Related papers: Hardy-Sobolev inequality with singularity a curve

A Schwarz function on an open domain $\Omega$ is a holomorphic function satisfying $S(\zeta)=\overline{\zeta}$ on $\Gamma$, which is part of the boundary of $\Omega$. Sakai in 1991 gave a complete characterization of the boundary of a…

We study the Dirichlet problem for a graph $\Sigma$ in $\mathbb{R}^{n+1}$ with normalized constant mean curvature $H>0$ and planar boundary $\Gamma=\partial \Omega$. Our main result is that the optimal solvability condition, namely that the…

We prove that, for any real $\lambda$, the system $-\Delta u +\lambda u = u^3-\beta uv^2$, $ -\Delta v+\lambda v =v^3-\beta vu^2$, $ u,v\in H^1_0(\Omega),$ where $\Omega$ is a bounded smooth domain of $R^3$, admits a bounded family of…

A property of smooth convex domains $\Omega \subset \mathbb{R}^n$ is that if two points on the boundary $x, y \in \partial \Omega$ are close to each other, then their normal vectors $n(x), n(y)$ point roughly in the same direction and this…

This paper concerns with the existence of nontrivial solution for the following problem \begin{equation} \left\{\begin{aligned} -\Delta u + V(x)u & = \gamma H_{e}(|u|-a)|u|^{q-2}u+|u|^{2^{*}-2}u\;\;\mbox{ in}\;\;\mathbb{R}^{N},\nonumber u…

In this note, we prove that the kernel of the linearized equation around a positive energy solution in $\mathbb{R}^n$, $n\geq 3$, to $-\Delta W-\gamma|x|^{-2}V=|x|^{-s}W^{2^\star(s)-1}$ is one-dimensional when $s+\gamma>0$. Here, $s\in…

We study properties of positive functions satisfying (E) --$\Delta$u+m|$\nabla$u| q -- u p = 0 is a domain $\Omega$ or in R N + when p > 1 and 1 < q < 2. We give sufficient conditions for the existence of a solution to (E) with a…

We consider an elliptic differential inequality: $\vert \Delta u(x) \vert \le C_0(\YYYY^{-\gamma}\vert u(x)\vert + \YYYY^{-\theta}\vert \nabla u(x)\vert)$ in an exterior domain $\R^n \setminus \ooo{U}$, where $U$ is a simply connected…

We consider the following Kirchhoff - Choquard equation \[ -M(\|\na u\|_{L^2}^{2})\De u = \la f(x)|u|^{q-2}u+ \left(\int_{\Om}\frac{|u(y)|^{2^*_{\mu}}}{|x-y|^{\mu}}dy\right)|u|^{2^*_{\mu}-2}u \; \text{in}\; \Om,\quad u = 0 \; \text{ on }…

We investigate the following quasilinear parabolic and singular equation, {equation} \tag{{\rm P$_t$}} \{{aligned} & u_t-\Delta_p u =\frac{1}{u^\delta}+f(x,u)\;\text{in}\,(0,T)\times\Omega, & u =0\,\text{on}…

We go further in the investigation of the Robin problem $(P_{\alpha})$: $-\Delta u=a(x)u^{q}$ in $\Omega$, $u\geq0$ in $\Omega$, $\partial_{\nu}u=\alpha u$ on $\partial \Omega$; on a bounded domain $\Omega\subset\mathbb{R}^{N}$, with $a$…

We study the existence and non-existence of positive solutions for the $(p,q)$-Laplace equation $-\Delta_p u -\Delta_q u = \alpha |u|^{p-2} u + \beta |u|^{q-2} u$, where $p \neq q$, under the zero Dirichlet boundary condition in $\Omega$.…

Let $N\geq 5$, $\Omega$ be a smooth bounded domain in $\mathbb{R}^{N}$, ${2^*}=\frac{2N}{N-2}$, $a>0$, $S=\inf\left\{\left. \int_{\mathbb{R}^{N}}|\nabla u|^2\,\right|\,u\in L^{2^*}(\mathbb{R}^{N}), \nabla u\in L^2(\mathbb{R}^{N}),…

We show that for a very general class of curvature functions defined in the positive cone, the problem of finding a complete strictly locally convex hypersurface in $H^n+1$ satisfying $f(\kappa)=\sigma\in(0, 1)$ with a prescribed asymptotic…

We consider weighted $L^p$-Hardy inequalities involving the distance to the boundary of a domain in the $n$-dimensional Euclidean space with nonempty boundary. Using criticality theory, we give an alternative proof of the following result…

Let $\Sigma$ be a closed Riemann surface, $h$ a positive smooth function on $\Sigma$, $\rho$ and $\alpha$ real numbers. In this paper, we study a generalized mean field equation \begin{align*} -\Delta u=\rho\left(\dfrac{he^u}{\int_\Sigma…

Given a smooth and bounded domain $\Omega(\subset\mathbf{R}^N)$, we prove the existence of two non-trivial, non-negative solutions for the semilinear degenerate elliptic equation \begin{align} \left. \begin{array}{l} -\Delta_\lambda u=\mu…

We consider the following nonlinear Choquard equation with Dirichlet boundary condition $$-\Delta u =\left(\int_{\Omega}\frac{|u|^{2_{\mu}^{\ast}}}{|x-y|^{\mu}}dy\right)|u|^{2_{\mu}^{\ast}-2}u+\lambda…

We review the indefinite sublinear elliptic equation $-\Delta u=a(x)u^{q}$ in a smooth bounded domain $\Omega\subset\mathbb{R}^{N}$, with Dirichlet or Neumann homogeneous boundary conditions. Here $0<q<1$ and $a$ is continuous and changes…

Suppose that a closed $1$-rectifiable set $\Gamma_0\subset \mathbb R^2$ of finite $1$-dimensional Hausdorff measure and a vector field $u$ in a dimensionally critical Sobolev space are given. It is proved that, starting from $\Gamma_0$,…