Duke Mathematical Journal

Singularity and decay estimates in superlinear problems via Liouville-type theorems, I: Elliptic equations and systems

Peter Poláčik, Pavol Quittner, and Philippe Souplet

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Abstract

In this article, we study some new connections between Liouville-type theorems and local properties of nonnegative solutions to superlinear elliptic problems. Namely, we develop a general method for derivation of universal, pointwise, a priori estimates of local solutions from Liouville-type theorems, which provides a simpler and unified treatment for such questions. The method is based on rescaling arguments combined with a key “doubling” property, and it is different from the classical rescaling method of Gidas and Spruck [16]. As an important heuristic consequence of our approach, it turns out that universal boundedness theorems for local solutions and Liouville-type theorems are essentially equivalent.

The method enables us to obtain new results on universal estimates of spatial singularities for elliptic equations and systems, under optimal growth assumptions that, unlike previous results, involve only the behavior of the nonlinearity at infinity. For semilinear systems of Lane-Emden type, these seem to be the first results on singularity estimates to cover the full subcritical range. In addition, we give an affirmative answer to the so-called Lane-Emden conjecture in three dimensions

Article information

Source
Duke Math. J., Volume 139, Number 3 (2007), 555-579.

Dates
First available in Project Euclid: 24 August 2007

Permanent link to this document
https://projecteuclid.org/euclid.dmj/1187916270

Digital Object Identifier
doi:10.1215/S0012-7094-07-13935-8

Mathematical Reviews number (MathSciNet)
MR2350853

Zentralblatt MATH identifier
1146.35038

Subjects
Primary: 35J60: Nonlinear elliptic equations 35B45: A priori estimates 35J45 35J70: Degenerate elliptic equations
Secondary: 35B40: Asymptotic behavior of solutions 35J65: Nonlinear boundary value problems for linear elliptic equations 35B33: Critical exponents

Citation

Poláčik, Peter; Quittner, Pavol; Souplet, Philippe. Singularity and decay estimates in superlinear problems via Liouville-type theorems, I: Elliptic equations and systems. Duke Math. J. 139 (2007), no. 3, 555--579. doi:10.1215/S0012-7094-07-13935-8. https://projecteuclid.org/euclid.dmj/1187916270


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