$ {W}^{1,N}$ versus $ {C}^{1}$ local minimizers for elliptic functionals with critical growth in $ {\mathbb{R}}^{N}$

Jacques Giacomoni; S. Prashanth; K. Sreenadh

doi:10.1016/j.crma.2009.01.010

Partial Differential Equations

W^{1, N}

versus

C^{1}

local minimizers for elliptic functionals with critical growth in

R^{N}

[Minima locaux relatifs à

C^{1}

et à

W^{1, N}

]

Présenté par : Haïm Brezis

Jacques Giacomoni ¹ ; S. Prashanth ² ; K. Sreenadh ³

¹ Université de Pau et des Pays de l'Adour, B.P. 576, 64012 Pau cedex, France
² T.I.F.R. CAM, P.B. No. 6503, Sharadanagar, Chikkabommasandra, Bangalore 560065, India
³ Department of Mathematics, Indian Institute of Technology Delhi, Hauz Khaz, New Delhi-16, India

Comptes Rendus. Mathématique, Volume 347 (2009) no. 5-6, pp. 255-260.

Résumé
Abstract

Soit Ω un ouvert borné régulier de $R^{N}$ , $f : Ω \times R \to R$ une fonction de Caratheodory vérifiant $s f (x, s) ⩾ 0 \forall (x, s) \in Ω \times R$ et $\sup_{x \in Ω} | f (x, s) | ⩽ C (1 + {| s |}^{p}) e^{{| s |}^{\frac{N}{(N - 1)}}}$ , $\forall s \in R$ et pour une constante $C > 0$ . Considérons la fonctionnelle $J : W^{1, N} (Ω) \to R$ , définie par

J (u) \overset{def}{=} \frac{1}{N} ‖ u ‖_{W^{1, N} (Ω)}^{N} - \int_{Ω} F (x, u) - \frac{1}{q + 1} ‖ u ‖_{L^{q + 1} (\partial Ω)}^{q + 1}

avec

F (x, u) = \int_{0}^{u} s f (x, s) d s

et

q > 0

. Nous démontrons que si

u_{0} \in C^{1} (\bar{Ω})

est un minimiseur local de J dans

C^{1} (\bar{Ω})

, alors il est aussi un minimiseur local de J dans

W^{1, N} (Ω)

.

Let $Ω \subset R^{N}$ be a bounded smooth domain, $f : Ω \times R \to R$ be a Caratheodory function with $s f (x, s) ⩾ 0 \forall (x, s) \in Ω \times R$ and $\sup_{x \in Ω} | f (x, s) | ⩽ C {(1 + | s |)}^{p} e^{{| s |}^{\frac{N}{(N - 1)}}}, \forall s \in R$ , for some $C > 0$ . Consider the functional J : $W^{1, N} (Ω) \to R$ , Ω defined as

J (u) \overset{def}{=} \frac{1}{N} ‖ u ‖_{W^{1, N} (Ω)}^{N} - \int_{Ω} F (x, u) - \frac{1}{q + 1} ‖ u ‖_{L^{q + 1} (\partial Ω)}^{q + 1},

where

F (x, u) = \int_{0}^{u} f (x, s) d s

and

q > 0

. We show that if

u_{0} \in C^{1} (\bar{Ω})

is a local minimum of J in the

C^{1} (\bar{Ω})

topology, then it is also a local minimum of J in

W^{1, N} (Ω)

topology.

Reçu le : 2008-09-02
Accepté le : 2009-01-12
Publié le : 2009-02-25

DOI : 10.1016/j.crma.2009.01.010

Affiliations des auteurs :

Jacques Giacomoni ¹ ; S. Prashanth ² ; K. Sreenadh ³

¹ Université de Pau et des Pays de l'Adour, B.P. 576, 64012 Pau cedex, France
² T.I.F.R. CAM, P.B. No. 6503, Sharadanagar, Chikkabommasandra, Bangalore 560065, India
³ Department of Mathematics, Indian Institute of Technology Delhi, Hauz Khaz, New Delhi-16, India

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     author = {Jacques Giacomoni and S. Prashanth and K. Sreenadh},
     title = {$ {W}^{1,N}$ versus $ {C}^{1}$ local minimizers for elliptic functionals with critical growth in $ {\mathbb{R}}^{N}$},
     journal = {Comptes Rendus. Math\'ematique},
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Jacques Giacomoni; S. Prashanth; K. Sreenadh. $ {W}^{1,N}$ versus $ {C}^{1}$ local minimizers for elliptic functionals with critical growth in $ {\mathbb{R}}^{N}$. Comptes Rendus. Mathématique, Volume 347 (2009) no. 5-6, pp. 255-260. doi : 10.1016/j.crma.2009.01.010. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2009.01.010/

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