Related papers: Uniqueness for linear integro-differential equatio…

We consider classical solutions to $-\Delta u = f(u)$ in half-spaces, under homogeneous Dirichlet boundary conditions. We prove that any positive solution is strictly monotone increasing in the direction orthogonal to the boundary, provided…

In the paper, we considered the existence and uniqueness of the global solution in the space of continuously differentiable functions for a nonlinear differential equation with the Caputo fractional derivative of general form. Our main…

In the present paper we derive Liouville type results and existence of periodic solutions for $\chi^{(2)}$ type systems with non-homogeneous nonlinearities. Moreover, we prove both universal bounds as well as singularity and decay estimates…

Unique continuation principles are fundamental properties of elliptic partial differential equations, giving conditions that guarantee that the solution to an elliptic equation must be uniformly zero. Since finite-element discretizations…

In this paper, we prove some Liouville theorem for the following elliptic equations involving nonlocal nonlinearity and nonlocal boundary value condition $$ \left\{ \begin{array}{ll} \displaystyle -\Delta u(y)=\intpr \frac{…

In this work, we present a systematic approach to investigate the existence, multiplicity, and local gradient regularity of solutions for nonlocal quasilinear equations with local gradient degeneracy. Our method involves an interactive…

We prove the existence and uniqueness of solutions to a Dirichlet problem \[ \begin{cases} Lu = f + v^{-1}\text{Div}(v{\bf e} h), & x \in \Omega; u = 0, & x \in \partial \Omega, \end{cases}\] where $L$ is a degenerate, linear, second order…

We study the existence and multiplicity of periodic weak solutions for a non-local equation involving an odd subcritical nonlinearity which is asymptotically linear at infinity. We investigate such problem by applying the the pseudo-index…

We consider fully nonlinear elliptic integro-differential operators with kernels of variable orders, which generalize the integro-differential operators of the fractional Laplacian type in \cite{CS}. Since the order of differentiability of…

We investigate Liouville-type results, existence, uniqueness and symmetry to the solution of nonlinear nonlocal elliptic equations of the form \[ Lu = |x|^{\gamma}\,H(u)\,G(\nabla u), \qquad x\in\R^n, \] where $L$ is a symmetric,…

Let $L$ be a second order elliptic operator $L$ with smooth coefficients defined on a domain $\Omega $ in $\mathbb{R}^d $, $d\geq3$, such that $L1\leq 0$. We study existence and properties of continuous solutions to the following problem…

We prove existence, symmetry and uniqueness of solutions to the fractional Gelfand equation $$ (-\Delta)^s u = e^u \quad \mbox{in $\mathbb{R}$} \quad \mbox{with} \quad \int_{\mathbb{R}} e^u dx < +\infty $$ for all exponents $s \in…

We show non-existence of solutions of the Cauchy problem in $\mathbb{R}^N$ for the nonlinear parabolic equation involving fractional diffusion $\partial_t u + (-\Delta)^s \phi(u)= 0,$ with $0<s<1$ and very singular nonlinearities $\phi$ .…

We deal with the uniqueness of distributional solutions to the continuity equation with a Sobolev vector field and with the property of being a Lagrangian solution, that means transported by a flow of the associated ordinary differential…

We prove a Liouville-type theorem for bounded stable solutions $v \in C^2(\R^n)$ of elliptic equations of the type (-\Delta)^s v= f(v)\qquad {in $\R^n$,} where $s \in (0,1)$ {and $f$ is any nonnegative function}. The operator $(-\Delta)^s$…

Non-existence and uniqueness results are proved for several local and non-local supercritical bifurcation problems involving a semilinear elliptic equation depending on a parameter. The domain is star-shaped but no other symmetry assumption…

We establish Liouville type theorems for elliptic systems with various classes of non-linearities on $\mathbb{R}^N$. We show among other things, that a system has no semi-stable solution in any dimension, whenever the infimum of the…

We show that a viscosity solution of a uniformly elliptic, fully nonlinear equation which vanishes on an open set must be identically zero, provided that the equation is $C^{1,1}$. We do not assume that the nonlinearity is convex or…

We consider the uniqueness of the following positive solutions of anisotropic elliptic equation: \begin{equation} \nonumber \left\{ \begin{aligned} -\Delta^F_p u&=u^q \quad \text{in} \quad \Omega, u&=0 \quad \text{on} \quad \partial \Omega,…

We extend existence and uniqueness results of [4] for nonlinear integro-differential equations of Volterra type between real locally complete vector spaces