Partial Differential Equations
$W1,N$ versus $C1$ local minimizers for elliptic functionals with critical growth in $RN$
Comptes Rendus. Mathématique, Volume 347 (2009) no. 5-6, pp. 255-260.

Let $Ω⊂RN$ be a bounded smooth domain, $f:Ω×R→R$ be a Caratheodory function with $sf(x,s)⩾0∀(x,s)∈Ω×R$ and $supx∈Ω|f(x,s)|⩽C(1+|s|)pe|s|N(N−1),∀s∈R$, for some $C>0$. Consider the functional J : $W1,N(Ω)→R$, Ω defined as

 $J(u)=def1N‖u‖W1,N(Ω)N−∫ΩF(x,u)−1q+1‖u‖Lq+1(∂Ω)q+1,$
where $F(x,u)=∫0uf(x,s)ds$ and $q>0$. We show that if $u0∈C1(Ω¯)$ is a local minimum of J in the $C1(Ω¯)$ topology, then it is also a local minimum of J in $W1,N(Ω)$ topology.

Soit Ω un ouvert borné régulier de $RN$, $f:Ω×R→R$ une fonction de Caratheodory vérifiant $sf(x,s)⩾0∀(x,s)∈Ω×R$ et $supx∈Ω|f(x,s)|⩽C(1+|s|p)e|s|N(N−1)$, $∀s∈R$ et pour une constante $C>0$. Considérons la fonctionnelle $J:W1,N(Ω)→R$, définie par

 $J(u)=def1N‖u‖W1,N(Ω)N−∫ΩF(x,u)−1q+1‖u‖Lq+1(∂Ω)q+1$
avec $F(x,u)=∫0usf(x,s)ds$ et $q>0$. Nous démontrons que si $u0∈C1(Ω¯)$ est un minimiseur local de J dans $C1(Ω¯)$, alors il est aussi un minimiseur local de J dans $W1,N(Ω)$.

Accepted:
Published online:
DOI: 10.1016/j.crma.2009.01.010

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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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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