Related papers: Reduced measures for semilinear elliptic equations…

In the paper, we first propose a definition of renormalized solution of semilinear elliptic equation involving operator corresponding to a general (possibly nonlocal) symmetric regular Dirichlet form satisfying the so-called absolute…

We study the existence of nonnegative solutions to the Dirichlet problem $\CL^{_{^M}}_{p,q}u:=-\Delta u+u^p-M|\nabla u|^q=\mu$ in a domain $\Omega\subset\BBR^N$ where $\mu$ is a nonnegative Radon measure, when $p>1$, $q>1$ and $M\geq 0$. We…

We introduce and study the Dirichlet problem for double divergence form elliptic equations with coefficients of low regularity and boundary conditions given by general Borel measures. Under broad assumptions we establish the solvability of…

We give a partial uniqueness result concerning comparable renormalized solutions of the nonlinear elliptic problem $-\diw(\aop(x,Du))=\mu$ in $\Omega$, $u=0$ on $\partial\Omega$, where $\mu$ is a Radon measure with bounded variation on…

We show that any general semilinear elliptic problem with Dirichlet or Neumann boundary conditions in an annulus A in R^2m ;m >1, invariant by the action of a certain symmetry group can be reduced to a nonhomogenous similar problem in an…

We study the singular semilinear equation $-Pu = \frac{f}{u^\gamma}$ on a bounded domain $\Omega$ with Dirichlet condition $u \equiv 0$ on $\partial \Omega$ , where $P$ is a second-order elliptic differential operator in nondivergence form.…

We prove the existence of a weak solution for a general class of Dirichlet anisotropic elliptic problems such as $\mathcal Au+\Phi(x,u,\nabla u)=\mathfrak{B}u+f$ in $\Omega$, where $\Omega$ is a bounded open subset of $\mathbb R^N$ and…

We study the existence and the properties of the reduced measures for the parabolic equations $\partial_tu-\Delta u+g(u)=0$ in $\Omega\times (0,\infty)$ subject to the conditions ($P$): $u=0$ on $\partial\Omega\times (0,\infty)$,…

In this paper, we study the semilinear subelliptic equation \[ \left\{ \begin{array}{cc} -\triangle_{X} u=f(x,u)+g(x,u) & \mbox{in}~\Omega, \\[2mm] u=0\hfill & \mbox{on}~\partial\Omega, \end{array} \right. \] where…

We study existence and stability for solutions of $Lu+g(x; u) = \omega$ in the closure of open set $\Omega$ where L is a second order elliptic operator, $g$ a Caratheodory function and $\omega$ a measure in $\bar\Omega$. We present a uni ed…

Let $\Omega \subset \mathbb{R}^N$ ($N \geq 3$) be a bounded smooth domain and $\delta(x)=\text{dist}(x,\partial \Omega)$. In this paper, we provide various necessary and sufficient conditions for the existence of weak solutions to $$…

Given a parabolic cylinder $Q =(0,T)\times\Omega$, where $\Omega\subset \mathbb{R}^{N}$ is a bounded domain, we prove new properties of solutions of \[ u_t-\Delta_p u = \mu \quad \text{in $Q$} \] with Dirichlet boundary conditions, where…

We study a class of fractional elliptic problems of the form $\Ds u= f(u)$ in the half space $\R^N_+:=\{x \in \R^N\::\: x_1>0\}$ with the complementary Dirichlet condition $u \equiv 0$ in $\R^N \setminus \R^N_+$. Under mild assumptions on…

We establish $L^p$ solvability of the Dirichlet problem, for some finite $p$, in a 1-sided chord-arc domain $\Omega$ (i.e., a uniform domain with Ahlfors-David regular boundary), for elliptic equations of the form \[ Lu=-\text{div}(A\nabla…

In this paper, we study the good-$\lambda$ type bounds for renormalized solutions to nonlinear elliptic problem: \begin{align*} \begin{cases} -\div(A(x,\nabla u)) &= \mu \quad \text{in} \ \ \Omega, \\ u &=0 \quad \text{on} \ \ \partial…

Let $A(D)$ be an elliptic homogeneous linear differential operator with complex constant coefficients, $ \mu $ be a vector-valued Borel measure and $w$ be a positive locally integrable function on $\mathbb{R}^N$. In this work, we present…

We study the Dirichlet boundary value problem for equations with absorption of the form $-\Delta u+g\circ u=\mu$ in a bounded domain $\Omega\subset R^N$ where $g$ is a continuous odd monotone increasing function. Under some additional…

We investigate optimal control problems governed by the elliptic partial differential equation $-\Delta u=f$ subject to Dirichlet boundary conditions on a given domain $\Omega$. The control variable in this setting is the right-hand side…

We study positive solutions to the fractional Lane-Emden system \begin{equation*} \tag{S}\label{S} \left\{ \begin{aligned} (-\Delta)^s u &= v^p+\mu \quad &&\text{in } \Omega \\ (-\Delta)^s v &= u^q+\nu \quad &&\text{in } \Omega\\ u = v &= 0…

In this paper we consider nonlinear elliptic PDEs of the type $$-\Delta_p u+a(x)|u|^{p-2}u=|u|^{p^*-2}u \qquad \mbox{ in }\Omega,$$ where $1<p<N$ and $p^*=Np/(N-p)$ is the critical Sobolev exponent, and allowing the asymptotic behavior of…