Semilinear equations for non-local operators: beyond the fractional Laplacian
type: Item
instance of: scholarly article
title: Semilinear equations for non-local operators: beyond the fractional Laplacian
zbMATH Open document ID: 1479.35914
author: Ivan Biočić
author: Zoran Vondraček
author: Vanja Wagner
published in: Nonlinear Analysis. Theory, Methods \& Applications. Series A: Theory and Methods
publication date: 2021-04-20T00:00:00Z
full work available at URL: https://arxiv.org/abs/2010.06240
review text: This paper deal with the following non-local semi-linear problem \[ \begin{cases} -Lu(x) = f(x,u(x)),&\mbox{in \(D\)},\\ u= \lambda, & \mbox{in \(D^c\)},\\ W_Du = \mu,&\mbox{on \(\partial D\)}, \end{cases}\tag{1} \] where \(D\subset \mathbb R^N\), \(N\geq 2\) is a bounded open set, \(f: D\times \mathbb R \to \mathbb R\) is continuous with respect to the second variable, and there is a function \(\rho:D\to [0, \infty)\) and a continuous function \(\Lambda: [0, \infty) \to [0, \infty)\) such that \[ |f(x,t)| \leq \rho (x)\Lambda(|t|). \] \(\lambda\) and \(\mu\) are signed measures in \(D^c=\mathbb R^N\setminus D\) and \(\partial D\) respectively. The operator \(L\) is a nonlocal operator defined as \[ Lu(x) = P.V. \int_{\mathbb R^N} (u(y)-u(x))j(|y-x|) dy, \] where the singular kernel \(j\) is completely determined by the function \(\phi: (0, \infty) \to (0, \infty)\) is a complete Bernstein function without drift satisfying certain weak scaling conditions. By using the sub and super solution the authors obtained the existence of weak dual solution to (1).
reviewed by: César Torres
Mathematics Subject Classification ID: 35R11
Mathematics Subject Classification ID: 31B25
Mathematics Subject Classification ID: 31C05
Mathematics Subject Classification ID: 35J61
Mathematics Subject Classification ID: 45K05
Mathematics Subject Classification ID: 60J35
Mathematics Subject Classification ID: 35R09
zbMATH DE Number: 7335764
zbMATH Keywords: semilinear differential equations
zbMATH Keywords: non-local operators
MaRDI profile type: Publication
OpenAlex ID: W3129279951
arXiv ID: 2010.06240
cites work: Large \(s\)-harmonic functions and boundary blow-up solutions for the fractional Laplacian
cites work: Very large solutions for the fractional Laplacian: towards a fractional Keller-Osserman condition
cites work: Existence and nonexistence of positive solutions to the fractional equation Δ^α/2 u = –u^γ in bounded domains
cites work: Blow up boundary solutions of some semilinear fractional equations in the unit ball
cites work: Nonnegative entire bounded solutions to some semilinear equations involving the fractional Laplacian
cites work: Q4888858
cites work: On overdetermined problems for a general class of nonlocal operators
cites work: Hopf's lemma for viscosity solutions to a class of non-local equations with applications
cites work: Relative Fatou theorem for harmonic functions of rotation invariant stable processes in smooth domains
cites work: Extension and trace for nonlocal operators
cites work: Density and tails of unimodal convolution semigroups
cites work: Semilinear Dirichlet problem for the fractional Laplacian
cites work: Estimates and structure of \(\alpha\)-harmonic functions
cites work: Blow-up phenomena in nonlocal eigenvalue problems: when theories of \(L^1\) and \(L^2\) meet
cites work: Large solutions to elliptic equations involving fractional Laplacian
cites work: Gaugeability and conditional gaugeability
cites work: Dirichlet heat kernel estimates for rotationally symmetric Lévy processes
cites work: General gauge and conditional gauge theorems
cites work: Markov Processes, Brownian Motion, and Time Symmetry
cites work: From Brownian Motion to Schrödinger’s Equation
cites work: Semilinear elliptic equations for the fractional Laplacian with Hardy potential
cites work: Boundary blow up solutions for fractional elliptic equations
cites work: On Harnack inequality and Hölder regularity for isotropic unimodal Lévy processes
cites work: Remarks on the nonlocal Dirichlet problem
cites work: On estimates of Poisson kernels for symmetric Lévy processes
cites work: Regularity for fully nonlinear integro-differential operators with kernels of variable orders
cites work: Generalized Evans-Krylov and Schauder type estimates for nonlocal fully nonlinear equations with rough kernels of variable orders
cites work: Two-sided Green function estimates for killed subordinate Brownian motions
cites work: Uniform boundary Harnack principle for rotationally symmetric Lévy processes in general open sets
cites work: Global uniform boundary Harnack principle with explicit decay rate and its application
cites work: Martin boundary of unbounded sets for purely discontinuous Feller processes
cites work: Minimal thinness with respect to subordinate killed Brownian motions
cites work: Scale invariant boundary Harnack principle at infinity for Feller processes
cites work: Accessibility, martin boundary and minimal thinness for Feller processes in metric measure spaces
cites work: Heat kernels of non-symmetric jump processes: beyond the stable case
cites work: An 𝐿_{𝑝}-theory for diffusion equations related to stochastic processes with non-stationary independent increment
cites work: Martin kernels for Markov processes with jumps
cites work: Suprema of Lévy processes
cites work: Nonlinear second order elliptic equations involving measures
cites work: Q4937701
cites work: Measures, Integrals and Martingales
cites work: Bernstein functions. Theory and applications
cites work: A probabilistic principle and generalized Schrödinger perturbation
DOI: 10.1016/J.NA.2021.112303
Recommended article: ON NONLOCAL NONLINEAR ELLIPTIC PROBLEMS WITH THE FRACTIONAL LAPLACIAN
Recommended article: Fractional evolution equation nonlocal problems with noncompact semigroups
Recommended article: Semilinear problems for the fractional Laplacian with a singular nonlinearity
Recommended article: Q4683392
Recommended article: Q5409264
Recommended article: Nonlocal Caucy problems for semilinear evolution equations involving almost sectorial operators
Recommended article: Q5409512
Recommended article: Nonlinear fractional Laplacian problems with nonlocal ‘gradient terms’
Recommended article: Nonlocal fractional elliptic equations and applications
Recommended article: A restricted nonlocal operator bridging together the Laplacian and the fractional Laplacian