Related papers: Entire $s$-harmonic functions are affine

In this paper we analyze the semi-linear fractional Laplace equation $$(-\Delta)^s u = f(u) \quad\text{ in } \mathbb{R}^N_+,\quad u=0 \quad\text{ in } \mathbb{R}^N\setminus \mathbb{R}^N_+,$$ where $\mathbb{R}^N_+=\{x=(x',x_N)\in…

We investigate symmetry properties of solutions to equations of the form $$ -\Delta u = \frac{a}{|x|^2} u + f(|x|, u)$$ in R^N for $N \geq 4$, with at most critical nonlinearities. By using geometric arguments, we prove that solutions with…

In this work, we deal with existence and uniqueness of positive solution $u_s$ for the singular quasilinear problem $(-\Delta_{\Phi})^su=u^{-\gamma}$ in the nonreflexive fractional Orlicz-Sobolev $ W^{s}_0L^{\Phi}(\Omega)$ for $0<s<1$.…

We investigate a version of the Phragm\'en-Lindel\"of theorem for solutions of the equation $\Delta_\infty u=0$ in unbounded convex domains. The method of proof is to consider this infinity harmonic equation as the limit of the $p$-harmonic…

We prove convergence of solutions of Dirichlet problems and Green's functions on Tutte harmonic embeddings to those of the linearized Monge--Amp\`ere equation $\mathcal{L}_\varphi h=0$. More precisely, we assume that piecewise linear…

There exists an exponentially decreasing function $f$ such that any singly $2\pi$-periodic positive solution $u$ of $-\Delta u +u-u^p=0$ in $[0,2\pi]\times \R^{N-1}$ verifies $u(x_1,x')\leq f(|x'|)$. We prove that with the same period and…

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 this paper the question of finding infinitely many solutions to the problem $-\Delta u+a(x)u=|u|^{p-2}u$, in $\mathbb{R}^N$, $u \in H^1(\mathbb{R}^N)$, is considered when $N\geq 2$, $p \in (2, 2N/(N-2))$, and the potential $a(x)$ is a…

We study the problem of deciding universal termination of linear and affine loops over the reals in the bit-model of real computation. We show that both problems are as close to decidable as one can expect them to be: there exist sound…

We prove Liouville type theorems for $p$-harmonic functions on exterior domains of the $d$-dimensional Euclidean space, where $1<p<\infty$ and $d\geq 2$. We show that every positive $p$-harmonic function satisfying zero Dirichlet, Neumann…

We show that the set of Lebesgue integrable functions in $[0,1]$ which are nowhere essentially bounded is spaceable, improving a result from [F. J. Garc\'{i}a-Pacheco, M. Mart\'{i}n, and J. B. Seoane-Sep\'ulveda. \textit{Lineability,…

We investigate the positive solutions of the semilinear elliptic equation \begin{align*} \sum^{N}_{i=1}\left(-\partial_{ii}\right)^{s}u=u^{p} \end{align*} with one-dimensional symmetric $2s$-stable operators. Firstly, in the whole space…

Starting from the symmetrical reflection functional equation of the zeta function, we have found that the sigma values satisfying zeta(s) = 0 must also satisfy both |zeta(s)| = |zeta(1 - s)| and |gamma(s/2)zeta(s)| = |gamma((1 - s)/2)zeta(1…

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),…

We study the equation $-\Delta u+u^q=0$, $q>1$, in a bounded $C^2$ domain $\Omega\subset R^N$. A positive solution of the equation is moderate if it is dominated by a harmonic function and $\sigma$-moderate if it is the limit of an…

We obtain necessary and sufficient conditions for the existence of a positive finite energy solution to the inhomogeneous quasilinear elliptic equation \[ -\Delta_{p} u = \sigma u^{q} + \mu \quad \text{on} \;\; \mathbb{R}^n \] in the…

In this paper, we prove that all the positive solutions for the PDE system (-\Delta)^{k}u_{i} = f_{i}(u_{1},..., u_{m}), x \in R^{n}, i = 1, 2,..., m are super polyharmonic, i.e. (-\Delta)^{j}u_{i} > 0, j = 1, 2,..., k - 1; i = 1, 2,...,m.…

Let $P_{2k}$ be a homogeneous polynomial of degree $2k$ and assume that there exist $C>0$, $D>0$ and $\alpha \ge 0$ such that \begin{equation*} \left\langle P_{2k}f_{m},f_{m}\right\rangle_{L^2(\mathbb{S}^{d-1})}\geq \frac{1}{C\left(…

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 $\phi\colon (0,1)\to (0,\infty)$…

It is proved that the dimension of the space of solutions of the Dirichlet problem for the harmonic functions in the unit disk with nontangential boundary limits 0 a.e. is infinite.