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We study the behavior near the origin of $C^2$ positive solutions $u(x)$ and $v(x)$ of the system $0\leq -\Delta u\leq f(v)$ $0\leq -\Delta v\leq g(u)$ in $B_1(0)\backslash\{0\}$ where $f,g:(0,\infty)\to (0,\infty)$ are continuous…

Let $U$ be an open set in $\mathbb{R}^d$. A continuous function $f\colon U \to \mathbb{R}$ is strongly nowhere differentiable if and only if for each $\gamma\in(0,1]$ and for each unit speed $C^{1,\gamma}$ curve $c\colon [a,b] \to U$, the…

We consider nonnegative solutions to $-\Delta u=f(u)$ in half-planes and strips, under zero Dirichlet boundary condition. Exploiting a rotating$\&$sliding line technique, we prove symmetry and monotonicity properties of the solutions, under…

\noi In this article, we study the existence of non-negative solutions of the following polyharmonic Kirchhoff type problem with critical singular exponential nolinearity $$ \quad \left\{ \begin{array}{lr} \quad…

We deal with the existence of $2\pi$-periodic solutions to the following non-local critical problem \begin{equation*} \left\{\begin{array}{ll} [(-\Delta_{x}+m^{2})^{s}-m^{2s}]u=W(x)|u|^{2^{*}_{s}-2}u+ f(x, u) &\mbox{in} (-\pi,\pi)^{N} \\…

We show that any 2-valued C^{1, \alpha} (\alpha \in (0, 1)) function u = {u_{1}, u_{2}} on an open ball B in {\mathbb R}^{n} with values u_{1}, u_{2} \in {\mathbb R}^{k} whose graph, viewed as a varifold with multiplicity 2 at points where…

Let $E$ be an infinite-dimensional separable Hilbert space. We show that for every $C^1$ function $f:E\to\mathbb{R}^d$, every open set $U$ with $C_f:=\{x\in E:\,Df(x)\; \text{is not surjective}\}\subset U$ and every continuous function…

\noindent In this paper we study existence of solution for a class of problem of the type $$ \left\{ \begin{array}{ll} -\Delta_{\Phi}{u}=f(u), \quad \mbox{in} \quad \Omega u=0, \quad \mbox{on} \quad \partial \Omega, \end{array} \right. $$…

We show non-existence of solutions of the Cauchy problem in $\mathbb{R}^N$ for the nonlinear parabolic equation involving fractional diffusion $\partial_t u + (-\Delta)^s \phi(u)= 0,$ with $0<s<1$ and very singular nonlinearities $\phi$ .…

We consider the $2m$-th order elliptic boundary value problem $Lu=f(x,u)$ on a bounded smooth domain $\Omega\subset\R^N$ with Dirichlet boundary conditions on $\partial\Omega$. The operator $L$ is a uniformly elliptic linear operator of…

Let $u\not\equiv -\infty$ be a subharmonic function on the complex plane $\mathbb C$. Then for any function $r\colon\mathbb C\to (0,1]$ satisfying the condition $$\inf_{z\in\mathbb C}\frac{\ln r(z)}{\ln(2+|z|)}>-\infty,$$ there is an entire…

Let $f$ be a continuous and non-decreasing function such that $f>0$ on $(0,\infty)$, $f(0)=0$, $\sup \_{s\geq 1}f(s)/s< \infty$ and let $p$ be a non-negative continuous function. We study the existence and nonexistence of explosive…

Let $\Omega=(a,b)\subset\mathbb{R}$, $0\leq m,n\in L^{1}(\Omega)$, $\lambda,\mu>0$ be real parameters, and $\phi:\mathbb{R}\rightarrow\mathbb{R}$ be an odd increasing homeomorphism. In this paper we consider the existence of positive…

Let $f$ be a real-valued function defined on $\mathbb{R}$, with $f(0) \neq 0$ and which is not constant in non empty open intervals. We prove the equations \begin{equation}\label{edif} \left\{ \begin{array}{rcll} (-\Delta )^{s}u & = & f(u),…

In this paper, we first prove some propositions of Sobolev spaces defined on a locally finite graph $G=(V,E)$, which are fundamental when dealing with equations on graphs under the variational framework. Then we consider a nonlinear…

Associated to a nonzero homomorphism $\varphi$ of a Banach algebra $A$, we regard special functionals, say $m_\varphi$, on certain subspaces of $A^\ast$ which provide equivalent statements to the existence of a bounded right approximate…

We investigate here the degenerate bi-harmonic equation: $$\Delta_{m}^2 u=f(x,u)\; \;\;\mbox{in} \O,\quad u = \Delta u = 0\quad \mbox{on }\; \p\Omega,$$ with $m\ge 2,$ and also the degenerate tri-harmonic equation: $$ -\Delta_{m}^3…

We consider the nonlocal Liouville type equation $$ (-\Delta)^{\frac{1}{2}} u = \varepsilon \kappa(x) e^u, \quad u > 0, \quad \mbox{in } I, \qquad u = 0, \quad \mbox{in } \mathbb{R} \setminus I, $$ where $I$ is a union of $d \geq 2$…

Let $C({\mathbb R}^n)$ denote the set of real valued continuous functions defined on ${\mathbb R}^n$. We prove that for every $n\ge 2$ there are positive numbers $\lambda _1 , \ldots , \lambda _n$ and continuous functions $\phi_1 ,\ldots ,…

Let $\phi(\cdot)$ and $\sigma(\cdot)$ denote the Euler function and the sum of divisors function, respectively. In this paper, we give a lower bound for the number of positive integers $m\le x$ for which the equation $m=n-\phi(n)$ has no…