For each α > 2 there is a binary word with critical exponent α.

Abstract We consider Hardy inequalities in IRn, n ≥ 3, with best constant that involve either distance to the boundary or distance to a surface of co-dimension k < n, and we show that they can still be improved by adding a multiple of a whole range of critical norms that at the extreme case become precisely the critical Sobolev norm. Résumé Nous étudions des inegalités de Hardy dans IRn, n ≥ 3,...

It is conjectured that M-theory in asymptotically flat spacetime must be supersymmetric, and that the observed SUSY breaking in the low energy world must be attributed to the existence of a nonzero cosmological constant. This would be consistent with experiment, if the critical exponent α in the relation MSUSY ∼ MP (Λ/M 4 P ) α took on the value 1/8 , rather than its classical value 1/4. We att...

For each α > 2 there is a binary word with critical exponent α.

This article concerns the existence of positive solutions for thesecond order equation involving a nonlocal term $$ -\Delta u=\gamma (-\Delta)^{-1} u+|u|^{p-1}u, under Dirichlet boundary conditions. We prove depending on real parameter \(\gamma&gt;0\), and up to critical value exponent \(p\), i.e. when \(1&lt;p\leq 2^*-1\), where \(2^*=\frac{2N}{N-2}\) is Sobolev exponent. For \(p=2^*-1\), this...

where 2* = 2 N / ( N 2) is the critical Sobolev exponent for the embedding H’(52) C L2* (0). Criticality and subcriticality play important rBles concerning the solvability of equation (1.1). Using (by now classical) variational methods one sees that equation (1.1) is solvable i f f has subcritical growth (and satisfies some additional hypotheses), while for f(u) = IuIP-~u the well-known Pohoiae...