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We obtain results for the following question where $m\ge 1$ and $n\ge 2$ are integers. {\bf Question.} For which continuous functions $f\colon [0,\infty)\to [0,\infty)$ does there exist a continuous function $\phi\colon (0,1)\to (0,\infty)$…

Let $m\ge 2$ be an integer. For any open domain $\Omega\subset\mathbb{R}^{2m}$, non-positive function $\varphi\in C^\infty(\Omega)$ such that $\Delta^m \varphi\equiv 0$, and bounded sequence $(V_k)\subset L^\infty(\Omega)$ we prove the…

We study the problem of the existence of increasing and continuous solutions $\varphi\colon[0,1]\to[0,1]$ such that $\varphi(0)=0$ and $\varphi(1)=1$ of the functional equation \begin{equation*}…

Let $L$ be a second order elliptic operator $L$ with smooth coefficients defined on a domain $\Omega $ in $\mathbb{R}^d $, $d\geq3$, such that $L1\leq 0$. We study existence and properties of continuous solutions to the following problem…

Let $\Omega$ be a smooth bounded domain in $\mathbb{R}^{N}$ and let $m$ be a possibly discontinuous and unbounded function that changes sign in $\Omega$. Let $f:\left[ 0,\infty\right) \rightarrow\left[ 0,\infty\right) $ be a continuous…

Let $L $ be a second order elliptic operator with smooth coefficients defined on a domain $\Omega \subset \mathbb{R}^d$ (possibly unbounded), $d\geq 3$. We study nonnegative continuous solutions $u$ to the equation $L u(x) - \varphi (x,…

Let $D$ be a smooth domain in $\mathbb{R}^N$, $N\geq 3$ and let $f$ be a positive continuous function on $\partial D$. Under some assumptions on $\varphi$, it is shown that the problem $\Delta u=2\varphi(u)$ in $D$ and $u=f$ on $\partial…

In this short note, we establish the following result: Let $f:[0,+\infty[\to [0,+\infty[$, $\alpha:[0,1]\to ]0,+\infty[$ be two continuous functions, with $f(0)=0$. Assume that, for some $a>0$, the function $\xi\to…

Almost perfect nonlinear (APN) functions play an important role in the design of block ciphers as they offer the strongest resistance against differential cryptanalysis. Despite more than 25 years of research, only a limited number of APN…

Equation $(-\Delta+k^2)u+f(u)=0$ in $D$, $u\mid_{\partial D}=0$, where $k=\const>0$ and $D\subset\R^3$ is a bounded domain, has a solution if $f:\R\to\R$ is a continuous function in the region $|u|\geq a$, piecewise-continuous in the region…

We study the existence of solutions of the non-linear differential equations on the compact Riemannian manifolds $(M^n,g), n\geq 2$, \Delta_p u + a(x)u^{p-1} = \lambda f(u,x), (E2) where $\Delta_p$ is the $p-$laplacian, with $1<p<n$. The…

This paper deals with the existence of positive solutions for the nonlinear system q(t)\phi(p(t)u'_{i}(t)))'+f^{i}(t,\textbf{u})=0,\quad 0<t<1,\quad i=1,2,...,n. This system often arises in the study of positive radial solutions of…

In this paper, we consider the existence (and nonexistence) of solutions to \[ -\mathcal{M}_{\lambda,\Lambda}^\pm (u'') + V(x) u = f(u) \quad {\rm in} \ \mathbf{R} \] where $\mathcal{M}_{\lambda,\Lambda}^+$ and…

For open sets $U$ in some space $X$, we are interested in positive solutions to semi-linear equations $ Lu=\varphi(\cdot,u)\mu$ on $U$. Here $L$ may be an elliptic or parabolic operator of second order (generator of a diffusion process) or…

By using the complete discrimination system for polynomials, we study the number of positive solutions in {\small $C[0,1]$} to the integral equation {\small $\phi (x)=\int_0^1k(x,y)\phi ^n(y)dy$}, where {\small…

We continue our study started in "On a problem of Janusz Matkowski and Jacek Weso{\l}owski" (see arXiv:1703.08459) of the functional equation \begin{equation*} \varphi(x)=\sum_{n=0}^{N}\varphi(f_n(x))-\sum_{n=0}^{N}\varphi(f_n(0))…

Let $\Omega:=\left( a,b\right) \subset\mathbb{R}$, $m\in L^{1}\left( \Omega\right) $ and $\lambda>0$ be a real parameter. Let $\mathcal{L}$ be the differential operator given by $\mathcal{L}u:=-\phi\left( u^{\prime}\right) ^{\prime}+r\left(…

: We establish existence of an infinite family of exponentially-decaying non-radial $C^2$ solutions to the equation $\Delta u + f(u) = 0$ on $R^2$ for a large class of nonlinearities $f$. These solutions have the form $u(r,\theta )=e^{i…

We study the existence/nonexistence of positive solution to the problem of the type: \begin{equation}\tag{$P_{\lambda}$} \begin{cases} \Delta^2u-\mu a(x)u=f(u)+\lambda b(x)\quad\textrm{in $\Omega$,}\\ u>0 \quad\textrm{in $\Omega$,}\\…

We construct a function $u \in W^{1,1}_{\mathrm{loc}} (B(0,1))$ which is a solution to $\Div (A \nabla u)=0$ in the sense of distributions, where $A$ is continuous and $u \not \in W^{1,p}_{\mathrm{loc}} (B(0,1))$ for $p > 1$. We also give a…