Related papers: Overdetermined problems with fractional Laplacian

Let $\textbf{A}$ be a symmetric convex quadratic form on $\mathbb{R}^{Nn}$ and $\Omega\Subset \mathbb{R}^n$ a bounded convex domain. We consider the problem of existence of solutions $u: \Omega \subset \mathbb{R}^n \longrightarrow…

This paper is devoted to the Laplacian operator of fractional order $s\in (0,1)$ in several dimensions. We consider the equation $(-\Delta)^su=f(x,u)$ in $\Omega$, $u=0$ in $\Omega^c$ and establish a representation formula for partial…

We consider numbers of the form $S_\beta(\boldsymbol{u}):=\sum_{n=0}^\infty \frac{u_n}{\beta^n}$ for $\boldsymbol{u}=\langle u_n \rangle_{n=0}^\infty$ a Sturmian sequence over a binary alphabet and $\beta$ an algebraic number with…

We consider the nonlinear Stefan problem $$ \left \{ \begin{array} {ll} -d \Delta u=a u-b u^2 \;\; & \mbox{for } x \in \Omega (t), \; t>0, \\ u=0 \mbox{ and } u_t=\mu|\nabla_x u |^2 \;\;&\mbox{for } x \in \partial\Omega (t), \; t>0, \\…

Let $\Omega\subset\mathbb{R}^\nu$, $\nu\ge 2$, be a $C^{1,1}$ domain whose boundary $\partial\Omega$ is either compact or behaves suitably at infinity. For $p\in(1,\infty)$ and $\alpha>0$, define \[…

Let $n \geq 2$ and $\Omega \subset \mathbb{R}^n$ be a bounded domain. Then by Trudinger-Moser embedding, $W_0^{1,n}(\Omega)$ is embedded in an Orlicz space consisting of exponential functions. Consider the corresponding semi linear…

In this article, we study the following fractional-Laplacian system with singular nonlinearity \begin{equation*} (P_{\lambda,\mu}) \left\{ \begin{array}{lr} (-\Delta)^s u = \lambda f(x) u^{-q}+ \frac{\alpha}{\alpha+\beta}b(x) u^{\alpha-1}…

We establish the existence of positive solutions for a system of coupled fourth-order partial differential equations on a bounded domain $\Omega \subset \mathbb{R}^n$\begin{align*} \left\{\begin{array}{l} \Delta^2u_1 +\beta_1 \Delta…

In an open bounded interval $\Omega$, the problem \[ u_{tt} = u_{xx} - \big(f(\Theta)\big)_x, \Theta_t = \Theta_{xx} - f(\Theta) u_{xt}, \] is considered under the boundary conditions $u|_{\partial\Omega}=\Theta_x|_{\partial\Omega}=0$, and…

We present a quantitative estimate for the radially symmetric configuration concerning a Serrin-type overdetermined problem for the torsional rigidity in a bounded domain $\Omega $, when the equation is known on $\Omega \setminus…

In this paper, we study the near-boundary behavior of functions $u\in\mathcal{F}(\Omega)$ in the case where $\Omega$ is strictly pseudoconvex. We also introduce a sufficient condition for belonging to $\mathcal{F}$ in the case where…

Suppose that $G=(V, E)$ is a connected locally finite graph with the vertex set $V$ and the edge set $E$. Let $\Omega\subset V$ be a bounded domain. Consider the following quasilinear elliptic equation on graph $G$ $$ \left \{…

We consider the problem $$(P_\eps)\qquad \Delta u +\lambda {e^{u}\over\int\limits_{\Omega\setminus B(\xi,\eps)}e^u}=0\ \hbox{in}\ \Omega\setminus B(\xi,\eps),\quad u =0\ \hbox{on}\ \partial\(\Omega\setminus B(\xi,\eps)\), $$ where $\Omega$…

In this paper we construct families of bounded domains $\Omega_\varepsilon$ and solutions $u_\varepsilon$ of \[\begin{cases} -\Delta u_\varepsilon=1&\text{ in }\ \Omega_\varepsilon\\ u_\varepsilon=0&\text{ on }\ \partial\Omega_\varepsilon…

We prove the solvability of the Dirichlet problem for the variable exponent $p$-Laplacian with boundary data in $W^{1,p(x)}(\Omega)$ on a bounded, smooth domain $\Omega \subset {\mathbb R}^n$. Our main focus will be on an a.e. finite…

In this article, we prove existence results of positive solutions for the following nonlinear elliptic problem with gradient terms: \begin{eqnarray*} \left\{\begin{array}{l@{\quad }l} (-\Delta)^\alpha u=f(x,u,v,\nabla u, \nabla v) &{\rm…

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…

In this paper, by variational and topological arguments based on linking and $\nabla$-theorems, we prove the existence of multiple solutions for the following nonlocal problem with mixed Dirichlet-Neumann boundary data, $$ \left\{…

In this paper, under very general assumptions, we prove existence and regularity of distributional solutions to homogeneous Dirichlet problems of the form $$\begin{cases} \displaystyle - \Delta_{1} u = h(u)f & \text{in}\, \Omega,\newline…

We consider the eigenvalue problem $\Delta^{\mathbb{S}^2} \xi + 2 \xi=0 $ in $ \Omega $ and $\xi = 0 $ along $ \partial \Omega $, being $\Omega$ the complement of a disjoint and finite union of smooth and bounded simply connected regions in…