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

By Philippe Souplet. 2009

282 citations recorded by the source corpus.

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.

Subfields: Applied Mathematics

View Paper: https://www.semanticscholar.org/paper/9fe9ede0fc5269a2e5436e3268508b54b272dd2d

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In Applied Mathematics

Citations in the corpus

Liouville theorems for stable solutions of biharmonic problem (2013, methodology): “Our proof borrows crucially an idea from Cowan-Esposito-Ghoussoub [2], who proved the regularity of extremal solutions for fourth order problems in bounded domains.”
Liouville type theorems for integral equations and integral systems (2013, result): “As for the non-radial case, this conjecture has not been solved completely, we refers the readers to [12,27,29,30,32] etc, and we don’t give some comments on these results here.”
Liouville-type theorems and bounds of solutions of Hardy-Hénon equations (2012, methodology): “By using additional interpolation and feedback arguments from [26], one could extend the result to higher dimensions N 4, but at the expense of the further restriction p < (N − 1)/(N − 3) pS .”
Hamiltonian elliptic systems: a guide to variational frameworks (2014, background): “ several ways of proving that (9.1) admits innitely many solutions (which have increasing energy). On the other hand, a priori bounds for positive solutions are known to hold under (H1) for N4 (see [86, 105]). For N5, there are some partial results which do not cover entirely the case (p;q) satisfyi”
Lane–Emden systems with negative exponents (2010, unknown): “, [1, 3, 6, 7, 16, 18, 19, 20, 21, 22, 23] and the references therein).”
A Monotonicity Formula and a Liouville-type Theorem for a Fourth Order Supercritical Problem (2013, methodology): “One reason is their use of the following inequality, due to Souplet [29] 1 A similar method was first announced in [7], and later published in the work by Farina, Sirakov and one of the authors [8].”
Local Boundedness and Harnack Inequality for Solutions of Linear Nonuniformly Elliptic Equations (2019, methodology): “A similar idea, namely the use of the Sobolev inequality on spheres and a smart choice of certain cutoff functions, was previously used in [10] to obtain a new div-curl lemma and in [31] to show nonexistence of solutions for the Lane-Emden equation.”