Related papers: Dirichlet problems for second order linear ellipti…

The present paper is devoted to study the asymptotic behavior of a sequence of linear elliptic equations with a varying drift term, whose coefficients are just bounded in $L^N(\Omega)$, with $N$ the dimension of the space. It is known that…

We consider a semilinear elliptic equation on a smooth bounded domain $\Om$ in $\R^2$, assuming that both the domain and the equation are invariant under reflections about one of the coordinate axes, say the y-axis. It is known that…

We show that if $u$ is a solution to a linear elliptic differential equation of order $2m\geq 2$ in the half-space with $t$-independent coefficients, and if $u$ satisfies certain area integral estimates, then the Dirichlet and Neumann…

We consider periodic homogenization of boundary value problems for second-order semilinear elliptic systems in 2D of the type $$ \partial_{x_i}\left(a_{ij}^{\alpha…

We consider the semilinear Dirichlet problem \[ \Delta u+kg(u)=\mu _1 \varphi _1+\cdots +\mu _n \varphi _n+e(x) \;\; \mbox{for $x \in \Omega$}, \;\; u=0 \;\; \mbox{on $\partial \Omega$}, \] where $\varphi _k$ is the $k$-th eigenfunction of…

We study connections between the problem of the existence of positive solutions for certain nonlinear equations and weighted norm inequalities. In particular, we obtain explicit criteria for the solvability of the Dirichlet problem…

In this paper we consider the Liouville equation $\Delta u +\lambda^2 e^{\,u}=0$ with Dirichlet boundary conditions in a two dimensional, doubly connected domain $\Omega$. We show that there exists a simple, closed curve $\gamma\subset…

We prove the existence of infinitely many solutions to a class of non-symmetric Dirichlet problems with exponential nonlinearities. Here the domain $\Omega \subset\subset \mathbb{R}^{2l}$ where $2l$ is the order of the equation. Considered…

We consider the nonlinear eigenvalue problem $ L u = \lambda f(u) $, posed in a smooth bounded domain $ \Omega \subseteq \Bbb{R}^{N} $ with Dirichlet boundary condition, where $ L $ is a uniformly elliptic second-order linear differential…

Let $\Omega$ be a $\mathcal C^2$-bounded domain of $\mathbb R^d$, $d=2,3$, and fix $Q=(0,T)\times\Omega$ with $T\in(0,+\infty]$. In the present paper we consider a Dirichlet initial-boundary value problem associated to the semilinear…

We consider a second order differential operator $A(\msx) = -\:\sum_{i,j=1}^d \partial_i a_{ij}(\msx) \partial_j \:+\: \sum_{j=1}^d \partial_j \big(b_j(\msx) \cdot \big)\:+\: c(\msx)$ on ${\bbR}^d$, on a bounded domain $D$ with Dirichlet…

The present paper is devoted to the study of the Dirichlet problem ${\rm{Re}}\,\omega(z)\to\varphi(\zeta)$ as $z\to\zeta,$ $z\in D,\zeta\in \partial D,$ with continuous boundary data $\varphi :\partial D\to\mathbb R$ for Beltrami equations…

We study the homogenization of the equation $-A(\frac{\cdot}{\varepsilon}):D^2 u_{\varepsilon} = f$ posed in a bounded convex domain $\Omega\subset \mathbb{R}^n$ subject to a Dirichlet boundary condition and the numerical approximation of…

We prove the existence of a solution to a singular anisotropic elliptic equation in a bounded open subset $\Omega$ of $\mathbb R^N$ with $N\ge 2$, subject to a homogeneous boundary condition: \begin{equation} \label{eq0} \left\{…

In this paper we give necessary and sufficient conditions on the compatibility of a $k$th order homogeneous linear elliptic differential operator $\mathbb{A}$ and differential constraint $\mathcal{C}$ for solutions of \begin{align*}…

Recently found all the fundamental solutions of a multidimensional singular elliptic equation are expressed in terms of the well-known Lauricella hypergeometric function in many variables. In this paper, we find a unique solution of the…

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,…

In this article, we prove the existence of solutions to a nonlinear nonlocal elliptic problem with a singualrity and a discontinuous critical nonlinearity which is given as follows. \begin{align} \begin{split}\label{main_prob}…

We consider periodic homogenization with localized defects for semilinear elliptic equations and systems of the type $$ \nabla\cdot\Big(\Big(A(x/\varepsilon)+B(x/\varepsilon)\Big)\nabla u(x)+c(x,u(x)\Big)=d(x,u(x)) \mbox{ in } \Omega $$…

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…