Related papers: Regularity of quasi-linear equations with H\"orman…

The multiplicity of positive weak solutions for a quasilinear Schr\"{o}dinger equations $-L_p u +(\lambda A(x)+1)|u|^{p-2}u= h(u)$ in $\mathbb{R}^N$ is established, where $L_p u\doteq \epsilon^{p}\Delta_p u +\epsilon^{p}\Delta_p (u^2)u$,…

Let $X$ be a separable Banach space endowed with a non-degenerate centered Gaussian measure $\mu$. The associated Cameron--Martin space is denoted by $H$. Consider two sufficiently regular convex functions $U:X\rightarrow\mathbb{R}$ and…

In this paper, we investigate the boundary H\"{o}lder regularity for elliptic equations (precisely, the Poisson equation, linear equations in divergence form and non-divergence form, the p-Laplace equations and fully nonlinear elliptic…

We establish general sufficient conditions for exact (and global) regularity in the $\bar\partial$-Neumann problem on $(p,q)$-forms, $0 \leq p \leq n$ and $1\leq q \leq n$, on a pseudoconvex domain $\Omega$ with smooth boundary $b\Omega$ in…

Let us consider a semilinear boundary value problem $ - \Delta u= f(x,u),$ in $\Omega,$ with Dirichlet boundary conditions, where $ \Omega \subset \mathbb{R}^N $, $N> 2,$ is a bounded smooth domain. We provide sufficient conditions…

We consider general nonlinear elliptic equations of the form \[ \operatorname{div}\, A(x,Du) = 0 \quad \text{in } \Omega, \] where $A:\Omega \times \mathbb R^n \to \mathbb R^n$ satisfies a quasi-isotropic $(p,q)$-growth condition, which is…

We study semilinear elliptic equations \begin{equation*} \begin{cases} -\Delta u = f(u) & \text{in } \Omega, \\ \partial_\nu u = 0 & \text{on } \partial\Omega, \end{cases} \end{equation*} with homogeneous Neumann boundary conditions in…

In this paper we prove the existence and regularity of weak solutions for the following system \begin{align*} \begin{cases} -\mbox{div}(M(x)\nabla u) + g(x,u,v) = f \ \ \mbox{in} \ \ \Omega\\ -\mbox{div}(M(x)\nabla v) = h(x,u,v) \ \…

We study the regularity of weak solutions for two elliptic systems involving the $n$-Laplacian and a critical nonlinearity in the right hand side: $H$-systems and $n$-harmonic maps into compact Riemannian manifolds. Under the assumptions…

Let $ 0\le f\in C^{0,1}(\mathbb R^n)$. Given a domain $\Omega\subset \mathbb R^n$, we prove that any stable solution to the equation $-\Delta u=f(u)$ in $\Omega$ satisfies a BMO interior regularity when $n=10$, and an Morrey…

Given a compact manifold $M$ equipped with smooth vector fields $X_1,\ldots, X_r$, we consider the generalized Dirichlet energy \[\mathbf{E}(f)= \sum_{j=1}^r\int_M |X_jf|^2\, dm,\] where $dm$ is a volume form, and ask if the set \[…

In this paper, we are interested in the regularity of weak solutions $u\colon\Omega_T\to\mathbb{R}$ to parabolic equations of the type \begin{equation*} \partial_t u - \mathrm{div} \nabla \mathcal{F}(x,t,Du) = f\qquad\mbox{in $\Omega_T$},…

This paper is concerned with two properties of positive weak solutions of quasilinear elliptic equations with nonlinear gradient terms. First, we show a Liouville-type theorem for positive weak solutions of the equation involving the…

This article studies a class of semilinear scalar field equations on the real line with variable coefficients in the linear terms. These coefficients are not necessarily small perturbations of a constant. We prove that under suitable…

In this manuscript, we investigate geometric regularity estimates for problems governed by quasi-linear elliptic models in non-divergence form, which may exhibit either degenerate or singular behavior when the gradient vanishes, under…

This paper studies Liouville properties for viscosity sub- and supersolutions of fully nonlinear degenerate elliptic PDEs, under the main assumption that the operator has a family of generalized subunit vector fields that satisfy the…

We obtain necessary and sufficient conditions with sharp constants on the distribution $\sigma$ for the existence of a globally finite energy solution to the quasilinear equation with a gradient source term of natural growth of the form…

In this article, we deal with the global regularity of weak solutions to a class of problems involving the fractional $(p,q)$-Laplacian, denoted by $(-\Delta)^{s_1}_{p}+(-\Delta)^{s_2}_{q}$, for $s_2, s_1\in (0,1)$ and $1<p,q<\infty$. We…

We establish sharp local $C^{1,\alpha}$-regularity for weak solutions to degenerate elliptic equations of $p$-Laplacian type with data in Morrey spaces. The proof relies on the Fefferman-Phong inequality and standard tools from regularity…

In this paper, we show that weak solutions of $$-\text{div} \mathbb{A}(x)\nabla u = 0 \qquad \text{where}\quad \mathbb{A}(x)= \mathbb{A}(x)^T \,\, \text{and} \,\, \lambda |\zeta|^2 \leq \langle \mathbb{A}(x)\zeta,\zeta\rangle \leq \Lambda…