A nonlocal supercritical Neumann problem

Abstract

We establish existence of positive non-decreasing radial solutions for a nonlocal nonlinear Neumann problem both in the ball and in the annulus. The nonlinearity that we consider is rather general, allowing for supercritical growth (in the sense of Sobolev embedding). The consequent lack of compactness can be overcome, by working in the cone of non-negative and non-decreasing radial functions. Within this cone, we establish some a priori estimates which allow, via a truncation argument, to use variational methods for proving existence of solutions. As a side result, we prove a strong maximum principle for nonlocal Neumann problems, which is of independent interest.

A supercritical elliptic equation in the annulus

“…Proof. The proof is analogous to the one in [3, Lemma 4.5] (see also [9,11]). We briefly sketch it below for the sake of completeness.…”

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

“…Let 1 < p < ∞ and let Ω ⊂ R n be an open and bounded set with smooth boundary. We consider (1.5) is the nonlocal normal derivative associated to (−∆) s p , see [1,4,8,10,24,25] and [14] for its introduction in the case p = 2 and c n,s,p is a suitable positive normalization constant only depending on n, s and p. Finally, β is a nonnegative given function. We would like to point out that the Neumann operator N s,2 u recovers the classical Neumann condition as a limit case, and has a clear probabilistic and variational interpretation as well, see [14] for the details.…”

The Neumann problem for the fractional Laplacian: regularity up to the boundary

“…The Neumann problem (1.1) was first introduced in [18,20], and has been subsequently studied in several papers; see for example [1,3,14,30,39]. As explained in detail in [18], (1.1) is a natural Neumann problem for the fractional Laplacian, for several reasons:…”