$ {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}

Presented by: 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

Abstract (Original language)
Résumé

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.

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)

.

Received: 2008-09-02
Accepted: 2009-01-12
Published online: 2009-02-25

DOI: 10.1016/j.crma.2009.01.010

Author's affiliations:

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

@article{CRMATH_2009__347_5-6_255_0,
     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},
     pages = {255--260},
     year = {2009},
     publisher = {Elsevier},
     volume = {347},
     number = {5-6},
     doi = {10.1016/j.crma.2009.01.010},
     language = {en},
}

TY  - JOUR
AU  - Jacques Giacomoni
AU  - S. Prashanth
AU  - K. Sreenadh
TI  - $ {W}^{1,N}$ versus $ {C}^{1}$ local minimizers for elliptic functionals with critical growth in $ {\mathbb{R}}^{N}$
JO  - Comptes Rendus. Mathématique
PY  - 2009
SP  - 255
EP  - 260
VL  - 347
IS  - 5-6
PB  - Elsevier
DO  - 10.1016/j.crma.2009.01.010
LA  - en
ID  - CRMATH_2009__347_5-6_255_0
ER  -

%0 Journal Article
%A Jacques Giacomoni
%A S. Prashanth
%A K. Sreenadh
%T $ {W}^{1,N}$ versus $ {C}^{1}$ local minimizers for elliptic functionals with critical growth in $ {\mathbb{R}}^{N}$
%J Comptes Rendus. Mathématique
%D 2009
%P 255-260
%V 347
%N 5-6
%I Elsevier
%R 10.1016/j.crma.2009.01.010
%G en
%F CRMATH_2009__347_5-6_255_0

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

References
Cited by

[1] A. Ambrosetti; H. Brezis; G. Cerami Combined effects of concave and convex nonlinearities in some elliptic problems, J. Funct. Anal., Volume 122 (1994), pp. 519-543

[2] H. Brezis; L. Nirenberg Minima locaux relatifs à $C^{1}$ et $H^{1}$ , C. R. Acad. Sci. Paris, Ser. I, Volume 317 (1993), pp. 465-472

[3] G. Azorero; J. Manfredi; I. Peral Sobolev versus Hölder local minimizers and global multiplicity for some quasilinear elliptic equations, Commun. Contemp. Math., Volume 2 (2000) no. 3, pp. 385-404

[4] F. Brock, L. Iturraga, P. Ubilla, A multiplicity result for the p-Laplacian involving a parameter, Ann. Inst. H. Poincaré Anal. Non Linéaire, in press

[5] P. Cherrier Meilleures constantes dans des inégalités relatives aux espaces de Sobolev, Bull. Sci. Math., Volume 108 (1984), pp. 225-262

[6] P. Cherrier Problèmes de Neumann nonlinéaires sur les variétés Riemanniennes, C. R. Acad. Sci. Paris, Sér. A, Volume 292 (1981), pp. 637-640

[7] D.G. De Figueiredo, J.P. Gossez, P. Ubilla, Local “superlinearity” and “sublinearity” for the p-Laplacian, in press

[8] J. Giacomoni; S. Prashanth; K. Sreenadh A global multiplicity result for N-Laplacian with critical nonlinearity of concave-convex type, J. Differential Equations, Volume 232 (2007), pp. 544-572

[9] G. Lieberman Boundary regularity for solutions of degenerate elliptic equations, Nonlinear Anal., Volume 12 (1988) no. 11, pp. 1203-1219

[10] G. Stampacchia Equations Elliptiques du Second Ordre à Coefficients Discontinues, Les Presses de l'Université de Montréal, 1966

Cited by Sources:

Comments - Policy