Comptes Rendus
Théorèmes d'existence pour des équations avec l'opérateur « 1-Laplacien », première valeur propre pour −Δ1
Comptes Rendus. Mathématique, Volume 334 (2002) no. 12, pp. 1071-1076.

On considère des équations de la forme

{-divσ=f(x,u),u0,u¬0,uBV(Ω),σ·u=|u|inΩ,|σ|L(Ω)1,σ·n(-u)=uonΩ,
Ω est un domaine borné de N, uBV(Ω) et f(x,u)LN(Ω). On s'interesse au cas où f est constante et en particulier on définit la première valeur propre λ1 pour −div(σ(u)), on étudie les premières fonctions propres. Ensuite on considère pour λ>λ1 des conditions nécessaires et suffisantes d'existence de solutions non triviales et non négatives pour l'équation
-divσ(u)=λ+fuq-1
(avec des conditions aux limites) et fL(Ω), 1<q⩽1=N/(N−1).

We consider partial differential equations of the form

{-divσ=f(x,u),u0,u¬0,uBV(Ω),σ·u=|u|inΩ,|σ|L(Ω)1,σ·n(-u)=uonΩ,
where Ω is some smooth bounded domain in N, uBV(Ω) and f(x,u)LN(Ω). We consider the case where f=cte, define the first eigenvalue λ1 for −div(σ(u)), and study eigenfunctions. We consider then for λλ1 some necessary and sufficient conditions on f and the first eigenfunctions for existence of nontrivial solutions to
-divσ(u)=λ+fuq-1
(with boundary conditions), fL(Ω), and 1<q⩽1=N/(N−1).

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/S1631-073X(02)02405-6

Françoise Demengel 1

1 University of Cergy-Pontoise, site de Saint-Martin, 2, avenue Adolphe Chauvin, 95 000 Cergy-Pontoise, France
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     title = {Th\'eor\`emes d'existence pour des \'equations avec l'op\'erateur {\guillemotleft} {1-Laplacien} {\guillemotright}, premi\`ere valeur propre pour {\ensuremath{-}\protect\emph{\ensuremath{\Delta}}\protect\textsubscript{1}}},
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Françoise Demengel. Théorèmes d'existence pour des équations avec l'opérateur « 1-Laplacien », première valeur propre pour −Δ1. Comptes Rendus. Mathématique, Volume 334 (2002) no. 12, pp. 1071-1076. doi : 10.1016/S1631-073X(02)02405-6. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/S1631-073X(02)02405-6/

[1] S. Alama; G. Tarantello On semilinear elliptic equations with indefinite nonlinearities, Calc. Var. and Partial Differential Equations, Volume 1 (1993), pp. 439-475

[2] H. Berestycki; I. Capuzzo Dolcetta; L. Nirenberg Problémes elliptiques indéfinis et théorème de Liouville non-linéaires, C. R. Acad. Sci. Paris, Série I, Volume 317 (1993), pp. 945-950

[3] I. Birindelli, F. Demengel, On some partial differential equation for noncoercive functional and critical Sobolev exponent, Adv. in Differential Equations, accepted

[4] I. Birindelli, F. Demengel, On some partial differential equation for non coercive functional and critical Sobolev exponent, Preprint, Universita di Roma La Sapienza

[5] F. Demengel On some nonlinear partial differential equations involving the 1-Laplacian and critical Sobolev exponent, Control Optim. Calc. Var., Mars 2000

[6] F. Demengel, Some existence's results for noncoercive 1-Laplacian operator, Prébublication de l'Université de Cergy-Pontoise, No. 21, 2001, soumis à Nonlinear Anal

[7] F. Demengel, Functions almost 1-harmonic, Prépublication de l'Université de Cergy Pontoise. No. 31, 2001

[8] P.-L. Lions La méthode de compacité concentration, I et II, Rev. Mat. Iberoamericana, Volume 1 (1985) no. 1, p. 145

[9] T. Ouyang On the positive solutions of semilinear equations of Δu+λu+hup=0 on compacts manifolds, II, Indiana Math. J., Volume 40 (1991), pp. 1083-1140

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