Liouville theorems and universal estimates for superlinear elliptic problems without scale invariance
Pavol Quittner, Philippe Souplet
TL;DR
This paper develops Liouville-type nonexistence results and universal bounds for superlinear elliptic problems that lack scale invariance, including Lane-Emden and Schrödinger-type systems. It introduces two complementary Liouville approaches: (i) a Pohozaev-identity framework with sphere-inequality tools, and (ii) a reduction to scalar equations via proportionality of components, extending known scalar results to multi-component systems and half-spaces. By connecting Liouville properties at 0, ∞, and in the whole space, the authors derive universal singularity and decay estimates for nonscale-invariant nonlinearities with regular variation, and they discuss implications for noncooperative systems where maximum-principle methods fail. The work also revisits the Gidas–Spruck integral Bernstein method in the scalar setting, provides a benchmark comparison of methods, and lays groundwork for parabolic extensions in a companion paper QSparab23.
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 Schrödinger 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.