Liouville theorems and universal estimates for superlinear elliptic problems without scale invariance

TL;DR

This paper develops new Liouville theorems and universal estimates for superlinear elliptic problems lacking scale invariance, broadening understanding of singularity and decay behaviors in complex nonlinear systems.

Contribution

It introduces novel Liouville theorems for elliptic systems without scale invariance and adapts rescaling and doubling techniques for broader classes of nonlinearities.

Findings

New Liouville theorems in whole space and half-space for non scale invariant problems

Universal singularity and decay estimates for Lane-Emden and Schrödinger systems

Comparison and improvement of classical methods for scalar equations

Abstract

We give applications of known and new Liouville type theorems to universal singularity and decay estimates for non scale invariant elliptic problems, including Lane-Emden and Schr\"odinger type systems. This applies to various classes of nonlinearities with regular variation and possibly different behaviors at $0$ and $\infty$. To this end, we adapt the method from [72] to elliptic systems, which relies on a generalized rescaling technique and on doubling arguments from [55]. This is in particular facilitated by new Liouville type theorems in the whole space and in a half-space, for elliptic problems without scale invariance, that we obtain. Our results apply to some non-cooperative systems, for which maximum principle based techniques such as moving planes do not apply. To prove these Liouville type theorems, we employ two methods, respectively based on Pohozaev-type identities combined with functional inequalities on the unit sphere, and on reduction to a scalar equation by proportionality of components. In turn we will survey the existing methods for proving Liouville-type theorems for superlinear elliptic equations and systems, and list some of the typical existing results for (Sobolev subcritical) systems. In the case of scalar equations, we also revisit the classical Gidas-Spruck integral Bernstein method, providing some improvements which turn out to be efficient for certain nonlinearities, and we next compare the performances of various methods on a benchmark example.