# A nonlocal supercritical Neumann problem

## TL;DR

This paper proves the existence of positive, non-decreasing radial solutions for a broad class of nonlocal nonlinear Neumann problems with supercritical growth, overcoming compactness issues via cone-based estimates and variational methods.

## Contribution

It introduces a novel approach to handle supercritical nonlinearities in nonlocal Neumann problems by working within a cone of radial functions, and establishes a strong maximum principle for such problems.

## Key findings

- Existence of solutions in supercritical regimes.
- Development of a priori estimates in a non-compact setting.
- Proof of a strong maximum principle for nonlocal Neumann problems.

## 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.