Related papers: Fully anisotropic elliptic problems with minimally…

We study the Dirichlet problem for a second order linear elliptic equation in a bounded smooth domain $\Omega$ in $\mathbb{R}^n$, $n \ge 3$, with the drift $\mathbf{b} $ belonging to the critical weak space $L^{n,\infty}(\Omega )$. We…

We give necessary and sufficient conditions for the existence of weak solutions to the model equation $$-\Delta_p u=\sigma \, u^q \quad \text{on} \, \, \, \R^n,$$ in the case $0<q<p-1$, where $\sigma\ge 0$ is an arbitrary locally integrable…

The present paper is devoted to study the asymptotic behavior of a sequence of linear elliptic equations with a varying drift term, whose coefficients are just bounded in $L^N(\Omega)$, with $N$ the dimension of the space. It is known that…

It is shown that for each finite number of Dirac measures supported at points $s_n$ in three-dimensional Euclidean space, with given amplitudes $a_n$, there exists a unique real-valued Lipschitz function $u$, vanishing at infinity, which…

Given a bounded smooth domain $\Omega$ in $\mathbb{R}^2$, we study the following anisotropic elliptic problem $$ \begin{cases} -\nabla\big(a(x)\nabla \upsilon\big)= a(x)\big[e^{\upsilon}-s\phi_1-4\pi\alpha\delta_q-h(x)\big]\,\,\,\,…

We generalize recent results on the monotonicity method, for inclusion detection in the partial data anisotropic Calder\'on problem, to very general non-self-adjoint perturbations. This involves a forward model that accounts for both the…

The present article investigates the existence, multiplicity and regularity of weak solutions of problems involving a combination of critical Hartree type nonlinearity along with singular and discontinuous nonlinearity. By applying…

We consider the obstacle problem with irregular barriers for semilinear elliptic equation involving measure data and operator corresponding to a general quasi-regular Dirichlet form. We prove existence and uniqueness of a solution as well…

We consider positive solutions to semilinear elliptic problems with singular nonlinearities, under zero Dirichlet boundary condition. We exploit a refined version of the moving plane method to prove symmetry and monotonicity properties of…

In this paper, we obtain a new Hsiung-Minkowski integral formula for anisotropic capillary hypersurfaces in the half-space, which includes the weighted Hsiung-Minkowski formula and classical anisotropic Minkowski identity for closed…

We consider a countably generated and uniformly closed algebra of bounded functions. We assume that there is a lower semicontinuous, with respect to the supremum norm, quadratic form and that normal contractions operate in a certain sense.…

This is a generalization of our prior work on the compact fixed point theory for the elliptic Rosseland-type equations. We obtain the maximum principle without the technical Steklov techniques. Inspired by the Rosseland equation in the…

We consider the Dirichlet problem for a class of elliptic and parabolic equations in the upper-half space $\mathbb{R}^d_+$, where the coefficients are the product of $x_d^\alpha, \alpha \in (-\infty, 1),$ and a bounded uniformly elliptic…

Most of lipschitz regularity results for nonlinear strictly elliptic equations are obtained for a suitable growth power of the nonlinearity with respect to the gradient variable (subquadratic for instance). For equations with superquadratic…

We consider the maximal regularity of a specific Vlasov-Fokker-Planck equation $\mathcal{A}u=f$ in the Euclidean space. The operator $\mathcal{A}=\Delta_{y}u-y\cdot \nabla_x{u}$ is an example of the Ornstein-Uhlenbeck operators. We prove…

We investigate elliptic irregular obstacle problems with $p$-growth involving measure data. Emphasis is on the strongly singular case $1 < p \le 2-1/n$, and we obtain several new comparison estimates to prove gradient potential estimates in…

In this paper, we are concerned with the following type of elliptic problems: $$ (-\Delta)^{\alpha} u+a(x) u=\frac{|u|^{2^*_{s}-2}u}{|x|^s}+k(x)|u|^{q-2}u, u\,\in\,H^\alpha({\mathbb R}^N), $$ where $2<q< 2^*$, $0<\alpha<1$, $0<s<2\alpha$,…

This is the final part of a series of papers where we study perturbations of divergence form second order elliptic operators $-\operatorname{div} A \nabla$ by first and zero order terms, whose complex coefficients lie in critical spaces,…

In this article, we prove the existence of solutions to a nonlinear nonlocal elliptic problem with a singualrity and a discontinuous critical nonlinearity which is given as follows. \begin{align} \begin{split}\label{main_prob}…

We study Sobolev spaces with weights in the half-space $\mathbb{R}^{N+1}_+=\{(x,y): x \in \mathbb{R}^N, y>0\}$, adapted to the singular elliptic operators \begin{equation*} \mathcal L =y^{\alpha_1}\Delta_{x}…