Related papers: Geometric approach to nonvariational singular elli…
On a bounded smooth domain we study solutions of a semilinear elliptic equation with an exponential nonlinearity and a Hardy potential depending on the distance to the boundary of the domain. We derive global a priori bounds of the…
We consider the boundary value problem $-\Delta_p u_\lambda -\Delta_q u_\lambda =\lambda g(x) u_\lambda^{-\beta}$ in $\Omega$ , $u_\lambda=0$ on $\partial \Omega$ with $u_\lambda>0$ in $\Omega.$ We assume $\Omega$ is a bounded open set in…
In this paper we obtain the existence of bounded positive entire radial solutions for the following nonlinear elliptic problem with a special nonlinear gradient term -\triangle_{p}u-b(x)|\nablau|^{p-1}=a(x)f(u), x\inR^{N} (N\geq3),…
We study singular metrics of constant negative $Q$-curvature in the Euclidean space $\mathbb{R}^n$ for every $n \geq 1$. Precisely, we consider solutions to the problem \[ (-\Delta)^{n/2}u=-e^{nu}\quad \text{on}\quad\mathbb{R}^{n}\backslash…
We study existence of nontrivial solutions to problem \begin{equation*} \left\lbrace \begin{array}{rcll} -\Delta u &=& \lambda u+f(u)&\text{ in }\Omega,\\ u&=&0&\text{ on }\partial \Omega, \end{array}\right. \end{equation*} where $\Omega…
The first three sections of this paper are a survey of the author's work on balanced metrics and stability notions in algebraic geometry. The last section is devoted to proving the well-known result that a geodesically convex function on a…
In this paper, we study the asymptotic behavior of positive solutions of the fractional Hardy-H\'enon equation $$ (-\Delta)^\sigma u = |x|^\alpha u^p ~~~~~~~~~~~ in ~~ B_1 \backslash \{0\} $$ with an isolated singularity at the origin,…
In this paper, we prove the existence of multiple solutions for a nonlinear nonlocal elliptic PDE involving a singularity which is given as \begin{eqnarray} (-\Delta_p)^s u&=& \frac{\lambda}{u^\gamma}+u^q~\text{in}~\Omega,\nonumber…
As a first result we prove higher order Schauder estimates for solutions to singular/degenerate elliptic equations of type: \[ -\mathrm{div}\left(\rho^aA\nabla w\right)=\rho^af+\mathrm{div}\left(\rho^aF\right) \quad\textrm{in}\; \Omega \]…
In this paper we deal with positive solutions for singular quasilinear problems whose model is $$ \begin{cases} -\Delta u + \frac{|\nabla u|^2}{(1-u)^\gamma}=g & \mbox{in $\Omega$,}\newline \hfill u=0 \hfill & \mbox{on $\partial\Omega$,}…
We study local asymptotics of solutions to fractional elliptic equations at boundary points, under some outer homogeneous Dirichlet boundary condition. Our analysis is based on a blow-up procedure which involves some Almgren type…
This paper studies a new gradient regularity in Lorentz spaces for solutions to a class of quasilinear divergence form elliptic equations with nonhomogeneous Dirichlet boundary conditions: \begin{align*} \begin{cases} div(A(x,\nabla u)) &=…
We study the boundary value problem with Radon measures for nonnegative solutions of $-\Delta u+Vu=0$ in a bounded smooth domain $\Gw$, when $V$ is a locally bounded nonnegative function. Introducing some specific capacity, we give…
Hybrid inverse problems are mathematical descriptions of coupled-physics (also called multi-waves) imaging modalities that aim to combine high resolution with high contrast. The solution of a high-resolution inverse problem, a first step…
Recent results using inverse scattering techniques interpret every solution $\phi (x,y)$ of the sine-Gordon equation as a non-linear superposition of solutions along the axes $x=0$ and $y=0$. Here we provide a geometric method of…
We study some properties of the solutions of (E) $\;-\Gd_p u+|\nabla u|^q=0$ in a domain $\Gw \sbs \BBR^N$, mostly when $p\geq q>p-1$. We give a universal priori estimate of the gradient of the solutions with respect to the distance to the…
We study operators on a singular manifold, here of conical or edge type, and develop a new general approach of representing asymptotics of solutions to elliptic equations close to the singularities. The idea is to construct so-called…
We prove well-posedness and regularity results for elliptic boundary value problems on certain domains with a smooth set of singular points. Our class of domains contains the class of domains with isolated oscillating conical singularities,…
We study existence of nonnegative solutions to a nonlinear parabolic boundary value problem with a general singular lower order term and a nonnegative measure as nonhomogeneous datum, of the form $$ \begin{cases} \displaystyle u_t -…
A famous theorem of Bers and Finn states that isolated singularities of solutions to the non-parametric minimal surface equation are removable. We show that this result remains valid, if the area functional is replaced by a general…