Neumann fractionalp-Laplacian: Eigenvalues and existence results

Abstract

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.

Neumann and Robin type boundary conditions in Fractional Orlicz-Sobolev spaces

“…In [31], under suitable conditions on the nonlinearities, the authors obtain existence of at most one positive solution by following the celebrated paper of Brezis-Oswald. The authors in [30], for the same problem but with β ≡ 0, and under suitable conditions on f , by using variational methods obtain existence of two positive solutions. It worths to be mention that the local counterpart of (1.2) for Orlicz functions in the Dirichlet case was studied in [12,24,32].…”

“…Very close to (1.2), as a second aim, we will study eigenvalues and minimizers of several nonlocal problems with non-standard growth involving different boundary conditions. For the case of powers, that is, for fractional p-Laplacian type operators, the Dirichlet case was studied for instance in [27,36], for the Neumann case see for instance [15,30], the Robin case was dealt in [19]. For general Orlicz functions and Dirichlet boundary conditions we refer to [35].…”

A priori estimates for the Fractional <i>p</i>-Laplacian with nonlocal Neumann boundary conditions and applications

“…Quite recently such an operator has been associated to a nonlocal Neumann boundary condition, namely |x − y| N +ps dy = 0 for every x ∈ R N \Ω. Such a condition was introduced in [8] for p = 2 and generalized in [2] and [14] for p > 1, and it turns out that it is the natural p-Neumann boundary condition associated to (−∆) s p . Indeed, this boundary condition allows us to give a formulation of weak solutions of problem (1.1) in a variational way, see [2,8,14].…”

“…Such a condition was introduced in [8] for p = 2 and generalized in [2] and [14] for p > 1, and it turns out that it is the natural p-Neumann boundary condition associated to (−∆) s p . Indeed, this boundary condition allows us to give a formulation of weak solutions of problem (1.1) in a variational way, see [2,8,14]. Moreover, in the case p = 2, it happens that among all functions in the associated fractional Sobolev space (to be defined below), the ones minimizing the Gagliardo seminorm, automatically satisfy the nonlocal Neumann boundary condition above, see [7].…”

“…The proof of Theorem 3.1 relies on a fractional version of De Giorgi's iteration method, which has been usefully employed in the case of fractional Dirichlet boundary conditions (for instance, see [10,11]), and also for eigenvalues of the Neumann fractional problem (see [14]). Of course, due to the non local nature of the problem, the classical steps cannot be followed verbatim and several novelties are needed.…”