Entire Extremals

Abstract

Extremal functions for the generalized Sobolev constant F* (Definition 3.1) are called entire extremals. Our main result concerning entire extremals is the second order decay estimate

$$ w(r) = cr^{p - n} (1 + o(r)) $$

in the C ¹-sense together with an integral representation for the factor c (Section 6.3). As a byproduct we solve a problem posed by Lions concerning radial symmetry of entire extremals. For smooth integrands an entire extremal is a ground state solution of

$$ \begin{gathered} - div\left( {\left| {\nabla w} \right|^{p - 2} \nabla w} \right) = \lambda f(w), \hfill \\ w(x) \to 0 as \left| x \right| \to \infty . \hfill \\ \end{gathered} $$

((6.1))