Towards a Brezis–Oswald-type result for fractional problems with Robin boundary conditions

Abstract

We consider a boundary value problem driven by the p−fractional Laplacian with nonlocal Robin boundary conditions and we provide necessary and sufficient conditions which ensure the existence of a unique positive (weak) solution. The results proved in this paper can be considered a first step towards a complete generalization of the classical result by Brezis and Oswald [6] to the nonlocal setting.

Four Solutions for Fractional p-Laplacian Equations with Asymmetric Reactions

“…Theorem 2.8 is a partial analogue for the fractional p-Laplacian of the classical results of [5,12]. Similar results in the fractional setting were obtained in [24] for p = 2, in [3] for any p > 1 and a pure power reaction, and in [32] for Robin boundary condition. In our case, we make a close connection to the regularity result of [21] in assuming that both u, v ∈ int(C 0 s (Ω) + ), which allows for a simpler proof.…”

Linking over cones for the Neumann Fractional $p-$Laplacian

“…By the maximum principle (see, for instance, [3] and [12] for the Robin problem and also [14] for some linear cases), we can conclude that u > 0 and v < 0 a.e. in R N .…”

“…x ∈ Ω 0 < µG(x, t) ≤ g(x, t)t, and there exist μ > p, a 3 > 0 and a 4 ∈ L 1 (Ω) such that for every t ∈ R and a.e. x ∈ Ω, (12) G(x, t) ≥ a 3 |t| μ − a 4 (x);…”

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

“…Nonlocal equations for the fractional p-Laplacian with boundary conditions involving non-local normal derivatives have been recently developed in the literature; see for instance [1,14,15,17,18,31,37,38].…”

“…Regarding existence of solutions to problem (1.2) in the particular case of the fractional p-Laplacian, there has been some recent develops. 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.…”