Continuity results with respect to domain perturbation for the fractional p -Laplacian (original) (raw)

Neumann fractionalp-Laplacian: Eigenvalues and existence results

Nonlinear Analysis, 2019

We develop some properties of the p−Neumann derivative for the fractional p−Laplacian in bounded domains with general p > 1. In particular, we prove the existence of a diverging sequence of eigenvalues and we introduce the evolution problem associated to such operators, studying the basic properties of solutions. Finally, we study a nonlinear problem with source in absence of the Ambrosetti-Rabinowitz condition.

On the fractional p-Laplacian equations with weight and general datum

Advances in Nonlinear Analysis, 2016

The aim of this paper is to study the following problem: \left\{\begin{aligned} \displaystyle(-\Delta)^{s}_{p,\beta}u&\displaystyle=f(x% ,u)&\displaystyle\phantom{}\text{in }\Omega,\\ \displaystyle u&\displaystyle=0&\displaystyle\phantom{}\text{in }\mathbb{R}^{% N}\setminus\Omega,\end{aligned}\right. where Ω is a smooth bounded domain of {\mathbb{R}^{N}} containing the origin, (-\Delta)^{s}_{p,\beta}u(x):=\mathrm{PV}\int_{\mathbb{R}^{N}}\frac{\lvert u(x)% -u(y)\rvert^{p-2}(u(x)-u(y))}{\lvert x-y\rvert^{N+ps}}\frac{dy}{\lvert x\rvert% ^{\beta}\lvert y\rvert^{\beta}} with {0\leq\beta<\frac{N-ps}{2}} , {1

A limiting obstacle type problem for the inhomogeneous p-fractional Laplacian

Calculus of Variations and Partial Differential Equations, 2019

In this manuscript we study an inhomogeneous obstacle type problem involving a fractional p-Laplacian type operator. First, we focus our attention in establishing existence and uniform estimates for any family of solutions {u p } p≥2 which depend on the data of the problem and universal parameters. Next, we analyze the asymptotic behavior of such a family as p → ∞. At this point, we prove that lim p→∞ u p (x) = u ∞ (x) there exists (up to a subsequence), verifies a limiting obstacle type problem in the viscosity sense, and it is an s-Hölder continuous function. We also present several explicit examples, as well as further features of the limit solutions and their free boundaries. In order to establish our results we overcome several technical difficulties and develop new strategies, which were not present in the literature for this type of problems. Finally, we remark that our results are new even for problems governed by fractional p-Laplacian operator, as well as they extend the previous ones by dealing with more general non-local operators, source terms and boundary data.

Solvability of Neumann boundary value problem for fractional p-Laplacian equation

Advances in Difference Equations

We consider the existence of solutions for a Neumann boundary value problem for the fractional p-Laplacian equation. Under certain nonlinear growth conditions of the nonlinearity, we obtain a new result on the existence of solutions by using the continuation theorem of coincidence degree theory. MSC: 34A08; 34B15

Non-resonant Fredholm alternative and anti-maximum principle for the fractional p-Laplacian

Journal of Fixed Point Theory and Applications, 2017

In this paper we extend two nowadays classical results to a nonlinear Dirichlet problem to equations involving the fractional p−Laplacian. The first result is an existence in a non-resonant range more specific between the first and second eigenvalue of the fractional p−Laplacian. The second result is the anti-maximum principle for the fractional p−Laplacian.

The Obstacle Problem at Zero for the Fractional p-Laplacian

Set-valued and Variational Analysis, 2020

In this paper we establish a multiplicity result for a class of unilateral, nonlinear, nonlocal problems with nonsmooth potential (variational-hemivariational inequalities), using the degree map of multivalued perturbations of fractional nonlinear operators of monotone type, the fact that the degree at a local minimizer of the corresponding Euler functional is equal one, and controlling the degree at small balls and at big balls. Keywords Obstacle problem • Fractional p-Laplacian • Operator of monotone type • Degree theory • Nonsmooth analysis Mathematics Subject Classification (2010) 47G20 • 47H05 • 47H11 • 49J40 • 49J52

On fractional p-Laplacian type equations with general nonlinearities

TURKISH JOURNAL OF MATHEMATICS, 2021

In this paper, we study the existence and multiplicity of solutions for a class of quasi-linear elliptic problems driven by a nonlocal integro-differential operator with homogeneous Dirichlet boundary conditions. As a particular case, we study the following problem: { (−∆) s p u = f (x, u) in Ω, u = 0 in R N \ Ω, where (−∆) s p is the fractional p-Laplacian operator, Ω is an open bounded subset of R N with Lipschitz boundary and f : Ω × R → R is a generic Carathéodory function satisfying either a p− sublinear or a p− superlinear growth condition.