Related papers: Higher order evolution inequalities with nonlinear…

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*}…

In this paper, we consider the following nonlinear parabolic equation \[ \partial_{t}u\,=\,\sum_{i=1}^{n}\partial_{x_{i}}\left[(\vert u_{x_{i}}\vert-\delta_{i})_{+}^{p-1}\frac{u_{x_{i}}}{\vert…

In this paper we deal with the equation \[-\Delta_p u+|u|^{p-2}u=|u|^{q-2}u\] for $1<p<2$ and $q>p$, under Neumann boundary conditions in the unit ball of $\mathbb R^N$. We focus on the three positive, radial, and radially non-decreasing…

We carry on the investigation started in [2] about the regularity of weak solutions to the strongly degenerate parabolic equation \[ u_{t}-\mathrm{div}\left[(\vert Du\vert-1)_{+}^{p-1}\frac{Du}{\vert…

We study the existence of nonnegative solutions to the following nonlocal elliptic problem involving singularity \begin{align} \mathfrak{M}\left(\int_{Q}\frac{|u(x)-u(y)|^p}{|x-y|^{N+ps}}dxdy\right)(-\Delta)_{p}^{s}…

We investigate the lifespan of solutions to the higher-order semilinear parabolic equation $$u_t+(-\Delta)^m u=|u|^p, \quad x \in \mathbb{R}^n, t>0 $$ with initial data. We focus on the precise asymptotic behavior of the lifespan of…

We deal with the following nonlinear problem involving fractional $p\&q$ Laplacians: \begin{equation*} (-\Delta)^{s}_{p}u+(-\Delta)^{s}_{q}u+|u|^{p-2}u+|u|^{q-2}u=\lambda h(x) f(u)+|u|^{q^{*}_{s}-2}u \mbox{ in } \mathbb{R}^{N},…

We study existence and multiplicity of nontrivial solutions of the following problem $$ \left\{ \begin{array}{rcll} -\Delta_p u+(-\Delta_p)^{s} u & = & \lambda|u|^{q-2}u+|u|^{p^{\ast}-2}u & \mbox{ in }\Omega,\\ u & = & 0 & \mbox{ on }…

We prove the existence of solutions for an evolution quasi-variational inequality with a first order quasilinear operator and a variable convex set, which is characterized by a constraint on the absolute value of the gradient that depends…

The aim of this paper is to treat the following problem $$ (P) \left\{ \begin{array}{rcll} (-\Delta)^s_{p, \beta} u &= & f(x,u) &\mbox{ in }\Omega, u & = & 0 &\mbox{ in } \mathds{R}^N\setminus\Omega, \end{array} \right. $$ where $$…

We study the regularity of weak solutions to evolution equations with distributed order fractional time derivative. We prove a weak Harnack inequality for nonnegative weak supersolutions and H\"older continuity of weak solutions to this…

Let $\Omega \subset \mathbb{R}^N$ ($N \geq 3$) be a bounded smooth domain and $\delta(x)=\text{dist}(x,\partial \Omega)$. In this paper, we provide various necessary and sufficient conditions for the existence of weak solutions to $$…

Let $\Omega\subset\mathbb{R}^{N}$ ($N\geq1$) be a bounded and smooth domain and $a:\Omega\rightarrow\mathbb{R}$ be a sign-changing weight satisfying $\int_{\Omega}a<0$. We prove the existence of a positive solution $u_{q}$ for the problem…

We consider periodic homogenization with localized defects of boundary value problems for semilinear ODE systems of the type $$ \Big((A(x/\varepsilon)+B(x/\varepsilon))u'(x)+c(x,u(x))\Big)'= d(x,u(x)) \mbox{ for } x \in (0,1),\;…

In this paper, we study the following quasi-linear elliptic inequality $\Delta_m u +u^p |\nabla u|^q \leqslant 0$ on weighted graphs, where $(m,p,q)\in (1,\infty)\times\mathbb{R}\times\mathbb{R}$. According to the ranges of parameters $(m,…

In this paper we shall investigate the nonexistence of positive solutions for the following nonlinear parabolic partial differential equation:\[ \begin{cases} \frac{\partial u}{\partial t}= \Delta_{\mathbb{G},p}u+V(x)u^{p-1} &…

We consider nonnegative solutions $u:\Omega\longrightarrow \mathbb{R}$ of second order hypoelliptic equations \begin{equation*} \mathscr{L} u(x) =\sum_{i,j=1}^n \partial_{x_i} \left(a_{ij}(x)\partial_{x_j} u(x) \right) + \sum_{i=1}^n b_i(x)…

In this paper, we study the regularity of weak solutions to the following strongly degenerate parabolic equation \begin{equation*} u_t-\div\left(\left(\left|Du\right|-1\right)_+^{p-1}\frac{Du}{\left|Du\right|}\right)=f\qquad\mbox{ in…

We prove existence of variational solutions for a class of doubly nonlinear nonlocal evolution equations whose prototype is the double phase equation \begin{align*} \partial_t u^m &+ \text{P.V.}\int_{\mathbb{R}^N}…

We study quasilinear degenerate singular elliptic equation of type -Delta_p u = \frac{u^{p^*(s)-1}}{|y|^t}$ in a smooth bounded domain \Omega in R^n=R^k \times R^{N-k}$, x=(y,z) in R^k \times R^{N-k}, 2 \leq k<N and N \geq 3, 1<p<2, 0\leq…