Related papers: An elliptic problem with degenerate coercivity and…

In this manuscript we deal with elliptic equations with superlinear first order terms in divergence form of the following type \[ -\mbox{div}(M(x)\nabla u)= -\mbox{div}(h(u)E(x))+f(x), \] where $M$ is a bounded elliptic matrix, the vector…

In this paper we deal with a Dirichlet problem for an elliptic equation involving the $1$-Laplacian operator and a source term. We prove that, when the growth of the source is subcritical, there exist two bounded nontrivial solutions to our…

In this work, we develop an efficient solver based on neural networks for second-order elliptic equations with variable coefficients and singular sources. This class of problems covers general point sources, line sources and the combination…

We investigate the problem $$ \left\{ \begin{array}{ll} -\Delta_p u = g(u)|\nabla u|^p + f(x,u) \ & \mbox{in} \ \ \Omega, \ \ \\ u>0 \ &\mbox{in} \ \ \Omega, \ \ u = 0 \ &\mbox{on} \ \ \partial\Omega, \end{array} \right. \leqno{(P)} $$ in a…

We obtain some fine gradient estimates near the boundary for solutions to fractional elliptic problems subject to exterior Dirichlet boundary conditions. Our results provide, in particular, the sign of the normal derivative of such…

We study an eigenvalue problem involving a degenerate and singular elliptic operator on the whole space $\mathbb{R}^N$. We prove the existence of an unbounded and increasing sequence of eigenvalues. Our study generalizes to the case of…

The goal of this article is to establish local Lipschitz continuity of weak solutions for a class of degenerated elliptic equations of divergence form, in the Heisenberg Group. The considered hypothesis for the growth and ellipticity…

We study non-variational degenerate elliptic equations with high order singular structures. No boundary data are imposed and singularities occur along an {\it a priori} unknown interior region. We prove that positive solutions have a…

We present recent advances in the regularity theory for weak solutions to some classes of elliptic and parabolic equations with strongly singular or degenerate structure. The equations under consideration satisfy standard $p$-growth and…

In this paper we are mainly concerned with nontrivial positive solutions to the Dirichlet problem for the degenerate elliptic equation \begin{gather} -\frac{\partial^2 u}{\partial x^2} -\left|x\right|^{2k}\frac{\partial^2 u}{\partial…

In this paper, exploiting variational methods, the existence of three weak solutions for a class of elliptic equations involving a general operator in divergence form and with Dirichlet boundary condition is investigated. Several special…

This paper studies a maximal $L^q$-regularity property for nonlinear elliptic equations of second order with a zero-th order term and gradient nonlinearities having superlinear and sub-quadratic growth, complemented with Dirichlet boundary…

Using a method developped in [1] and [2], we prove the existence of weak non trivial solutions to fourth order elliptic equations with singularities and with critical Sobolev growth.

We prove the existence of a weak solution for a general class of Dirichlet anisotropic elliptic problems such as $\mathcal Au+\Phi(x,u,\nabla u)=\mathfrak{B}u+f$ in $\Omega$, where $\Omega$ is a bounded open subset of $\mathbb R^N$ and…

We consider solutions to degenerate anisotropic elliptic equations in order to study their regularity. In particular we establish second-order estimates and enclose regularity results for the stress field. All our results are new even in…

We study a general class of quasilinear elliptic equations with nonstandard growth to prove the existence of a very weak solution to such a problem. A key ingredient in the proof is a priori global weighted gradient estimate of a very weak…

This paper is devoted to prove existence of renormalized solutions for a class of non--linear degenerate elliptic equations involving a non--linear convection term, which satisfies a growth properties, and a Hardy potential. Additionally,…

Let $L$ be an infinitely degenerate second-order linear operator defined on a bounded smooth Euclidean domain. Under weaker conditions than those of H\"ormander, we show that the Dirichlet problem associated with $L$ has a unique smooth…

We consider Dirichlet problems for linear elliptic equations of second order in divergence form on a bounded or exterior smooth domain $\Omega$ in $\mathbb{R}^n$, $n \ge 3$, with drifts $\mathbf{b}$ in the critical weak $L^n$-space…

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…