On the Pohozaev identity for the fractional $p$-Laplacian operator in $\mathbb{R}^N$

TL;DR

This paper establishes the existence of nontrivial solutions for a nonlinear fractional p-Laplacian problem and proves that these solutions, as well as sufficiently smooth ones, satisfy a Pohozaev identity.

Contribution

It demonstrates the existence of solutions and the validity of the Pohozaev identity for fractional p-Laplacian equations, extending previous results to this nonlocal operator.

Findings

Existence of nontrivial weak solutions

Solutions satisfy the Pohozaev identity

Pohozaev identity holds for sufficiently smooth solutions

Abstract

In this paper, we show the existence of a nontrivial weak solution for a nonlinear problem involving the fractional $p$-Laplacian operator and a Berestycki-Lions type nonlinearity. This solution satisfies a Pohozaev identity. Moreover, we prove that any sufficiently smooth solution fulfills the Pohozaev identity.