Related papers: On linear elliptic equations with drift terms in c…

Let $p\in(0,\frac{N}{N-2\alpha})$, $\alpha\in(0,1)$ and $\Omega\subset \R^N$ be a bounded $C^2$ domain containing $0$. If $\delta_0$ is the Dirac measure at $0$ and $k>0$, we prove that the weakly singular solution $u_k$ of $(E_k)$ $…

Here we study the Dirichlet problem for first order linear and quasi-linear hyperbolic PDEs on a simply connected bounded domain of $\R^2$, where the domain has an interior outflow set and a mere inflow boundary. By means of a Lyapunov…

We are interested in regularity properties of semi-stable solutions for a class of singular semilinear elliptic problems with advection term defined on a smooth bounded domain of a complete Riemannian manifold with zero Dirichlet boundary…

In this paper, we study the Dirichlet problem for the implicit degen- erate nonlinear elliptic equation with variable exponent in a bounded domain. We obtain sufficient conditions for the existence of a solution with- out regularization and…

This paper studies the existence of positive normalized solutions to the singular elliptic equation \[ -\Delta u + \lambda u = u^{-r} + u^{p-1} \quad \text{in } \Omega, \] with the Dirichlet boundary condition $u=0$ on $\partial\Omega$ and…

This work deals with the system $(-\Delta)^m u= a(x) v^p$, $(-\Delta)^m v=b(x) u^q$ with Dirichlet boundary condition in a domain $\Omega\subset\RR^n$, where $\Omega$ is a ball if $n\ge 3$ or a smooth perturbation of a ball when $n=2$. We…

This paper deals with existence and regularity of positive solutions of singular elliptic problems on a smooth bounded domain with Dirichlet boundary conditions involving the $\Phi$-Laplacian operator. The proof of existence is based on a…

We study the Dirichlet problem for systems of the form -\Delta u^k=f^k(x,u)+\mu^k, x\in\Omega, k=1,...,n, where \Omega\subset R^d$ is an open (possibly nonregular) bounded set, \mu^1,...,\mu^n are bounded diffuse measures on \Omega,…

Let $\mathcal{O} \subset \mathbb{R}^d$ be a bounded domain of class $C^2$. In the Hilbert space $L_2(\mathcal{O};\mathbb{C}^n)$, we consider a matrix elliptic second order differential operator $\mathcal{A}_{D,\varepsilon}$ with the…

We consider the weakly coupled elliptic system of logistic type, \begin{equation}\label{LS} \begin{cases} -\Delta u &=\lambda_1 u- |u|^{p-2}u+ \beta |u|^{\frac{p}{2}-2}u |v|{^{\frac{p}{2}-1}}v\mbox{ in }\Omega, -\Delta v & =\lambda_2 v-…

In this paper, we provide a new means of establishing solvability of the Dirichlet problem on Lipschitz domains, with measurable data, for second order elliptic, non-symmetric divergence form operators. We show that a certain optimal…

We investigate the existence and multiplicity of weak solutions for a nonlinear Kirchhoff type quasilinear elliptic system on the whole space $\mathbb{R}^N$. We assume that the nonlinear term satisfies the locally super-$(m_1,m_2)$…

Using analysis for 2-admissible functions in weighted Sobolev spaces and stochastic calculus for possibly degenerate symmetric elliptic forms, we construct weak solutions to a wide class of stochastic differential equations starting from an…

Solvability of Cauchy's problem in $\mathbb{R}^2$ for subcritical quasi-geostrophic equation is discussed here in two phase spaces; $L^p(\mathbb{R}^2)$ with $p> \frac{2}{2\alpha-1}$ and $H^s(\mathbb{R}^2)$ with $s>1$. A solution to that…

Let $\mathcal{L}$ be a second-order linear elliptic operator with complex coefficients. We show that if the $L^p$ Dirichlet problem for the elliptic system $\mathcal{L}(u)=0$ in a fixed Lipschitz domain $\Omega$ in $\mathbb{R}^d$ is…

We establish some existence and regularity results to the Dirichlet problem, for a class of quasilinear elliptic equations involving a partial differential operator, depending on the gradient of the solution. Our results are formulated in…

This paper considers the Dirichlet problem $$ -\mathrm{div}(a\nabla u_a)=f \quad \hbox{on}\,\,\ D, \qquad u_a=0\quad \hbox{on}\,\,\partial D, $$ for a Lipschitz domain $D\subset \mathbb R^d$, where $a$ is a scalar diffusion function. For a…

We consider divergence form elliptic equations $Lu:=\nabla\cdot(A\nabla u)=0$ in the half space $\mathbb{R}^{n+1}_+ :=\{(x,t)\in \mathbb{R}^n\times(0,\infty)\}$, whose coefficient matrix $A$ is complex elliptic, bounded and measurable. In…

We study the Dirichlet problem for the following prescribed mean curvature PDE $$ \begin{cases} -\operatorname{div}\dfrac{\nabla v}{\sqrt{1+|\nabla v|^{2}}}=f(x,v) \text{ in }\Omega\\ v=\varphi \text{ on }\partial\Omega. \end{cases} $$…

We extend and improve the results in \cite{DK16}: showing that weak solutions to full elliptic equations in divergence form with zero Dirichlet boundary conditions are continuously differentiable up to the boundary when the leading…