Related papers: Fractional differentiability for solutions of nonl…

In this paper, we study quadratic growth solutions $u$ of fully nonlinear elliptic equations of the form $F(D^2u,x)=f$ in $\mathbb{R}^n$, where $f$ is periodic and $F$ has the periodicity in $x$. Under the assumption that the oscillation of…

By using some recent results for divergence form equations, we study the $L_p$-solvability of second-order elliptic and parabolic equations in nondivergence form for any $p\in (1,\infty)$. The leading coefficients are assumed to be in…

Let $L$ be a second order uniformly elliptic differential operator in a domain $D$ of $\mathbb{R}^{d}$, $\psi:\mathbb{R}_+\to \mathbb{R}_+$ be a nondecreasing continuous function and let $\xi,g:D\to\mathbb{R}_+$ be locally bounded Borel…

This paper is devoted to the study of the existence of positive solutions for a problem related to a higher order fractional differential equation involving a nonlinear term depending on a fractional differential operator,…

We consider the mixed local and nonlocal functionals with nonstandard growth \begin{eqnarray*} u\mapsto\int_{\Omega}(|Du|^p-f(x)u)\,dx+\int_{\mathbb{R}^N}\int_{\mathbb{R}^N}\frac{|u(x)-u(y)|^q}{|x-y|^{N+sq}}\,dxdy \end{eqnarray*} with…

In this paper, we consider the dual fractional parabolic problem in the right half space. We prove that the positive solutions are strictly increasing in $x_1$ direction without assuming the solutions be bounded. So far as we know, this is…

We study the well-posedness of triply nonlinear degenerate elliptic-parabolic-hyperbolic problem $$ b(u)_t - {\rm div} \tilde{\mathfrak a}(u,\nabla\phi(u))+\psi(u)=f, \quad u|_{t=0}=u_0 $$ in a bounded domain with homogeneous Dirichlet…

This note studies local integral gradient bounds for distributional solutions of a large class of partial differential inequalities with diffusion in divergence form and power-like first-order terms. The applications of these estimates are…

We define the fractional powers $L^s=(-a^{ij}(x)\partial_{ij})^s$, $0 < s < 1$, of nondivergence form elliptic operators $L=-a^{ij}(x)\partial_{ij}$ in bounded domains $\Omega\subset\mathbb{R}^n$, under minimal regularity assumptions on the…

Let $\textbf{A}$ be a symmetric convex quadratic form on $\mathbb{R}^{Nn}$ and $\Omega\Subset \mathbb{R}^n$ a bounded convex domain. We consider the problem of existence of solutions $u: \Omega \subset \mathbb{R}^n \longrightarrow…

In this article, we prove existence results of positive solutions for the following nonlinear elliptic problem with gradient terms: \begin{eqnarray*} \left\{\begin{array}{l@{\quad }l} (-\Delta)^\alpha u=f(x,u,v,\nabla u, \nabla v) &{\rm…

We study the semilinear elliptic equation \begin{equation*} -\Delta u=u^\alpha |\log u|^\beta\quad\text{in }B_1\setminus\{0\}, \end{equation*} where $B_1\subset\mathbb{R}^n$ with $n\geq 3$, $\frac{n}{n-2} < \alpha < \frac{n+2}{n-2}$ and…

We consider a class of nonlinear non-diagonal elliptic systems with $p$-growth and establish the $L^q$-integrability for all $q\in [p,p+2]$ of any weak solution provided the corresponding right hand side belongs to the corresponding…

We study fully nonlinear elliptic equations such as \[ F(D^2u) = u^p, \quad p>1, \] in $\R^n$ or in exterior domains, where $F$ is any uniformly elliptic, positively homogeneous operator. We show that there exists a critical exponent,…

We consider the $2m$-th order elliptic boundary value problem $Lu=f(x,u)$ on a bounded smooth domain $\Omega\subset\R^N$ with Dirichlet boundary conditions on $\partial\Omega$. The operator $L$ is a uniformly elliptic linear operator of…

We consider a class of {energy integrals}, associated to nonlinear and non-uniformly elliptic equations, with integrands $f(x,u,\xi)$ satisfying anisotropic $p_i,q$-growth conditions of the form $$ \sum_{i=1}^n \lambda_i (x)|\xi_i|^{p_i}\le…

We consider fractional wave equations with exponential or arbitrary polynomial nonlinearities. We prove the global well-posedness on the support of the corresponding Gibbs measures. We provide ill-posedness constructions showing that the…

We prove well-posedness for doubly nonlinear parabolic stochastic partial differential equations of the form $dX_t-\text{div}\,\gamma(\nabla X_t)\,dt+\beta(X_t)\,dt\ni B(t,X_t)\,dW_t$, where $\gamma$ and $\beta$ are the two nonlinearities,…

We prove the theorem of linearized asymptotic stability for fractional differential equations. More precisely, we show that an equilibrium of a nonlinear Caputo fractional differential equation is asymptotically stable if its linearization…

This article investigates the existence, nonexistence, and multiplicity of positive solutions to the sublinear fractional elliptic problem $(P_{\lambda}^s)$. We begin by establishing several a priori estimates that provide regularity…