Related papers: Regularity results for a class of mixed local and …

This article deals with the study of the following singular quasilinear equation: \begin{equation*} (P) \left\{ \ -\Delta_{p}u -\Delta_{q}u = f(x) u^{-\delta},\; u>0 \text{ in }\; \Om; \; u=0 \text{ on } \pa\Om, \right. \end{equation*}…

This paper considers a local and non-local problem characterized by singular nonlinearity and a source term. Specifically, we focus on the following problem: \begin{equation}\label{A}\tag{P} -\Delta_{p} u + (-\Delta)^{s}_{q} u = f(x)…

We establish local boundedness and local H\"older continuity of weak solutions to the following prototype problem: $$ -\operatorname{div}\left(|x|^{-2 \beta}|\nabla u|^{\mathbf{q}-2} \nabla u\right)+(-\Delta)_{p(\cdot, \cdot),…

In this article, we examine the H\"older regularity of solutions to equations involving a mixed local-nonlocal nonlinear nonhomogeneous operator $\fp + \fqs$ with singular data, under the minimal assumption that $p> sq$. The regularity…

We study local H\"older regularity of bounded, weak solutions for the nonlocal quasilinear equations of the form \[ (|u|^{q-2}u)_t + \text{P.V.} \int_{\mathbb{R}^n} \frac{|u(x,t) - u(y,t)|^{p-2}(u(x,t)-u(y,t))}{|x-y|^{n+sp}} dy = 0, \] with…

It is establish regularity results for weak solutions of quasilinear elliptic problems driven by the well known $\Phi$-Laplacian operator given by \begin{equation*} \left\{\ \begin{array}{cl} \displaystyle-\Delta_\Phi u= g(x,u), &…

Let $\Omega$ be a bounded $C^2$ domain in $\mathbb{R}^n$ and $u\in C(\mathbb{R}^n)$ solves \begin{equation*} \begin{aligned} \Delta u + a Iu + C_0|Du| \geq -K\quad \text{in}\; \Omega, \quad \Delta u + a Iu - C_0|Du|\leq K \quad \text{in}\;…

This article deals with the study of the following nonlinear doubly nonlocal equation: \begin{equation*} (-\Delta)^{s_1}_{p}u+\ba(-\Delta)^{s_2}_{q}u = \la a(x)|u|^{\delta-2}u+ b(x)|u|^{r-2} u,\; \text{ in }\; \Om, \; u=0 \text{ on }…

We discuss the H\"older regularity of solutions to the semilinear equation involving the fractional Laplacian $(-\Delta)^s u=f(u)$ in one dimension. We put in evidence a new regularity phenomenon which is a combined effect of the…

In this work we analyze the existence of solutions to the fractional quasilinear problem, $$ (P) \left\{ \begin{array}{rcll} u_t+(-\Delta )^s u &=&|\nabla u|^{\alpha}+ f &\inn \Omega_T\equiv\Omega\times (0,T),\\ u(x,t)&=&0 &…

In this article, we investigate the existence, uniqueness, nonexistence, and regularity of weak solutions to the nonlinear fractional elliptic problem of type $(P)$ (see below) involving singular nonlinearity and singular weights in smooth…

In the present paper we investigate the following semilinear singular elliptic problem: \begin{equation*} (\rm P)\qquad \left \{\begin{array}{l} -\Delta u = \dfrac{p(x)}{u^{\alpha}}\quad \text{in} \Omega \\ u = 0\ \text{on} \Omega,\ u>0…

Given $\Omega(\subseteq\;R^{1+m})$, a smooth bounded domain and a nonnegative measurable function $f$ defined on $\Omega$ with suitable summability. In this paper, we will study the existence and regularity of solutions to the quasilinear…

We introduce a new class of quasi-linear parabolic equations involving nonhomogeneous degeneracy or/and singularity $$ \partial_t u=[|D u|^q+a(x,t)|D u|^s]\left(\Delta u+(p-2)\left\langle D^2 u\frac{D u}{|D u|},\frac{D u}{|D…

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…

We consider the mixed local-nonlocal semi-linear elliptic equations driven by the superposition of Brownian and L\'evy processes \begin{equation*} \left\{ \begin{array}{ll} - \Delta u + (-\Delta)^s u = g(x,u) & \hbox{in $\Omega$,} u=0 &…

Let $D$ be an bounded region in ${\bf R}^n$. The regularity of solutions of a family of quasilinear elliptic partial differential equations is studied, one example being $\Delta_nu=Vu^{n-1}$. The coefficients are assumed to be in the space…

We study the existence, multiplicity and regularity results of weak solutions for the Dirichlet problem of a semi-linear elliptic equation driven by the mixture of the usual Laplacian and fractional Laplacian \begin{equation*} \left\{%…

In this paper, we prove the existence of weak, veryweak and duality solutions to a class of elliptic problems involving singularity and measure data which is given by: $-\Delta u+(-\Delta)^s u = \frac{f(x)}{u^\gamma} +\mu$ in $\Omega$ with…

We study the regularity properties of a weak solution to the boundary value problem for the equation $-\Delta \rho +a u=f$ in a bounded domain $\Omega\subset \mathbb{R}^N$, where $\rho=e^{-\mbox{div}\left(|\nabla u|^{p-2}\nabla…