathcal {B}}({\mathcal {H}})$$ , such that $$\{f_n\}_{n=1}^\infty $$ is a K-frame. We show that $${\mathcal {KF}}$$ is a right ideal of $${\mathcal {B}}({\mathcal {H}})$$ . We indicate by an example that $${\mathcal {KF}}$$ is not necessarily a left ideal. Moreover, we provide some sufficient conditions for the finite sum of K-frames to be a K-frame. We also use some examples to compare our results with existing ones. These examples demonstrate that our achievements do not depend on the available results. Furthermore, we study the same subject for K-g-frames and controlled frames and get some similar significant results.

Let f and g be two holomorphic curves of a ball $$\Delta (R)$$ into $${\mathbb {P}}^n({\mathbb {C}})$$ with finite growth index, and let $$H_1,\ldots ,H_{2n+2}$$ be $$2n+2$$ hyperplanes in general position. In this paper, our first purpose is to show that if f and g have the same inverse images for all $$H_i\ (1\le i\le 2n+2)$$ with multiplicities counted to level $$l_i$$ satisfying an explicitly estimate concerning $$c_f$$ and $$c_g$$ , then the map $$f\times g$$ into $${\mathbb {P}}^n({\mathbb {C}})\times {\mathbb {P}}^n({\mathbb {C}})$$ must be algebraically degenerated. Our second purpose is to prove that $$f=g$$ if they share $$2n+2$$ hyperplanes with some certain conditions (in particular, they share $$2n+2$$ hyperplanes with multiplicities counted to level $$n+1$$ ). Our results extend and improve the previous results for the case of holomorphic curve from $${\mathbb {C}}$$ on these directions.

In this paper, we give a generalization of the well-known monotone convergence theorem by replacing the usual hypothesis on monotonicity of the measurable functions, namely $$0\le f_1\le f_2\le \cdots \le f_n\le \cdots \le \infty $$ , by an inequality of Copson type, to wit, $$f_{n+k}\ge \sum _{j=1}^k\alpha _j f_{n+k-j}$$ , where $$f_j$$ are non-negative measurable functions, k is a fixed positive integer, $$0

The notions of r-Bell polynomials and their generalization, the r-Dowling polynomials are due to Mező and Cheon, Jung. Recently, Nyul and Rácz defined the r-Lah polynomials, which are close relatives of r-Bell polynomials. In the present paper, we introduce the Dowling type generalization of r-Lah polynomials, the r-Dowling–Lah polynomials. We give a comprehensive study of them using the results of the author and Nyul on r-Whitney–Lah numbers, which are the coefficients of these polynomials.

In this paper, we introduce a new conjugation $$C_{\xi }$$ on the weighted Hardy space $$H_{\rho }(\mathbb {D})$$ , where $$C_{\xi }$$ is given by (2.1) in Theorem 2.2. In particular, we prove that $$C_{\xi }$$ and $$C_{\mu ,\lambda }$$ are unitarily equivalent where $$C_{\mu ,\lambda }$$ is given in Ko and Lee (J Math Anal Appl 434:20–34, 2016). Using this, we investigate a complex symmetric Toeplitz operator $$T_{\varphi }$$ with respect to the conjugation $$C_{\xi }$$ on the weighted Hardy space $$H_{\rho }(\mathbb {D})$$ . Finally, we consider $$C_{\mu ,\lambda }$$ -invariant of Berezin transform.

In this paper, we introduce a new class of operators: absolute-(k, m) paranormal operators. We will show basic structural properties and some spectral properties of this class of operators. We will be studying the conditions under which composition operators and weighted composition operators on $$L^{2}(\mu )$$ spaces become absolute-(k, m)-paranormal operators and absolute- $$(k^{*},m)$$ -paranormal operators, in terms of Radon–Nikodym derivative $$h_{m}.$$ Some necessary and sufficient conditions for a composition operator $$C_{\phi }$$ on Fock Spaces to be an absolute-(1, m)-paranormal operators and absolute- $$(1^{*},m)$$ -paranormal operators have also been explored.

In this paper, we are concerned in establishing properties about the function $$\vartheta $$ and versions of the classical Keller–Osserman condition to prove existence of solutions to the Schrodinger quasilinear elliptic problem $$\begin{aligned} \left\{ \begin{array}{l} \displaystyle \mathrm{div}\left( \vartheta (u)\nabla u\right) -\frac{1}{2}\vartheta '(u)|\nabla u|^2=a(x)g(u)~ \text{ in }~ \Omega ,\\ u\ge 0\ \text{ in }~\Omega ,\ u(x){\mathop {\longrightarrow }\limits ^{d(x)\rightarrow 0}} \infty , \end{array} \right. \end{aligned}$$ where $$\Omega \subset {\mathbb {R}}^N$$ , with $$N\ge 3$$ , is a bounded domain, $$a:{\bar{\Omega }} \rightarrow [0,\infty )$$ and $$g:[0,\infty ) \rightarrow [0,\infty )$$ are suitable nonnegative continuous functions, $$\vartheta :{\mathbb {R}}\rightarrow (0,\infty )$$ is a $$C^1$$ -function satisfying appropriated hypotheses, and $$d(x)=\mathrm{dist}(x,\partial \Omega )$$ stands for the distance function to the boundary of $$\Omega $$ . By exploring a dual approach and the relationship among the properties of $$\vartheta $$ with its corresponding Keller–Osserman condition, we were able to show existence of solutions for this problem.

This paper is devoted to the following p-Kirchhoff type of problems: $$\begin{aligned} \left\{ \begin{array}{ll} -(a+b\int _{\Omega }|\nabla u|^{p}\,\text{ d }x)\Delta _{p} u=-\lambda |u|^{q-2}u+f(x,u),x\in \Omega \\ u=0,x\in \partial \Omega . \end{array} \right. \end{aligned}$$ Without assuming the standard subcritical polynomial growth condition ensuring the compactness of a bounded (P.S.) sequence, we show that the Dirichlet boundary value problem has at least a weak nontrivial solution by using Morse theory.

In this paper, we study the regularity of $$\log (\partial \phi )$$ when $$\phi $$ is the principal solution of the homogeneous Beltrami equation and the coefficient $$\mu $$ has compact support and is in the Sobolev space $$W^{1,p}_c({\mathbb {C}})$$ with $$1

Let R be a commutative ring with non-zero identity and $$W^*(R)$$ be the set of all non-zero and non-unit elements of R. The cozero-divisor graph of R, denoted by $$\Gamma ^{\prime }(R)$$ , is a graph with the vertex set $$W^*(R)$$ and two distinct vertices a and b are adjacent if and only if $$a\not \in Rb$$ and $$b\not \in Ra$$ . In this paper, the metric dimension and strong metric dimension of $$\Gamma ^{\prime }(R)$$ are investigated. We compute the exact values of strong metric and metric dimension in cozero-divisor graphs of reduced rings. Moreover, the metric dimension in cozero-divisor graphs of non-reduced rings is discussed.

In this paper, we investigate an existence result of the nonlinear elliptic system of the type: $$\begin{aligned} \left\{ \begin{array}{ll} \displaystyle -div\Big (A(x,v)\left| \nabla u\right| ^{p(x)-2}\nabla u\Big ) + \left| u\right| ^{p(x)-2} u =\mu &{}\ \ \text{ in }\ \Omega \\ \displaystyle -div\Big (B(x,v)\left| \nabla v\right| ^{p(x)-2}\nabla v\Big ) + \left| v\right| ^{p(x)-2} v =\gamma |\nabla u|^{q_{0}(x)} &{}\ \ \text{ in }\ \Omega ,\\ \end{array} \right. \end{aligned}$$ where $$\Omega $$ is a bounded open subset of $${\mathbb {R}}^{N},\ N\ge 2,\ 2-\frac{1}{N}0$$ ) and $$\gamma $$ is a positive constant and $$q_{0}(x)\in [1, \frac{N(p(x)-1)}{N-1}[$$ .

The aim of this paper is to present new results related to the convergence of the sequence of the complex q-Bernstein polynomials $$ \{B_{n,q}(f_\alpha ;z)\},$$ where $$01.$$ In addition, the asymptotic behavior of the polynomials $$ \{B_{n,q}(f_\alpha ;z)\},$$ with $$q>1$$ has been investigated and the obtained results are illustrated by numerical examples.

In this paper we study a system of boundary value problems involving weak p-Laplacian on the Sierpi\'nski gasket in $\mathbb{R}^2$. Parameters $\lambda, \gamma, \alpha, \beta$ are real and $11$ we show the existence of at least two nontrivial weak solutions to the system of equations for some $(\lambda,\gamma) \in \mathbb{R}^2.$, Comment: 25 pages, 2 figures

By Cartan’s version of Nevanlinna’s theory, we prove the following result: let m and n be two positive integers satisfying $$n\ge 2+m,$$ n ≥ 2 + m , let $$p\not \equiv 0$$ p ≢ 0 be a polynomial, let $$\eta \ne 0$$ η ≠ 0 be a finite complex number, let $$\omega _{1}, \omega _{2}, \ldots , \omega _{m}$$ ω 1 , ω 2 , … , ω m be m distinct finite nonzero complex numbers, and let $$H_{j}$$ H j be either exponential polynomials of degree less than q, or an ordinary polynomial in z for $$0\le j\le m$$ 0 ≤ j ≤ m , such that $$H_{j}\not \equiv 0$$ H j ≢ 0 for $$1\le j\le m.$$ 1 ≤ j ≤ m . Suppose that $$f\not \equiv \infty $$ f ≢ ∞ is a meromorphic solution of the difference equation: $$\begin{aligned} f^n(z)+p(z)f(z+\eta )&=H_0(z)+H_1(z)e^{\omega _{1}z^{q}}+H_2(z)e^{\omega _{2}z^{q}}\\&\quad +\cdots +H_m(z)e^{\omega _{m}z^{q}}, \end{aligned}$$ f n ( z ) + p ( z ) f ( z + η ) = H 0 ( z ) + H 1 ( z ) e ω 1 z q + H 2 ( z ) e ω 2 z q + ⋯ + H m ( z ) e ω m z q , such that the hyper-order of f satisfies $$\rho _2(f) ρ 2 ( f ) < 1 . Then, f reduces to a transcendental entire function, such that either $$n=m+2$$ n = m + 2 with $$H_0\not \equiv 0$$ H 0 ≢ 0 and $$\lambda (f)=\rho (f)=q,$$ λ ( f ) = ρ ( f ) = q , or $$m=2,$$ m = 2 , $$H_0=0$$ H 0 = 0 and: $$\begin{aligned} f(z)=\frac{H_1(z-\eta )e^{\omega _{1}(z-\eta )^{q}}}{p(z-\eta )} \end{aligned}$$ f ( z ) = H 1 ( z - η ) e ω 1 ( z - η ) q p ( z - η ) with $$\begin{aligned} H^n_1(z)=p^n(z)H_2(z+\eta )e^{\omega _2P_{q-1}(z)}\quad \text {and}\quad P_{q-1}(z)=\sum \limits _{k=1}^q\left( {\begin{array}{c}q\\ k\end{array}}\right) \eta ^kz^{q-k}. \end{aligned}$$ H 1 n ( z ) = p n ( z ) H 2 ( z + η ) e ω 2 P q - 1 ( z ) and P q - 1 ( z ) = ∑ k = 1 q q k η k z q - k . This result improves Theorems 1.1 and 1.3 from [19] by removing some assumptions of theirs. An example is provided to show that some results obtained in this paper, in a sense, are the best possible.

This paper considers a class of fractional Schrodinger–Poisson type systems with doubly critical growth $$\begin{aligned} \left\{ \begin{array}{ll} (-\Delta )^su+V(x)u-\phi |u|^{2^*_s-3}u=K(x)|u|^{2^*_s-2}u,&{} \text{ in } {\mathbb {R}}^3,\\ (-\Delta )^s\phi =|u|^{2^*_s-1},&{} \text{ in } {\mathbb {R}}^3, \end{array}\right. \end{aligned}$$ where $$s\in (3/4,1)$$ , $$2^*_s=\frac{6}{3-2s}$$ , $$V\in L^{\frac{3}{2s}}({\mathbb {R}}^{3})$$ , $$K\in L^{\infty }({\mathbb {R}}^{3})$$ . By applying the concentration-compactness principle and variational method, the existence of ground state solutions to the systems is derived.

In this paper, we study the single-valued extension property of hyponormal operators on Banach spaces $${\mathcal {X}}$$ . In particular, we prove that if a bounded linear operator T on $${\mathcal {X}}$$ has the property (II) or the property (I $$')$$ (see Definition 2.3), then T has the single-valued extension property. Moreover, we show that for strictly convex (resp., smooth) $$\mathcal {{\mathcal {X}}}$$ , if $$T\in {{\mathcal {L}}}(\mathcal {{\mathcal {X}}})$$ is hyponormal (resp., $$\,^*$$ -hyponormal) on $$\mathcal {{\mathcal {X}}}$$ , then T has the single-valued extension property.

In this paper, we are interested in the existence result of solutions for the nonlinear Dirichlet problem of the type: $$\begin{aligned} \left\{ \begin{aligned}&-\mathrm{div} (M(x) \nabla u )+ \gamma u^p= B \frac{|\nabla u|^q}{u^\theta }+f\ \ \mathrm{in}\ \Omega ,\\&u> 0\ \ \mathrm{in}\ \Omega ,\\&u=0\ \ \mathrm{on}\ {\partial \Omega },\\ \end{aligned} \right. \end{aligned}$$ where $$\Omega $$ is a bounded open subset of $$\mathbb {R}^N$$ , $$N>2$$ , M(x) is a uniformly elliptic and bounded matrix, $$\gamma > 0$$ , $$B> 0$$ , $$1\le q

We prove in this paper an existence result of entropy solutions for nonlinear parabolic equations of the form: $$\begin{aligned} \displaystyle \frac{\partial u}{\partial t}-{\text {div}}\, a(x,t,u,\nabla u)-\mathrm{div}\Phi (x,t,u)= f \quad \text {in }{Q_T=\Omega \times (0,T)}, \end{aligned}$$ where the lower order term $$\Phi $$ satisfies only a natural growth condition prescribed by the N-function M defining the Orlicz spaces framework and the data f are an element of $$L^1(Q_T)$$ . We do not assume any restriction neither on M nor on its complementary $$\overline{M}$$ . No particular growth is considered on $$\Phi $$ .

We introduce a pointwise definition of Lipschitz (also called Holder) spaces adapted to the parabolic Hermite operator $$\mathbb {H}= \partial _t- \Delta _x+|x|^2$$ on $$\mathbb {{R}}^{n+1}$$ . Also for every $$\alpha >0$$ , we define the following spaces by means of the Poisson semigroup of $$\mathbb {H}$$ , $$\mathcal {P}_y^{\mathbb {H}}=e^{-y\sqrt{\mathbb {H}}}$$ : $$\begin{aligned} \Lambda _\alpha ^{\mathcal {P}^\mathbb {H}}= & {} \left\{ f: \;f\in L^\infty (\mathbb {R}^{n+1})\, \mathrm{and} \, \left\| \partial _y^k e^{-y\sqrt{\mathbb {H}}} f \right\| _{L^\infty (\mathbb {R}^{n+1})}\right. \\&\left. \le C_k y^{-k+\alpha },\, \mathrm {for}\, k=[\alpha ]+1,\;y>0 \right\} , \end{aligned}$$ with the obvious norm. We prove that both spaces do coincide and their norms are equivalent. For the harmonic oscillator, $$\mathcal {{H}}=-\Delta _x+|x|^2$$ , Stinga and Torrea introduced in 2011 adapted Holder classes. Parallel to the parabolic case, we characterize these pointwise Holder spaces via the $$L^\infty $$ norm of the derivatives of the Poisson and heat semigroups, $$e^{-y\sqrt{\mathcal {{H}}}}$$ and $$e^{-\tau \mathcal {{H}}}$$ , respectively. As important applications of these semigroups characterizations, we get regularity results regarding the boundedness in these adapted Lipschitz spaces of operators related to $$\mathbb {H}$$ and $$\mathcal {{H}}$$ as fractional (positive and negative) powers, Bessel potentials, Hermite Riesz transforms, and Laplace transform multipliers, in a more direct way. The proofs use in a fundamental way the semigroup definition of the operators considered along the paper. The non-convolution structure of the operators produces an extra difficulty on the arguments.

We consider the nonlinear elliptic equation with quadratic convection $$ -\Delta u + g(u) |\nabla u|^2=\lambda f(u) $$ - Δ u + g ( u ) | ∇ u | 2 = λ f ( u ) in a smooth bounded domain $$ \Omega \subset {\mathbb {R}}^N $$ Ω ⊂ R N ($$ N \ge 3$$ N ≥ 3 ) with zero Dirichlet boundary condition. Here, $$ \lambda $$ λ is a positive parameter, $$ f:[0, \infty ):(0\infty ) $$ f : [ 0 , ∞ ) : ( 0 ∞ ) is a strictly increasing function of class $$C^1$$ C 1 , and g is a continuous positive decreasing function in $$ (0, \infty ) $$ ( 0 , ∞ ) and integrable in a neighborhood of zero. Under natural hypotheses on the nonlinearities f and g, we provide some new regularity results for the extremal solution $$u^*$$ u ∗ . A feature of this paper is that our main contributions require neither the convexity (even at infinity) of the function $$ h(t)=f(t)e^{-\int _0^t g(s)ds}$$ h ( t ) = f ( t ) e - ∫ 0 t g ( s ) d s , nor that the functions $$ gh/h'$$ g h / h ′ or $$ h'' h/h'^2$$ h ′ ′ h / h ′ 2 admit a limit at infinity.

Consider the system of functional differential inequalities: $$\begin{aligned} \mathcal {D}\big (\sigma \big )\big [u'(t)-\ell (u)(t)\big ]\ge 0\qquad \text{ for } \text{ a. } \text{ e. } \,t\in [a,b],\quad \varphi (u)\ge 0, \end{aligned}$$ where $$\ell :C\big ([a,b];\mathbb {R}^n\big )\rightarrow L\big ([a,b];\mathbb {R}^n\big )$$ is a linear bounded operator, $$\varphi :C\big ([a,b];\mathbb {R}^n\big )\rightarrow \mathbb {R}^n$$ is a linear bounded functional, $$\sigma =(\sigma _i)_{i=1}^n$$ , where $$\sigma _i\in \{-1,1\}$$ , and $$\mathcal {D}\big (\sigma \big )={\text {diag}}(\sigma _1,\dots ,\sigma _n)$$ . In the present paper, we establish conditions guaranteeing that every absolutely continuous vector-valued function u satisfying the above-mentioned inequalities admits also the inequalities $$u(t)\ge 0$$ for $$t\in [a,b]$$ and $$\mathcal {D}\big (\sigma \big )u'(t)\ge 0$$ for a. e. $$t\in [a,b]$$ .

Let $$(M,\langle \;,\;\rangle _{TM})$$ be a Riemannian manifold. It is well known that the Sasaki metric on TM is very rigid, but it has nice properties when restricted to $$T^{(r)}M=\{u\in TM,|u|=r \}$$ . In this paper, we consider a general situation where we replace TM by a vector bundle $$E\longrightarrow M$$ endowed with a Euclidean product $$\langle \;,\;\rangle _E$$ and a connection $$\nabla ^E$$ which preserves $$\langle \;,\;\rangle _E$$ . We define the Sasaki metric on E and we consider its restriction h to $$E^{(r)}=\{a\in E,\langle a,a\rangle _E=r^2 \}$$ . We study the Riemannian geometry of $$(E^{(r)},h)$$ generalizing many results first obtained on $$T^{(r)}M$$ and establishing new ones. We apply the results obtained in this general setting to the class of Euclidean Atiyah vector bundles introduced by the authors in Boucetta and Essoufi J Geom Phys 140:161–177, 2019). Finally, we prove that any unimodular three dimensional Lie group G carries a left invariant Riemannian metric, such that $$(T^{(1)}G,h)$$ has a positive scalar curvature.

This paper deals with the existence of positive radial solutions of the elliptic equation with nonlinear gradient term $$\begin{aligned} \left\{ \begin{array}{ll} -\Delta u = f(|x|,\,u,\,|\nabla u|)\,,\qquad x\in \Omega \,,\qquad \qquad \\ u|_{\partial \Omega }=0\,, \end{array}\right. \end{aligned}$$ where $$\Omega =\{x\in \mathbb {R}^N:\;|x

Let p be a prime number, let G be a finite group, let N be a normal subgroup of G, and let $$\theta $$ be a G-invariant irreducible character of N. In Rizo (J Algebra 514:254–272, 2018), we introduced a canonical partition of the set $$\mathrm{Irr}(G|\theta )$$ of irreducible constituents of the induced character $$\theta ^G$$ , relative to the prime p. We call the elements of this partition the $$\theta $$ -blocks. In this paper, we construct a canonical basis of the complex space of class functions defined on $$\{ x \in G \, |\, x_p \in N\}$$ , which supersedes previous non-canonical constructions. This allows us to define $$\theta $$ -decomposition numbers in a natural way. We also prove that the elements of the partition of $${\text {Irr}}(G|\theta )$$ established by these $$\theta $$ -decomposition numbers are the $$\theta $$ -blocks.

The purpose of this paper is to study the nonlocal elliptic equation involving critical Hardy–Sobolev exponents as follows, $$\begin{aligned}(\mathrm{P}) {\left\{ \begin{array}{ll} (-\Delta )^s u -\mu \frac{u}{|x|^{2s}}= \lambda |u|^{q-2}u +\frac{|u|^{2_\alpha ^*-2}u}{|x|^\alpha } &{} \text {in} \ \Omega ,\\ u=0 &{} \text {in} \ \mathbb {R}^n\setminus \Omega , \end{array}\right. } \end{aligned}$$ where $$\Omega \subset \mathbb {R}^N$$ is a bounded domain with Lipschitz boundary, $$00$$ is a parameter, $$0\le \mu

In this paper, we study the following Dirichlet problem for a parabolic equation involving fractional p-Laplacian with logarithmic nonlinearity $$\begin{aligned} \left\{ \begin{array}{llc} u_{t}+(-\Delta )^{s}_{p}u+|u|^{p-2}u=|u|^{p-2}u\log (|u|) &{} \text {in}\ &{} \Omega ,\;t>0 , \\ u =0 &{} \text {in} &{} {\mathbb {R}}^{N}\backslash \Omega ,\;t > 0, \\ u(x,0)=u_{0}(x), &{} \text {in} &{}\Omega , \end{array}\right. \end{aligned}$$ where $$\Omega \subset {\mathbb {R}}^N \, ( N\ge 1)$$ is a bounded domain with Lipschitz boundary and $$2\le p< \infty $$ . The local existence will be done using the Galerkin approximations. By combining the potential well theory with the Nehari manifold, we establish the existence of global solutions. Then by virtue of a differential inequality technique, we prove that the local solutions blow-up in finite time with arbitrary negative initial energy and suitable initial values. Moreover, we give decay estimates of global solutions. The main difficulty here is the lack of logarithmic Sobolev inequality concerning fractional p-Laplacian.

In this paper, we consider the problem of the approximation of the integral of a function f over a d-dimensional simplex S of $$\mathbb {R}^{d}$$ R d by some quadrature formulas which use only the functional and derivative values of f on the boundary of the simplex S or function data at the vertices of S, at points on its facets and at its center of gravity. The quadrature formulas are computed by integrating over S a polynomial approximant of f which uses functional and derivative values at the vertices of S.

In this paper, we show the following result: if C is an n-dimensional 0-symmetric convex compact set, $$f:C\rightarrow [0,\infty )$$ is concave, and $$\phi :[0,\infty )\rightarrow [0,\infty )$$ is not identically zero, convex, with $$\phi (0)=0$$ , then $$\begin{aligned} \frac{1}{|C|}\int _C\phi (f(x)){\text {d}}x\le \frac{1}{2}\int _{-1}^1\phi (f(0)(1+t)){\text {d}}t, \end{aligned}$$ where |C| denotes the volume of C. If $$\phi $$ is strictly convex, equality holds if and only if f is affine, C is a generalized symmetric cylinder and f becomes 0 at one of the basis of C. We exploit this inequality to answer a question of Francisco Santos on estimating the volume of a convex set by means of the volume of a central section of it. Second, we also derive a corresponding estimate for log-concave functions.

In this paper, we establish gradient estimates for positive solutions to the following equation with respect to the $p$-Laplacian $$\Delta_{p}u=-\lambda |u|^{p-2}u$$ with $p>1$ on a given complete Riemannian manifold. Consequently, we derive upper bound estimates of the first nontrivial eigenvalue of the $p$-Laplacian., Comment: All comments are welcome

Let G be a locally compact group and $$(\Phi , \Psi )$$ be a complementary pair of N-functions. In this paper, the Fig $$\grave{\hbox {a}}$$ –Talamanca–Herz algebras and the space of p-pseudomeasures on G are extended to Orlicz spaces. Indeed, the Banach algebra of $$\Phi $$ -pseudomeasures $${\mathop {\mathrm{PM}}}_{\Phi }(G)$$ and the Fig $$\grave{\hbox {a}}$$ –Talamanca–Herz–Orlicz algebras $${\mathop {\mathrm{A}}}_{\Phi }(G)$$ are defined. Then, it is shown that $${\mathop {\mathrm{A}}}_{\Phi }(G)^*={\mathop {\mathrm{PM}}}_{\Psi }(G)$$ . Furthermore, we characterize $$\mathop {\mathrm{Cv}}_{\Phi }(G)$$ , the space of $$\Phi $$ -convoluters, in terms of right translation invariant operators on $$M^{\Phi }(G)$$ . Then when G is amenable, we show that $$\mathop {\mathrm{Cv}}_{\Phi }(G)$$ , is equal to $$\mathop {\mathrm{PM}}_{\Phi }(G)$$ , a generalization of the classical p-version. Finally, we study $$B_{\Phi }(G)$$ , the space of pointwise multipliers of $$\mathop {\mathrm{A}}_{\Phi }(G)$$ when G is amenable.

Let $${\mathbb {B}}_X$$ and $${\mathbb {B}}_Y$$ be bounded symmetric domains realized as the unit balls of $$\hbox {JB}^*$$ -triples X and Y, respectively. In this paper, we generalize the Landau theorem to holomorphic mappings on $${\mathbb {B}}_X$$ using the Schwarz–Pick lemma for holomorphic mappings on $${\mathbb {B}}_X$$ . Next, we give a necessary condition for the composition operators $$C_{\varphi }$$ between the Bloch spaces on $${\mathbb {B}}_X$$ and $${\mathbb {B}}_Y$$ to be bounded below by using a sampling set for the Bloch space, where $$\varphi $$ is a holomorphic mapping from $${\mathbb {B}}_X$$ to $${\mathbb {B}}_Y$$ . We also obtain other necessary conditions for the composition operators $$C_{\varphi }$$ between the Bloch spaces in the case $${\mathbb {B}}_Y$$ is a complex Hilbert ball $${\mathbb {B}}_H$$ . We give a sufficient condition for the composition operators $$C_{\varphi }$$ between the Bloch spaces on $${\mathbb {B}}_X$$ and $${\mathbb {B}}_Y$$ to be bounded below using a sampling set for the Bloch space. In the case $$\dim X=\dim Y

Let A be the local ring, at a singular point P, of an algebraic variety $$V\subset {\mathbb {A}}^{r+1}_k$$ of multiplicity $$e=e(A)>1$$. If V is a curve in Orecchia (Can Math Bull 24:423–431, 1981) P was said to be an ordinary singularity when V has e (simple) tangents at P or equivalently when the projectivized tangent cone $$\mathrm{{Proj}}(G(A))$$ of V at P is reduced (in which case consists of e points). In this paper, we show that the definition of ordinary singularity has a natural extension to higher dimensional varieties, in the case in which P is an isolated singularity and the normalization $$\overline{A}$$ of A is regular. In fact, we define P to be an ordinary singularity if the projectivized tangent cone $$\mathrm{{Proj}}(G(A))$$ of V at P is reduced, i.e. is a variety in $${\mathbb {P}}^{r}_k$$. We prove that an ordinary singularity has multilinear projectivized tangent cone that is a union of e linear varieties $$L_1,\ldots ,L_e$$. In the case in which $$L_1,\ldots ,L_e$$ are in generic position, we show that the affine tangent cone is also reduced and then multilinear. Finally, we show how to construct wide classes of parametric varieties with regular normalization at a singular isolated ordinary point.