Sharp criteria of Liouville type for some nonlinear systems

TL;DR

This paper establishes precise conditions under which positive solutions do not exist for certain nonlinear PDE systems, using innovative techniques like iteration schemes, shooting methods, and Pohozaev identities.

Contribution

It introduces new analytical methods to derive sharp Liouville type nonexistence criteria for various nonlinear systems, including Hardy-Littlewood-Sobolev and Wolff type equations.

Findings

Sharp nonexistence criteria for HLS type systems

Extension of results to Wolff type and γ-Laplace systems

Dichotomy between existence and nonexistence for finite energy solutions

Abstract

In this paper, we establish the sharp criteria for the nonexistence of positive solutions to the Hardy-Littlewood-Sobolev (HLS) type system of nonlinear equations and the corresponding nonlinear differential systems of Lane-Emden type equations. These nonexistence results, known as Liouville type theorems, are fundamental in PDE theory and applications. A special iteration scheme, a new shooting method and some Pohozaev type identities in integral form as well as in differential form are created. Combining these new techniques with some observations and some critical asymptotic analysis, we establish the sharp criteria of Liouville type for our systems of nonlinear equations. Similar results are also derived for the system of Wolff type integral equations and the system of $\gamma$-Laplace equations. A dichotomy description in terms of existence and nonexistence for solutions with finite energy is also obtained.