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 S n − 1 , and feedback and measure arguments. Such Liouville-type nonexistence results have many applications in the study of nonvariational elliptic systems.