Related papers: Global compactness results for nonlocal problems

We study the existence of non-zero positive solutions of a class of systems of differential equations driven by fractional powers of the Laplacian. Our approach is based on the notion of fixed point index, and allows us to deal with…

In the first part of this paper, we prove the global regularity, in an adequate parabolic Bessel-Potential space and then in the corresponding parabolic fractional Sobolev space, of the unique solution to following fractional heat equation…

We study a nonlinear elliptic problem defined in a bounded domain involving fractional powers of the Laplacian operator together with a concave-convex term. We characterize completely the range of parameters for which solutions of the…

We classify global Lipschitz solutions to two-phase free boundary problems governed by concave fully nonlinear equations, as either two-plane solutions or solutions to a one-phase problem.

The paper addresses questions of existence and regularity of solutions to linear partial differential equations whose coefficients are generalized functions or generalized constants in the sense of Colombeau. We introduce various new…

We prove trace and extension results for Sobolev-type function spaces that are well suited for nonlocal Dirichlet and Neumann problems including those for the fractional $p$-Laplacian. Our results are robust with respect to the order of…

By using a perturbation technique in critical point theory, we prove the existence of solutions for two types of nonlinear equations involving fractional differential operators.

We give blow-up behavior for solutions to an elliptic system with Dirichlet condition, and, weight and boundary singularity. Also, we have a compactness result for this elliptic system with regular H{\"o}lderian weight and boundary…

In this paper, we investigate the existence of weak solution for a fractional type problems driven by a nonlocal operator of elliptic type in a fractional Orlicz-Sobolev space, with homogeneous Dirichlet boundary conditions. We first extend…

This paper aims to investigate a Harnack inequality for non-negative solutions of the normalized infinity Laplacian with nonlinear absorption and gradient terms. More specifically, we establish a Harnack inequality for non-negative…

In the present work, we consider a fractional p-Kirchhoff equation in the entire space R^N featuring doubly nonlinearities, involving a generalized nonlocal Choquard subcritical term together with a local critical Sobolev term; the problem…

In this paper, we study different notions of solutions of nonlocal and nonlinear equations of fractional $p$-Laplace type $${\rm P.V.} \int_{\mathbb R^n}\frac{|u(x)-u(y)|^{p-2}(u(x)-u(y))}{|x-y|^{n+sp}}\,dy = 0.$$ Solutions are defined via…

In the present paper, we establish a multiplicity result for a following class of nonlocal Neumann eigenvalue problems involving the fractional p-Laplacian. \begin{align} \begin{cases} (-\Delta)^{s}_{p}u + a(x) \abs{u}^{p-2}u =\lambda…

We prove the existence of ground state solutions to critical growth $p$-Laplacian and fractional $p$-Laplacian problems that are nonresonant at zero.

We prove nonexistence of nontrivial bounded solutions to some nonlinear problems involving nonlocal operators of the form \[Lu(x)=\sum a_{ij}\partial_{ij}u+{\rm PV}\int_{\R^n}(u(x)-u(x+y))K(y)dy.\] These operators are infinitesimal…

We study regularity properties of solutions to nonlinear and nonlocal evolution problems driven by the so-called \emph{$0$-order fractional $p-$Laplacian} type operators: $$ \partial_t u(x,t)=\mathcal{J}_p u(x,t):=\int_{\mathbb{R}^n}…

We establish existence, uniqueness as well as quantitative estimates for solutions to the fractional nonlinear diffusion equation, $\partial_t u +{\mathcal L}_{s,p} (u)=0$, where ${\mathcal L}_{s,p}=(-\Delta)_p^s$ is the standard fractional…

We consider a Br\'ezis-Nirenberg type critical growth $p$-Laplacian problem involving a parameter $\mu > 0$ in a smooth bounded domain $\Omega$. We prove the existence of multiple nontrivial solutions if either $\mu$ or the volume of…

We study an initial value problem with fractional Laplacian and a singular drift term, and obtain local and global existence theorems similar to the results in Jourdain et al.(2005).

This paper deals with the lack of compactness in nonlinear elliptic problems $(P)$. In particular, a domain $\Omega$ is provided where not converging Palais-Smale sequences exist at every energy level. Nevertheless, it is proved that…