In this paper, we consider solutions to the following chemotaxis system with general sensitivity $$\left\{\begin{array}{c}\tau u_t=\Delta u-\nabla ·\left(u\nabla \chi \left(v\right)\right)\text{in}\Omega \times (0,\infty ),\hfill \\ \eta v_t=\Delta v-v+u\text{in}\Omega \times (0,\infty ),\hfill \\ {\displaystyle \frac{\partial u}{\partial \nu }=\frac{\partial u}{\partial \nu }=0\text{on}\partial \Omega \times (0,\infty ).}\hfill \end{array}\right.$$ Here, $\tau$ and $\eta$ are positive constants, $\chi$ is a smooth function on $(0,\infty )$ satisfying ${\chi }^{\text{'}}(·)>0$ and $\Omega$ is a bounded domain of ${𝐑}^n$ ($n\ge 2$). It is well known that the chemotaxis system with direct sensitivity ($\chi \left(v\right)={\chi }_{0}v$, ${\chi }_0>0$) has blowup solutions in the case where $n\ge 2$. On the other hand, in the case where $\chi \left(v\right)={\chi }_{0}logv$ with $0<{\chi }_0\ll 1$, any solution to the system exists globally in time and is bounded. We present a sufficient condition for the boundedness...

Let $\mathbb{N}$ be the set of positive integers and let $s\in \mathbb{N}$. We denote by $d^s$ the arithmetic function given by $d^s\left(n\right)={\left(d\left(n\right)\right)}^s$, where $d\left(n\right)$ is the number of positive divisors of $n$. Moreover, for every $\ell ,m\in \mathbb{N}$ we denote by ${\delta }^{s,\ell ,m}\left(n\right)$ the sequence $$\underset{m\text{-times}}{\underbrace{d^s(d^s(...d^s(d^s\left(n\right)+\ell )+\ell ...)+\ell )}}=\left\{\begin{array}{cc}{d}^s\left(n\right)\hfill & \text{for}m=1,\hfill \\ d^s(d^s\left(n\right)+\ell )\hfill & \text{for}m=2,\hfill \\ d^s(d^s(d^s\left(n\right)+\ell )+\ell )\hfill & \text{for}m=3,\hfill \\ ⋮\hfill \end{array}\right.$$ We present classical and nonclassical notes on the sequence ${\left({\delta }^{s,\ell ,m}\left(n\right)\right)}_{m\ge 1}$, where $\ell$, $n$, $s$ are understood as parameters.

We investigate some nonlinear elliptic problems of the form $${\Delta }^{2}v+\sigma \left(x\right)v=h(x,v)\text{in}\Omega ,v=\Delta v=0\text{on}\partial \Omega ,\left(\mathrm{P}\right)$$ where $\Omega$ is a regular bounded domain in ${ℝ}^N$, $N\ge 2$, $\sigma \left(x\right)$ a positive function in $L^{\infty }\left(\Omega \right)$, and the nonlinearity $h(x,t)$ is indefinite. We prove the existence of solutions to the problem (P) when the function $h(x,t)$ is asymptotically linear at infinity by using variational method but without the Ambrosetti-Rabinowitz condition. Also, we consider the case when the nonlinearities are superlinear and subcritical.

We study the existence and nonexistence of positive solutions of the nonlinear equation $$-{\Delta }_{p\left(x\right)}u=\lambda k\left(x\right)u^q\pm h\left(x\right)u^r\text{in}\Omega ,u=0\text{on}\partial \Omega$$ where $\Omega \subset {ℝ}^N$, $N\ge 2$, is a regular bounded open domain in ${ℝ}^N$ and the $p\left(x\right)$-Laplacian $${\Delta }_{p\left(x\right)}u:={\text{div}\left(\right|\nabla u|}^{p\left(x\right)-2}\nabla u)$$ is introduced for a continuous function $p\left(x\right)>1$ defined on $\Omega$. The positive parameter $\lambda$ induces the bifurcation phenomena. The study of the equation (Q) needs generalized Lebesgue and Sobolev spaces. In this paper, under suitable assumptions, we show that some variational methods still work. We use them to prove the existence of positive...

Let $\mathrm{T}$ be the family of all typically real functions, i.e. functions that are analytic in the unit disk $\Delta :=\{z\in ℂ:|z|<1\}$, normalized by $f\left(0\right)=f^{\text{'}}\left(0\right)-1=0$ and such that Im $z$ Im $f\left(z\right)$ $\ge 0$ for $z\in \Delta$. Moreover, let us denote: ${\mathrm{T}}^{\left(2\right)}:=\{f\in \mathrm{T}:f\left(z\right)=-f(-z)\text{for}z\in \Delta \}$ and ${\mathrm{T}}^{M,g}:=\{f\in \mathrm{T}:f\prec Mg\text{in}\Delta \}$, where $M>1$, $g\in \mathrm{T}\cap \mathrm{S}$ and $\mathrm{S}$ consists of all analytic functions, normalized and univalent in $\Delta$.We investigate  classes in which the subordination is replaced with the majorization and the function $g$ is typically real but does not necessarily univalent, i.e. classes $\{f\in \mathrm{T}:f\ll Mg\text{in}\Delta \}$, where $M>1$, $g\in \mathrm{T}$, which we denote...

We consider the problem $$\left\{\begin{array}{cc}-\Delta u=u^{\frac{N+2}{N-2}+\lambda }\hfill & \text{in}\Omega \setminus \epsilon \omega ,\hfill \\ u\&gt;0\hfill & \text{in}\Omega \setminus \epsilon \omega ,\hfill \\ u=0\hfill & \text{on}\partial \left(\Omega \setminus \epsilon \omega \right),\hfill \end{array}\right.$$ where $\Omega$ and $\omega$ are smooth bounded domains in ${ℝ}^N$, $N\ge 3$, $\epsilon \&gt;0$ and $\lambda \in ℝ.$ We prove that if the size of the hole $\epsilon$ goes to zero and if, simultaneously, the parameter $\lambda$ goes to zero at the appropriate rate, then the problem has a solution which blows up at the origin.

A graph $G$ is called $\{H_1,H_2,\cdots ,H_k\}$-free if $G$ contains no induced subgraph isomorphic to any graph $H_i$, $1\le i\le k$. We define $${\sigma }_k=min\left\{\sum _{i=1}^{k}d\left(v_i\right):\{v_1,\cdots ,v_k\}\text{is}\text{an}\text{independent}\text{set}\text{of}\text{vertices}\text{in}G\right\}.$$ In this paper, we prove that (1) if $G$ is a connected $\{K_{1,4},K_{1,4}+e\}$-free graph of order $n$ and ${\sigma }_3\left(G\right)\ge n-1$, then $G$ is traceable, (2) if $G$ is a 2-connected $\{K_{1,4},K_{1,4}+e\}$-free graph of order $n$ and $|N\left(x_1\right)\cup N\left(x_2\right)|+|N\left(y_1\right)\cup N\left(y_2\right)|\ge n-1$ for any two distinct pairs of non-adjacent vertices $\{x_1,x_2\}$, $\{y_1,y_2\}$ of $G$, then $G$ is traceable, i.e., $G$ has a Hamilton path, where $K_{1,4}+e$ is a graph obtained by joining a pair of non-adjacent vertices in a $K_{1,4}$.

For any positive integer $k\ge 2$, let ${\left(P_n^{\left(k\right)}\right)}_{n\ge 2-k}$ be the $k$-generalized Pell sequence which starts with $0,\cdots ,0,1$ ($k$ terms) with the linear recurrence $$P_n^{\left(k\right)}=2P_{n-1}^{\left(k\right)}+P_{n-2}^{\left(k\right)}+\cdots +P_{n-k}^{\left(k\right)}\text{for}n\ge 2.$$ Let ${\left(N_n\right)}_{n\ge 0}$ be Narayana’s sequence given by $$N_0=N_1=N_2=1\text{and}{N}_{n+3}=N_{n+2}+N_n.$$ The purpose of this paper is to determine all $k$-Pell numbers which are sums of two Narayana’s numbers. More precisely, we study the Diophantine equation $$P_p^{\left(k\right)}=N_n+N_m$$ in nonnegative integers $k$, $p$, $n$ and $m$.

The author proved in 2018 that if $G$ is an open subset of a Hilbert space, $f_1,f_2:G\to ℝ$ continuous functions and $\omega$ a nontrivial modulus such that $f_1\le f_2$, $f_1$ is locally semiconvex with modulus $\omega$ and $f_2$ is locally semiconcave with modulus $\omega$, then there exists $f\in C_{\text{loc}}^{1,\omega }\left(G\right)$ such that $f_1\le f\le f_2$. This is a generalization of Ilmanen’s lemma (which deals with linear modulus and functions on an open subset of ${ℝ}^n$). Here we extend the mentioned result from Hilbert spaces to some superreflexive spaces, in particular to $L^p$ spaces, $p\in [2,\infty )$. We...

Let $X$ be a proper, smooth, geometrically connected curve over a $p$-adic field $k$. Lichtenbaum proved that there exists a perfect duality: $$\text{Br}\left(X\right)\times \text{Pic}\left(X\right)\to \mathbb{Q}/\mathbb{Z}$$ between the Brauer and the Picard group of $X$, from which he deduced the existence of an injection of $\text{Br}\left(X\right)$ in ${\displaystyle \prod _{P\in X}\text{Br}\left(k_P\right)}$ where $P\in X$ and $k_P$ denotes the residual field of the point $P$. The aim of this paper is to prove that if $G=\tilde{G}$ is an $X_{et}$- scheme of semi-simple simply connected groups (s.s.s.c groups), then we can deduce from Lichtenbaum’s results...

The paper deals with the existence of a Kneser solution of the $n$-th order nonlinear differential inclusion $$\begin{array}{cc}\hfill x^{\left(n\right)}\left(t\right)\in -A_1(t,x\left(t\right),...,x^{(n-1)}\left(t\right))x^{(n-1)}\left(t\right)-...-A_n(t,x\left(t\right),...,& x^{(n-1)}\left(t\right))x\left(t\right)\\ & \text{for}\text{a.a.}t\in [a,\infty ),\end{array}$$ where $a\in (0,\infty )$, and $A_i:[a,\infty )\times {ℝ}^n\to ℝ$, $i=1,...,n,$ are upper-Carathéodory mappings. The derived result is finally illustrated by the third order Kneser problem.

This paper is devoted to the study of some nonlinear degenerated elliptic equations, whose prototype is given by $$\begin{array}{cccc}\hfill t2& -{\mathrm{div}\left(b\right(\left|u\right|\left)\right|\nabla u|}^{p-2}{\nabla u)+d(\left|u\right|\left)\right|\nabla u|}^p=f-{\mathrm{div}\left(c\left(x\right)\right|u|}^{\alpha })\hfill & \hfill & \text{in}\Omega ,\hfill \\ & u=0\hfill & \hfill & \text{on}\partial \Omega ,\hfill \end{array}t$$ where $\Omega$ is a bounded open set of ${ℝ}^N$ ($N\ge 2$) with $1<p<N$ and $f\in L^1\left(\Omega \right),$ under some growth conditions on the function $b(·)$ and $d(·),$ where $c(·)$ is assumed to be in $L^{\frac{N}{(p-1)}}\left(\Omega \right).$ We show the existence of renormalized solutions for this non-coercive elliptic equation, also, some regularity results will be concluded.

We prove a density version of the Carlson–Simpson Theorem. Specifically we show the following. For every integer $k\ge 2$ and every set $A$ of words over $k$ satisfying $\lim {\sup }_{n\to \infty }|A\cap {\left[k\right]}^n|/k^n>0$ there exist a word $c$ over $k$ and a sequence $\left(w_n\right)$ of left variable words over $k$ such that the set $c\cup \{c^{}{w}_0{\left(a_0\right)}^{}..{.}^{}{w}_n\left(a_n\right):n\in \mathbb{N}\text{and}{a}_0,...,a_n\in \left[k\right]\}$ is contained in $A$. While the result is infinite-dimensional its proof is based on an appropriate finite and quantitative version, also obtained in the paper.

We construct an upper bound for the following family of functionals ${\left\{E_{\epsilon }\right\}}_{\epsilon \&gt;0}$, which arises in the study of micromagnetics: $$E_{\epsilon }\left(u\right)={\int }_{\Omega }{\epsilon \left|\nabla u\right|}^2+\frac{1}{\epsilon }{\int }_{{ℝ}^2}{\left|H_u\right|}^2.$$ Here $\Omega$ is a bounded domain in ${ℝ}^2$, $u\in H^1(\Omega ,S^1)$ (corresponding to the magnetization) and $H_u$, the demagnetizing field created by $u$, is given by $$\left\{\begin{array}{cc}\mathrm{div}(\tilde{u}+H_u)=0\hfill & \text{in}{ℝ}^2,\hfill \\ \mathrm{curl}{H}_u=0\hfill & \text{in}{ℝ}^2,\hfill \end{array}\right.$$ where $\tilde{u}$ is the extension of $u$ by $0$ in ${ℝ}^2\setminus \Omega$. Our upper bound coincides with the lower bound obtained by Rivière and Serfaty.