Related papers: Multiplicity theorems involving functions with non…

The paper deals with the equation $-\Delta u+a(x) u =|u|^{p-1}u $, $u \in H^1(\mathbb{R}^N)$, with $N\ge 2$, $p>1,\ p<{N+2\over N-2}$ if $N\ge 3$, $a\in L^{N/2}_{loc}(\mathbb{R}^N)$, $\inf a>0$, $\lim_{|x| \to \infty} a(x)= a_\infty$.…

In this paper, we consider the following nonlinear elliptic equation with gradient term: \[ \left\{ \begin{gathered} - \Delta u - \frac{1}{2}(x \cdot \nabla u) + (\lambda a(x)+b(x))u = \beta u^q +u^{2^*-1}, \hfill 0<u \in…

For an analytic function f(z)=z+\sum_{n=2}^\infty a_n z^n satisfying the inequality \sum_{n=2}^\infty n(n-1)|a_n|\leq \beta, sharp bound on $\beta$ is determined so that $f$ is either starlike or convex of order $\alpha$. Several other…

We study the singularity (multifractal) spectrum of continuous convex functions defined on $[0,1]^{d}$. Let $E_f({h}) $ be the set of points at which $f$ has a pointwise exponent equal to $h$. We first obtain general upper bounds for the…

The following extension of Bohr's theorem is established: If a somewhere convergent Dirichlet series $f$ has an analytic continuation to the half-plane $\mathbb{C}_\theta = \{s = \sigma+it\,:\, \sigma>\theta\}$ that maps $\mathbb{C}_\theta$…

In this paper, we deal with a problem of the type $$\cases{(\phi(u'))'=\nabla_xF(t,u) & in $[0,T]$\cr & \cr u(0)=u(T)\ , \hskip 3pt u'(0)=u'(T)\ ,\cr}$$ where, in particular, $\phi$ is a homeomorphism from an open ball of ${\bf R}^n$ onto…

We consider several old problems involving the number of prime divisors function $\omega(n)$, as well as the related functions $\Omega(n)$ and $\tau(n)$. Firstly, we show that there are infinitely many positive integers $n$ such that…

Let $G \subset {\mathbb R}^{n}$ be an open convex set which is either bounded or contains a translation of a convex cone with nonempty interior. It is known that then, for every modulus $\omega$, every function on $G$ which is both…

We introduce and analyse a general class of not necessarily bounded multiplicative functions, examples of which include the function $n \mapsto \delta^{\omega (n)}$, where $\delta \neq 0$ and where $\omega$ counts the number of distinct…

Supposing only that $\displaystyle\lim_{t \to 0} \frac{f(t)}{t} = 0$ and $\displaystyle\lim_{t \to \infty} \frac{f(t)}{t^{p}} = 0$, for some $p \in \left(1,\frac{N+1}{N-1}\right)$, we prove that solutions to the extension problem…

We show that there is always a uniformly antisymmetric f:A-> {0,1} if A subset R is countable. We prove that the continuum hypothesis is equivalent to the statement that there is an f:R-> omega with |S_x| <= 1 for every x in R. If the…

In order to obtain solutions to problem $$ {{array}{c} -\Delta u=\dfrac{A+h(x)} {|x|^2}u+k(x)u^{2^*-1}, x\in {\mathbb R}^N, u>0 \hbox{in}{\mathbb R}^N, {and}u\in {\mathcal D}^{1,2}({\mathbb R}^N), {array}. $$ $h$ and $k$ must be chosen…

The aim of this paper is to prove the existence of multiple solutions for a family of nonlinear elliptic systems in divergence form coupled with a pointwise gradient constraint: \begin{align*} \left\{ \begin{array}{ll}…

This is a conitunation of [1] and [2]. We prove that if function $f$ belongs to the class $\Lambda_{\omega} \overset{\text{def}}{=} \{f: \omega_{f}(\delta)\leq \text{const} \omega(\delta)\} $ for an arbitrary modulus of continuity $\omega$,…

In this note, we extend the well-known theorems of M. Riesz and Zygmund on conjugate functions as follows. Let $\Omega$ be a domain in $\mathbb C^n$. Suppose that $f=u+iv\in \mathcal O(\Omega)$ satisfies $v(z_0)=0$ for some $z_0\in \Omega$.…

It is proved that whenever two aperiodic repetitive tilings with finite local complexity have homeomorphic tiling spaces, their associated complexity functions are asymptotically equivalent in a certain sense (which implies, if the…

In this work we prove that certain entire $q$-functions have infinitely many nonzero roots $\left\{ \rho_{n}\right\} _{n=1}^{\infty}$, as $n\to+\infty$ the moduli $\left|\rho_{n}\right|$ grow at least exponentially. Applications to entire…

We study the set of continuous functions that admit no spurious local optima (i.e. local minima that are not global minima) which we term \textit{global functions}. They satisfy various powerful properties for analyzing nonconvex and…

We consider the inhomogeneous (or density dependent) incompressible Euler equations in a three-dimensional periodic domain. We construct density $\varrho$ and velocity $u$ such that, for any $\alpha<1/7$, both of them are $\alpha $-H\"older…

Let $D_j\subset\mathbb C^{n_j}$ be a pseudoconvex domain and let $A_j\subset D_j$ be a locally pluriregular set, $j=1,...,N$. Put $$ X:=\bigcup_{j=1}^N A_1\times...\times A_{j-1}\times D_j\times A_{j+1}\times...\times A_N. $$ Let $M\subset…