Related papers: Symmetry results for nonlinear elliptic operators …

We obtain symmetry results for solutions of an elliptic system of equation possessing a cooperative structure. The domain in which the problem is set may possess "holes" or "small vacancies" (measured in terms of capacity) along which the…

We study the nonlinear fractional reaction diffusion equation $\partial_{t}u + (-\Delta)^{s} u= f(t,x,u)$, $s\in(0,1)$ in a bounded domain $\Omega$ together with Dirichlet boundary conditions on $\R^N \setminus \Omega$. We prove asymptotic…

We prove that, if $g$ is a continuous asymptotically convex function, any solution $u$ of $-\Delta u+g(u)=0$ in a ball B which tends to infinity on $\partial B$ is radially symmetric.

This paper deals with solutions of semilinear elliptic equations of the type \[ \left\{\begin{array}{ll} -\Delta u = f(|x|, u) \qquad & \text{ in } \Omega, \\ u= 0 & \text{ on } \partial \Omega, \end{array} \right. \] where $\Omega$ is a…

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}…

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…

Assume that $f(s) = F'(s)$ where $F$ is a double-well potential. Under certain conditions on the Lipschitz constant of $f$ on $[-1,1]$, we prove that arbitrary bounded global solutions of the semilinear equation $\Delta u = f(u)$ on…

The paper proves Liouville-type results for stable solutions of semilinear elliptic PDEs with convex nonlinearity, posed on the entire Euclidean space. Extensions to solutions which are stable outside a compact set are also presented.

We give necessary and sufficient conditions for the existence of a positive solution with zero boundary values to the elliptic equation \[ \mathcal{L}u = \sigma u^{q} + \mu \quad \text{in} \;\; \Omega, \] in the sublinear case $0<q<1$, with…

In this paper, we prove the existence of non-radial solutions to the problem $-\triangle u=f(z,u)$, $u|_{\partial D}=0$ on the unit disc $D:=\{z\in \mathbb C : |z|<1\}$ with $u(z)\in \mathbb R^k$, where $f$ is a sub-linear continuous…

For open radial sets $\Omega\subset \mathbb{R}^N$, $N\geq 2$ we consider the nonlinear problem \[ (P)\quad Iu=f(|x|,u) \quad\text{in $\Omega$,}\quad u\equiv 0\quad \text{on $\mathbb{R}^N\setminus \Omega$ and }\lim_{|x|\to\infty} u(x)=0, \]…

In this paper we consider positive supersolutions of the nonlinear elliptic equation \[- \Delta u = \rho(x) f(u)|\nabla u|^p, \qquad \hfill \mbox{ in } \Omega,\] where $0\le p<1$, $ \Omega$ is an arbitrary domain (bounded or unbounded) in $…

Exact self-consistent particle-like solutions with spherical and/or cylindrical symmetry to the equations governing the interacting system of scalar, electromagnetic and gravitational fields have been obtained. As a particular case it is…

Boundary differentiability is shown for solutions of nondivergence elliptic equations with unbounded drift

In this paper, we investigate the asymptotic symmetry and monotonicity of positive solutions to a reaction-diffusion equation in the unit ball, utilizing techniques from elliptic geometry. Firstly, we discuss the properties of solutions in…

We study spectral stability estimates of elliptic operators in divergence form $-\textrm{div} [A(w) \nabla g(w)]$ with the Neumann boundary condition in non-Lipschitz domains $\Omega \subset \mathbb C$. The suggested method is based on…

In this paper we prove radial symmetry for solutions to a free boundary problem with a singular right hand side, in both elliptic and parabolic regime. More exactly, in the unit ball $B_1$ we consider a solution to the fully nonlinear…

We show radial symmetry of positive solutions to the H\'{e}non equation $-\Delta u = |x|^{-\ell} u^q $ in $\mathbb{R}^N \setminus \{ 0\} $, where $\ell \geq 0$, $q>0$ and satisfy further technical conditions. A new ingredient is a maximum…

We consider the following equations: \begin{equation*} \left\{\begin{array}{ll} (-\triangle)^{\alpha/2}u(x)=f(v(x)), \\ (-\triangle)^{\beta/2}v(x)=g(u(x)), &x \in R^{n},\\ u,v\geq 0, &x \in R^{n}, \end{array} \right. \end{equation*} for…

In this article, we study the existence of positive solutions to elliptic equation (E1) $$(-\Delta)^\alpha u=g(u)+\sigma\nu \quad{\rm in}\quad \Omega,$$ subject to the condition (E2) $$u=\varrho\mu\quad {\rm on}\quad \partial\Omega\ \ {\rm…