Pohozaev-type identities for classes of quasilinear elliptic local and nonlocal equations and systems, with applications

TL;DR

This paper derives Pohozaev-type identities for a broad class of quasilinear elliptic equations and systems involving local and nonlocal $p$-Laplace operators, with novel results in mixed anisotropic and fractional cases.

Contribution

It introduces new Pohozaev identities for mixed local and nonlocal $p$-Laplace equations and systems, including cases previously unexplored, such as when $p=2$.

Findings

Derived identities for anisotropic $p$-Laplace equations

Extended identities to fractional and mixed operators

Presented applications demonstrating the identities' utility

Abstract

In this article, we establish Pohozaev-type identities for a class of quasilinear elliptic equations and systems involving both local and nonlocal $p$-Laplace operators. Specifically, we obtain these identities in $\mathbb{R}^n$ for the purely anisotropic $p$-Laplace equations, the purely fractional $p$-Laplace equations, as well as for equations that incorporate both anisotropic and fractional $p$-Laplace features. We also extend these results to the corresponding systems. To the best of our knowledge, the identities we derive in the mixed case are new even when $p=2$. Finally, we illustrate some of the applications of our main results.