A non-linear problem involving a critical Sobolev exponent.

Soohyun Bae (1) , Rejeb Hadiji (2) , Francois Vigneron (2) , Habib Yazidi (3)

1 Hanbat National University
2 LAMA - Laboratoire d'Analyse et de Mathématiques Appliquées
3 ESSTT - Ecole Supérieure des Sciences et Techniques [Tunis]

We study the quasi-linear minimization problem on $H^1_0(\Omega)\subset L^q$ with $q=\frac{2n}{n-2}$~: $$\inf_{\|u\_{|L^q}=1}\int_\Omega (1+|x|^\beta |u|^k)|\nabla u|^2.$$. We show that minimizers exist only in the range β < kn/q which corresponds to a dominant nonlinear term. On the contrary, the linear influence for β>= kn/q prevents their existence.

Domaines: Equations aux dérivées partielles [math.AP]

Soohyun Bae, Rejeb Hadiji, Francois Vigneron, Habib Yazidi. A non-linear problem involving a critical Sobolev exponent.. Journal of Mathematical Analysis and Applications, 2012, 396 (1), pp.98-107. ⟨hal-00794199⟩