Related papers: Oscillation results of higher order linear differe…

We show two results. First, a refinement of Freiman's theorem: if A is a finite set of integers and |A+A| < K|A|, then A is contained in a multidimensional progression of dimension at most O(K^{7/4} log^3K) and size at most exp(O(K^{7/4}…

We study singularity formation in nonlinear differential equations of order $m\leqslant 2$, $y^{(m)}=A(x^{-1},y)$. We assume $A$ is analytic at $(0,0)$ and $\partial_y A(0,0)=\lambda\ne 0$ (say, $\lambda=(-1)^m$). If $m=1$ we assume…

I begin from a particular field of generalised Puiseux series and investigate a class of nonlinear differential equations in the field. It is appeared that the main part of differential equation determines solvability and positions of…

Let $p$ and $q$ be locally H\"{o}lder functions in $\RR^N$, $p>0$ and $q\geq 0$. We study the Emden-Fowler equation $-\Delta u+ q(x)|\nabla u|^a=p(x)u^{-\gamma}$ in $\RR^N$, where $a$ and $\gamma$ are positive numbers. Our main result…

We study the $\epsilon \to 0$ behavior of recurrence relations of the type $\sum_{j=0}^l a_j(k\epsilon,\epsilon)y_{k+j}=0,$ $k\in \zdd$ ($l$ fixed). The $a_j$ are $C^{\infty}$ functions in each variable on $I\times [0,\e_0]$ for a bounded…

We study infinite order differential operators acting in the spaces of exponential type entire functions. We derive conditions under which such operators preserve the set of Laguerre entire functions which consists of the polynomials…

We study the periodic and the Neumann boundary value problems associated with the second order nonlinear differential equation \begin{equation*} u'' + c u' + \lambda a(t) g(u) = 0, \end{equation*} where $g \colon…

It is a classical fact that the exponential function is solution of the integral equation $ \int_0^X f(x)dx + f(0) =f(X)$. If we slightly modify this equation to $ \int_0^X f(x)dx+f(0)=f(\alpha X)$ with $\alpha\in ]0,1[$, it seems that no…

In this paper we study the dynamical behaviour of the differential equation \begin{equation*} x''+ax^+ -bx^-=f(t), \end{equation*} where $x^+=\max\{x,0\}$,\ $x^-=\max\{-x,0\}$, $a$ and $b$ are two different positive constants, $f(t)$ is a…

We show that for any $k$-times continuously differentiable function $f:[a,\infty)\longrightarrow{\mathbb R}$, any integer $q\ge 0$ and any $\alpha>1$ the inequality $$\liminf_{x\to\infty} \frac{x^k \cdot\log x\cdot \log_2 x\cdot\dots\cdot…

We give upper and lower bounds for the number of solutions of the equation $p(z)\log|z|+q(z)=0$ with polynomials $p$ and $q$.

In this paper we study the existence of ground state solution and concentration of maxima for a class of strongly indefinite problem like $$ \left\{\begin{array}{l} -\Delta u+V(x)u=A(\epsilon x)f(u) \quad \mbox{in} \quad \R^{N}, \\ u\in…

We consider the differential equations y''=\lambda_0(x)y'+s_0(x)y, where \lambda_0(x), s_0(x) are C^{\infty}-functions. We prove (i) if the differential equation, has a polynomial solution of degree n >0, then \delta_n=\lambda_n…

In this paper, without any assumption on $v$ and under extremely mild assumption $u(x)=O(|x|^{K})$ at $\infty$ for some $K\gg1$ arbitrarily large, we prove classification of solutions to the following conformally invariant system with mixed…

In the present paper by the Fourier transform we show that every linear differential equations of $n$-th order has a solution in $L^1(\Bbb{R})$ which is infinitely differentiable in $\Bbb{R} \setminus \{0\}$. Moreover the Hyers-Ulam…

This paper deals with the prescribed mean curvature equations both in the Euclidean case and in the Lorentz-Minkowski case in presence of a nonlinearity $g$ such that $g'(0)>0$. We show the existence of oscillating solutions, namely with an…

A Bank--Laine function $E$ is written as $E=f_1f_2$ for two normalized solutions $f_1$ and $f_2$ of the second order differential equation $f''+Af=0$, where $A$ is an entire function. In this paper, we first complete the construction of…

As a consequence of the main result of this paper efficient conditions guaranteeing the existence of a $T-$periodic solution to the second order differential equation \begin{equation*} u"=\frac{h(t)}{u^{\lambda}} \end{equation*} are…

We are interested in the qualitative properties of solutions of the H\'enon type equations with exponential nonlinearity. First, we classify the stable at infinity solutions of $\Delta u +|x|^\alpha e^u=0$ in $\mathbb{R}^N$, which gives a…

In this paper, we study the following biharmonic equations:% $$ \left\{\aligned&\Delta^2u-a_0\Delta u+(\lambda b(x)+b_0)u=f(u)&\text{ in }\bbr^N,\\% &u\in\h,\endaligned\right.\eqno{(\mathcal{P}_{\lambda})}% $$ where $N\geq3$,…