Existence and uniqueness results for nonlinear elliptic problems with a lower order term and measure datum

Betta, M.Francesca; Mercaldo, Anna; Murat, François; Porzio, M.Michaela

Comptes Rendus. Mathématique, Tome 334 (2002) no. 9, pp. 757-762.

Dans cette Note nous considérons une classe de problèmes non linéaires et non coercifs dont le prototype est

$$-{△}_p\mathrm{u}+\mathrm{b}\left(\mathrm{x}\right){\left|\nabla \mathrm{u}\right|}^{\lambda }=\mu \mathrm{dans}\Omega ,\mathrm{u}=0\mathrm{sur}\partial \Omega ,$$

où $\Omega$ est un ouvert borné de ${ℝ}^N$ (N⩾2), △_p est le p-Laplacien (1<p<N) ou une variante de cet opérateur, μ est une mesure de Radon bornée ou une function de ${\mathrm{L}}^1\left(\Omega \right)$, λ⩾0 et b appartient à l'espace de Lorentz ${\mathrm{L}}^{\mathrm{N},1}\left(\Omega \right)$ ou à l'espace de Lebesgue ${\mathrm{L}}^{\infty }\left(\Omega \right)$. Nous démontrons l'existence et l'unicité de solutions renormalisées de ce problème.

In this Note we consider a class of noncoercive nonlinear problems whose prototype is

$$-{△}_p\mathrm{u}+\mathrm{b}\left(\mathrm{x}\right){\left|\nabla \mathrm{u}\right|}^{\lambda }=\mu \mathrm{in}\Omega ,\mathrm{u}=0\mathrm{on}\partial \Omega ,$$

where $\Omega$ is a bounded open subset of ${ℝ}^N$ (N⩾2), △_p is the so called p-Laplace operator (1<p<N) or a variant of it, μ is a Radon measure with bounded variation on $\Omega$ or a function in ${\mathrm{L}}^1\left(\Omega \right)$, λ⩾0 and b belongs to the Lorentz space ${\mathrm{L}}^{\mathrm{N},1}\left(\Omega \right)$ or to the Lebesgue space ${\mathrm{L}}^{\infty }\left(\Omega \right)$. We prove existence and uniqueness of renormalized solutions.

[1] Bénilan, P.; Boccardo, L.; Gallouët, T.; Gariepy, R.; Pierre, M.; Vazquez, J.L. An L¹-theory of existence and uniqueness of solutions of nonlinear elliptic equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci., Volume 22 (1995), pp. 241-273

[2] M.F. Betta, A. Mercaldo, F. Murat, M.M. Porzio, Existence of renormalized solutions to nonlinear elliptic equations with a lower order term and right-hand side a measure, J. Math. Pures Appl., to appear

[3] M.F. Betta, A. Mercaldo, F. Murat, M.M. Porzio, Uniqueness of renormalized solutions to nonlinear elliptic equations with a lower order term and right-hand side ${\mathrm{L}}^1\left(\Omega \right)$, ESAIM Control Optim. Calc. Var., Special issue dedicated to the memory of Jacques-Louis Lions (2002), to appear

[4] M.F. Betta, A. Mercaldo, F. Murat, M.M. Porzio, Uniqueness results for nonlinear elliptic equations with a lower order term, in preparation

[5] Boccardo, L.; Gallouët, T. Nonlinear elliptic and parabolic equations involving measure data, J. Funct. Anal., Volume 87 (1989), pp. 149-169

[6] Boccardo, L.; Gallouët, T.; Orsina, L. Existence and uniqueness of entropy solutions for nonlinear elliptic equations with measure data, Ann. Inst. H. Poincaré Anal. Non Linéaire, Volume 13 (1996), pp. 539-551

[7] Bottaro, G.; Marina, M.E. Problema di Dirichlet per equazioni ellittiche di tipo variazionale su insiemi non limitati, Boll. Un. Mat. Ital., Volume 8 (1973), pp. 46-56

[8] Dal Maso, G.; Murat, F.; Orsina, L.; Prignet, A. Renormalized solutions for elliptic equations with general measure data, Ann. Scuola Norm. Sup. Pisa Cl. Sci., Volume 28 (1999), pp. 741-808

[9] Del Vecchio, T. Nonlinear elliptic equations with measure data, Pot. Analysis, Volume 4 (1995), pp. 185-203

[10] P.-L. Lions, F. Murat, Solutions renormalisées d'équations elliptiques non linéaires, to appear

[11] F. Murat, Soluciones renormalizadas de EDP elipticas no lineales, Preprint 93023, Laboratoire d'Analyse Numérique de l'Université Paris VI, 1993

[12] Murat, F. Équations elliptiques non linéaires avec second membre L¹ ou mesure, Actes du 26ème Congrès National d'Analyse Numérique, Les Karellis, France, 1994, p. A12-A24

[13] Serrin, J. Pathological solutions of elliptic differential equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci., Volume 18 (1964), pp. 385-387

[14] Stampacchia, G. Le problème de Dirichlet pour les équations elliptiques du second ordre à coefficients discontinus, Ann. Inst. Fourier (Grenoble), Volume 15 (1965), pp. 189-258