We show that the functional

$$
\mathrm{I}(u) = \int _{\mathrm{B}}a(|x|)\:u\:(x)\:dx + \int _{\mathrm{B}}h(|x|,\mathrm{\Delta }u\:(x)−\mathrm{\lambda }u\:(x))\:dx
$$

attains a minimum on the space $\mathrm{W}_{0}^{2,p}\left(\mathrm{B}\right)$.


### Résumé

Nous montrons que le fonctionnel

$$
\mathrm{I}(u) = \int _{\mathrm{B}}a(|x|)\:u\:(x)\:dx + \int _{\mathrm{B}}h\:(|x|,\mathrm{\Delta }u\:(x)−\mathrm{\lambda }u\:(x))\:dx
$$

atteint le minimum sur l’espace $\mathrm{W}_{0}^{2,p}\left(\mathrm{B}\right)$.


Radially symmetric solutions of a class of problems of the calculus of variations without convexity assumptions

Arrigo Cellina

Fabián Flores

Abstract

Résumé

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