The proof of the Lane-Emden conjecture in four space dimensions

Abstract

We partially solve a well-known conjecture about the nonexistence of positive entire solutions to elliptic systems of Lane-Emden type when the pair of exponents lies below the critical Sobolev hyperbola. Up to now, the conjecture had been proved for radial solutions, or in n ≤ 3 space dimensions, or in certain subregions below the critical hyperbola for n ≥ 4. We here establish the conjecture in four space dimensions and we obtain a new region of nonexistence for n ≥ 5. Our proof is based on a delicate combination involving Rellich-Pohozaev type identities, a comparison property between components via the maximum principle, Sobolev and interpolation inequalities on Sn - 1, and feedback and measure arguments. Such Liouville-type nonexistence results have many applications in the study of nonvariational elliptic systems. © 2009 Elsevier Inc. All rights reserved.