Related papers: Sharp estimates and existence for anisotropic elli…

We study the existence problem for positive solutions $u \in L^{r}(\mathbb{R}^{n})$, $0<r<\infty$, to the quasilinear elliptic equation \[ -\Delta_{p} u = \sigma u^{q} \quad \text{in} \;\; \mathbb{R}^n \] in the sub-natural growth case…

We investigate nonlinear elliptic Dirichlet problems whose growth is driven by a general anisotropic $N$-function, which is not necessarily of power type and need not satisfy the $\Delta_2$ nor the $\nabla _2$-condition. Fully anisotropic,…

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 investigate the existence and nonexistence of positive solutions for the quasilinear elliptic inequality $L_\mathcal{A} u= -{\rm div}[\mathcal{A}(x, u, \nabla u)]\geq (I_\alpha\ast u^p)u^q$ in $\Omega$, where $\Omega\subset \mathbb{R}^N,…

We study the obstacle problem for fully nonlinear elliptic operators with an anisotropic degeneracy on the gradient: \[ \min \left\{f-|Du|^\gamma F(D^2u),u-\phi\right\} = 0 \quad\textrm{ in }\quad \Omega. \] We obtain existence of solutions…

We consider fully nonlinear uniformly elliptic cooperative systems with quadratic growth in the gradient, such as $$ -F_i(x, u_i, Du_i, D^2 u_i)- \langle M_i(x)D u_i, D u_i \rangle =\lambda c_{i1}(x) u_1 + \cdots + \lambda c_{in}(x) u_n…

In this manuscript, we investigate regularity estimates for a class of quasilinear elliptic equations in the non-divergence form that may exhibit degenerate behavior at critical points of their gradient. The prototype equation under…

Schauder estimates hold in nonuniformly elliptic problems under optimal assumptions on the growth of the ellipticity ratio.

In this paper, we study the following degenerate critical elliptic equations with anisotropic coefficients $$ -div(|x_{N}|^{2\alpha}\nabla u)=K(x)|x_{N}|^{\alpha\cdot 2^{*}(s)-s}|u|^{2^{*}(s)-2}u {in} \mathbb{R}^{N} $$ where…

We study the semilinear elliptic problem \[ -\Delta u = Q_{\Omega} |u|^{p-2}u \quad \text{in } \mathbb{R}^N, \] where \( Q_{\Omega} = \chi_{\Omega} - \chi_{\mathbb{R}^N \setminus \Omega} \) for a bounded smooth domain \( \Omega \subset…

Estimation problems with constrained parameter spaces arise in various settings. In many of these problems, the observations available to the statistician can be modelled as arising from the noisy realization of the image of a random linear…

We use "generalized" version of total variation, coarea formulas, isoperimetric inequalities to obtain sharp estimates for solutions (and for their gradients) to anisotropic elliptic equations with a lower order term, comparing them with…

We obtain a local estimate for the gradient of solutions to a second-order elliptic equation in divergence form with bounded measurable coefficients that are square-Dini continuous at the single point x=0. In particular, we treat the case…

We study the solvability in the whole Euclidean space of coercive quasi-linear and fully nonlinear elliptic equations modeled on $\Delta u\pm g(|\nabla u|)= f(u)$, $u\ge0$, where $f$ and $g$ are increasing continuous functions. We give…

We study quasilinear elliptic equations of the form $\text{div} \mathbf{A}(x,u,\nabla u) = \text{div}\mathbf{F} $ in bounded domains in $\mathbb{R}^n$, $n\geq 1$. The vector field $\mathbf{A}$ is allowed to be discontinuous in $x$,…

Let $p_i\geq 2$ and consider the following anisotropic $p$-Laplace equation $$ -\sum_{i=1}^{N}\frac{\partial}{\partial x_i}\Big(\Big|\frac{\partial u}{\partial x_i}\Big|^{p_i-2}\frac{\partial u}{\partial x_i}\Big)=g(x)f(u),\,\,u>0\text{ in…

We prove sharp asymptotic estimates for the gradient of positive solutions to certain nonlinear $p$-Laplace equations in Euclidean space by showing symmetry and uniqueness of positive solutions to associated limiting problems.

We study a class of Dirichlet boundary value problems whose prototype is \begin{equation}\label{1.2abs} \left\{\begin{array}{ll} -\Delta_p u =h(u)|\nabla u|^p+u^{q-1}+f(x)\, &\quad\hbox{in } \ \Omega\,,\\ u\ge 0\,,&{\quad\hbox{in } \…

In this article we introduce an asymptotic preserving scheme designed to compute the solution of a two dimensional elliptic equation presenting large anisotropies. We focus on an anisotropy aligned with one direction, the dominant part of…

Using a classical technique introduced by Achi E. Brandt for elliptic equations, we study a general class of nonlocal equations obtained as a superposition of classical and fractional operators in different variables. We obtain that the…