Related papers: Remarks on nonlinear equations with measures

In this work, we consider the Dirichlet boundary value problem for nonlinear triharmonic equation. Due to the reduction of the nonlinear boundary value problem to operator equation for the nonlinear term and the unknown second normal…

If $\Omega$ is a bounded domain in $\mathbb R^N$ and $f$ a continuous increasing function satisfying a super linear growth condition at infinity, we study the existence and uniqueness of solutions for the problem (P): $\partial_tu-\Delta…

We consider the following boundary value problem -\Delta u= g(x,u) + f(x,u) x\in \Omega u=0 x\in \partial \Omega where $g(x,-\xi)=-g(x,\xi)$ and $g$ has subcritical exponential growth in $\mathbb{R} ^2$. Using the method developed by Bolle,…

Let us consider a semilinear boundary value problem $ - \Delta u= f(x,u),$ in $\Omega,$ with Dirichlet boundary conditions, where $ \Omega \subset \mathbb{R}^N $, $N> 2,$ is a bounded smooth domain. We provide sufficient conditions…

We investigate the scalar Chern-Simons equation $-\Delta u + e^u(e^u-1) = \mu$ in cases where there is no solution for a given nonnegative finite measure $\mu$. Approximating $\mu$ by a sequence of nonnegative $L^1$ functions or finite…

We study existence and uniqueness of solutions of (E 1) --$\Delta$u + $\mu$ |x| ^{-2} u + g(u) = $\nu$ in $\Omega$, u = $\lambda$ on $\partial$$\Omega$, where $\Omega$ $\subset$ R N + is a bounded smooth domain such that 0 $\in$…

We study the first vanishing time for solutions of the Cauchy-Dirichlet problem to the semilinear $2m$-order ($m \geq 1$) parabolic equation $u_t+Lu+a(x) |u|^{q-1}u=0$, $0<q<1$ with $a(x) \geq 0$ bounded in the bounded domain $\Omega…

Let $\Omega$ be a bounded domain in ${\mathbb R}^N$ and $T>0$. We study the problem \begin{equation} (P)\left\{ \begin{array}{lll} u_t - \Delta u \pm g(u) &= \mu \quad &\text{in } Q_T:=\Omega \times (0,T) \\ \phantom{------,} u&=0 &\text{on…

We prove a result of existence of positive solutions of the Dirichlet problem for $-\Delta_p u=\mathrm{w}(x)f(u,\nabla u)$ in a bounded domain $\Omega\subset\mathbb{R}^N$, where $\Delta_p$ is the $p$-Laplacian and $\mathrm{w}$ is a weight…

This paper deals with various questions related to the isoperimetic problem for smooth positive measure $d\mu = \varphi(x)dx$, with $x \in \Omega \subset \mathbb{R}^N$. Firstly we find some necessary conditions on the density of the measure…

In this paper, we study the following singular problem, under mixed Dirichlet-Neumann boundary conditions, and involving the fractional Laplacian \begin{equation*} \label{1} \begin{cases} (-\Delta)^{s}u = \lambda u^{-q} + u^{2^*_s-1}, \quad…

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 note we study the boundary regularity of solutions to nonlocal Dirichlet problems of the form $Lu=0$ in $\Omega$, $u=g$ in $\mathbb R^N\setminus\Omega$, in non-smooth domains $\Omega$. When $g$ is smooth enough, then it is easy to…

We are mainly concerned with equations of the form $-Lu=f(x,u)+\mu$, where $L$ is an operator associated with a quasi-regular possibly nonsymmetric Dirichlet form, $f$ satisfies the monotonicity condition and mild integrability conditions,…

We investigate the weak solvability and properties of weak solutions to the Dirichlet problem for a scalar elliptic equation $-\Delta u + b^{(\alpha)}\cdot \nabla u= f$ in a bounded domain $\Omega\subset {\mathbb R^2}$ containing the…

In this paper, we investigate boundary estimates for the Dirichlet problem for a class of fully nonlinear elliptic equations with general boundary conditions, including nonzero boundary conditions. Given specific structural conditions on…

For a domain $\Omega\subset\dR^N$ we consider the equation $ -\Delta u + V(x)u = Q_n(x)\abs{u}^{p-2}u$ with zero Dirichlet boundary conditions and $p\in(2,2^*)$. Here $V\ge 0$ and $Q_n$ are bounded functions that are positive in a region…

We study the interior regularity of solutions to the Dirichlet problem $Lu=g$ in $\Omega$, $u=0$ in $\R^n\setminus\Omega$, for anisotropic operators of fractional type $$ Lu(x)= \int_{0}^{+\infty}\,d\rho \int_{S^{n-1}}\,da(\omega)\, \frac{…

In this paper, we study the boundary H\"older regularity for solutions to the fractional Dirichlet problem in unbounded domains with boundary \begin{equation*} \begin{cases} (-\Delta)^s u(x) = g(x),&\text{in } \Omega, u(x)=0, &\text{in }…

We study a mixed boundary value problem for the quasilinear elliptic equation $\mathop{\rm div}\mathcal{A}(x,\nabla u(x))=0$ in an open infinite circular half-cylinder with prescribed continuous Dirichlet data on a part of the boundary and…