Uniqueness of renormalized solutions to nonlinear elliptic equations with a lower order term and right-hand side in L 1 (Ω)
ESAIM: Control, Optimisation and Calculus of Variations, Volume 8 (2002), pp. 239-272.

In this paper we prove uniqueness results for the renormalized solution, if it exists, of a class of non coercive nonlinear problems whose prototype is

-div(a(x)(1+|u| 2 ) p-2 2 u)+b(x)(1+|u| 2 ) λ 2 =finΩ,u=0onΩ,
where Ω is a bounded open subset of N , N2, 2-1/N<p<N, a belongs to L (Ω), a(x)α 0 >0, f is a function in L 1 (Ω), b is a function in L r (Ω) and 0λ<λ * (N,p,r), for some r and λ * (N,p,r).

DOI: 10.1051/cocv:2002051
Classification: 35J25, 35J60
Keywords: uniqueness, nonlinear elliptic equations, noncoercive problems, data in $L^1$
@article{COCV_2002__8__239_0,
     author = {Betta, M. F. and Mercaldo, A. and Murat, F. and Porzio, M. M.},
     title = {Uniqueness of renormalized solutions to nonlinear elliptic equations with a lower order term and right-hand side in $L^1(\Omega )$},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     pages = {239--272},
     publisher = {EDP-Sciences},
     volume = {8},
     year = {2002},
     doi = {10.1051/cocv:2002051},
     mrnumber = {1932952},
     zbl = {1092.35032},
     language = {en},
     url = {http://archive.numdam.org/articles/10.1051/cocv:2002051/}
}
TY  - JOUR
AU  - Betta, M. F.
AU  - Mercaldo, A.
AU  - Murat, F.
AU  - Porzio, M. M.
TI  - Uniqueness of renormalized solutions to nonlinear elliptic equations with a lower order term and right-hand side in $L^1(\Omega )$
JO  - ESAIM: Control, Optimisation and Calculus of Variations
PY  - 2002
SP  - 239
EP  - 272
VL  - 8
PB  - EDP-Sciences
UR  - http://archive.numdam.org/articles/10.1051/cocv:2002051/
DO  - 10.1051/cocv:2002051
LA  - en
ID  - COCV_2002__8__239_0
ER  - 
%0 Journal Article
%A Betta, M. F.
%A Mercaldo, A.
%A Murat, F.
%A Porzio, M. M.
%T Uniqueness of renormalized solutions to nonlinear elliptic equations with a lower order term and right-hand side in $L^1(\Omega )$
%J ESAIM: Control, Optimisation and Calculus of Variations
%D 2002
%P 239-272
%V 8
%I EDP-Sciences
%U http://archive.numdam.org/articles/10.1051/cocv:2002051/
%R 10.1051/cocv:2002051
%G en
%F COCV_2002__8__239_0
Betta, M. F.; Mercaldo, A.; Murat, F.; Porzio, M. M. Uniqueness of renormalized solutions to nonlinear elliptic equations with a lower order term and right-hand side in $L^1(\Omega )$. ESAIM: Control, Optimisation and Calculus of Variations, Volume 8 (2002), pp. 239-272. doi : 10.1051/cocv:2002051. http://archive.numdam.org/articles/10.1051/cocv:2002051/

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

[2] M.F. Betta, A. Mercaldo, F. Murat and M.M. Porzio, Existence and uniqueness results for nonlinear elliptic problems with a lower order term and measure datum. C. R. Acad. Sci. Paris Sér. I Math. 332 (to appear). | MR | Zbl

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

[4] M.F. Betta, A. Mercaldo, F. Murat and M.M. Porzio, Uniqueness results for nonlinear elliptic equations with a lower order term (to appear). | MR | Zbl

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

[6] G. Bottaro and M.E. Marina, Problema di Dirichlet per equazioni ellittiche di tipo variazionale su insiemi non limitati. Boll. Un. Mat. Ital. 8 (1973) 46-56. | MR | Zbl

[7] A. Dall'Aglio, Approximated solutions of equations with L 1 data. Application to the H-convergence of quasi-linear parabolic equations. Ann. Mat. Pura Appl. 170 (1996) 207-240. | Zbl

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

[9] G. Dolzmann, N. Hungerbühler and S. Müller, Uniqueness and maximal regularity for nonlinear elliptic systems of n-Laplace type with measure valued right-hand side. J. Reine Angew. Math. 520 (2000) 1-35. | MR | Zbl

[10] A. Fiorenza and C. Sbordone, Existence and uniqueness results for solutions of nonlinear equations with right-hand side in L 1 (Ω). Studia Math. 127 (1998) 223-231. | MR | Zbl

[11] L. Greco, T. Iwaniec and C. Sbordone, Inverting the p-harmonic operator. Manuscripta Math. 92 (1997) 249-258. | MR | Zbl

[12] O. Guibé, Remarks on the uniqueness of comparable renormalized solutions of elliptic equations with measure data. Ann. Mat. Pura Appl. Ser. IV 180 (2002) 441-449. | MR | Zbl

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

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

[15] F. Murat, Équations elliptiques non linéaires avec second membre L 1 ou mesure, in Actes du 26 e Congrès National d'Analyse Numérique. Les Karellis, France (1994) A12-A24.

[16] A. Prignet, Remarks on existence and uniqueness of solutions of elliptic problems with right-hand side measures. Rend. Mat. Appl. 15 (1995) 321-337. | MR | Zbl

[17] J. Serrin, Pathological solutions of elliptic differential equations. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 18 (1964) 385-387. | Numdam | MR | Zbl

[18] G. Stampacchia, Le problème de Dirichlet pour les équations elliptiques du second ordre à coefficients discontinus. Ann. Inst. Fourier (Grenoble) 15 (1965) 189-258. | Numdam | MR | Zbl

Cited by Sources: