Breaking of resonance and regularizing effect of a first order quasi-linear term in some elliptic equations

Abdellaoui, Boumediene; Peral, Ireneo; Primo, Ana

Abdellaoui, Boumediene ; Peral, Ireneo ; Primo, Ana

Annales de l'I.H.P. Analyse non linéaire, Tome 25 (2008), p. 969-985 / Harvested from Numdam

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Publié le : 2008-01-01

MR 2457819 | Zbl 1156.35036

DOI : https://doi.org/10.1016/j.anihpc.2007.06.003

@article{AIHPC_2008__25_5_969_0,
     author = {Abdellaoui, Boumediene and Peral, Ireneo and Primo, Ana},
     title = {Breaking of resonance and regularizing effect of a first order quasi-linear term in some elliptic equations},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     volume = {25},
     year = {2008},
     pages = {969-985},
     doi = {10.1016/j.anihpc.2007.06.003},
     mrnumber = {2457819},
     zbl = {1156.35036},
     language = {en},
     url = {http://dml.mathdoc.fr/item/AIHPC_2008__25_5_969_0}
}

Abdellaoui, Boumediene; Peral, Ireneo; Primo, Ana. Breaking of resonance and regularizing effect of a first order quasi-linear term in some elliptic equations. Annales de l'I.H.P. Analyse non linéaire, Tome 25 (2008) pp. 969-985. doi : 10.1016/j.anihpc.2007.06.003. http://gdmltest.u-ga.fr/item/AIHPC_2008__25_5_969_0/

Bibliographie

[1] Abdellaoui B., Dall'Aglio A., Peral I., Some remarks on elliptic problems with critical growth on the gradient, J. Differential Equations 222 (1) (2006) 21-62. | MR 2200746 | Zbl pre05013584

[2] Abdellaoui B., Peral I., The equation $- Δ u - λ \frac{u}{{|x|}^{2}} = {|\nabla u|}^{p} + c f (x)$ : The optimal power, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) VI (2007) 1-25. | Numdam | MR 2341519 | Zbl pre05177568

[3] Abdellaoui B., Peral I., Primo A., Some elliptic problems with Hardy potential and critical growth in the gradient: non-resonance and blow-up results, J. Differential Equations 239 (2007) 386-416. | MR 2344278 | Zbl pre05182214

[4] Alaa N., Pierre M., Weak solutions of some quasilinear elliptic equations with data measures, SIAM. J. Math. Anal. 24 (1) (1993) 23-35. | MR 1199524 | Zbl 0809.35021

[5] Boccardo L., Gallouët T., Orsina L., Existence and nonexistence of solutions for some nonlinear elliptic equations, J. Anal. Math. 73 (1997) 203-223. | MR 1616410 | Zbl 0898.35035

[6] Boccardo L., Murat F., Almost everywhere convergence of the gradients of solutions to elliptic and parabolic equations, Nonlinear Anal. TMA 19 (6) (1992) 581-597. | MR 1183665 | Zbl 0783.35020

[7] Boccardo L., Murat F., Puel J.P., Existence de solutions faibles pour des équations elliptiques quasi-linéaires à croissance quadratique, in: Lions L., Brezis H. (Eds.), Nonlinear Partial Differential Equations and their Applications, Collège de France Seminar, vol. IV, Research Notes in Math, vol. 84, Pitman, London, 1983, pp. 19-73. | MR 716511 | Zbl 0588.35041

[8] Boccardo L., Murat F., Puel J.P., Resultats d'existence pour certains problèmes elliptiques quasi-linéaires, Ann. Sc. Norm. Sup. Pisa 11 (2) (1984) 213-235. | Numdam | MR 764943 | Zbl 0557.35051

[9] Boccardo L., Murat F., Puel J.P., Existence des solutions non bornées pour certains équations quasi-linéaires, Portugal. Math. 41 (1982) 507-534. | MR 766873 | Zbl 0524.35041

[10] Boccardo L., Orsina L., Peral I., A remark on existence and optimal summability of solutions of elliptic problems involving Hardy potential, Discrete and Continuous Dynamical Systems 16 (3) (2006) 513-523. | MR 2257147 | Zbl pre05141172

[11] Brezis H., Kamin S., Sublinear elliptic equations in $R^{N}$ , Manuscripta Math. 74 (1992) 87-106. | MR 1141779 | Zbl 0761.35027

[12] Brezis H., Lieb E.H., A relation between pointwise convergence of functions and convergence of functionals, Proc. Amer. Math. Soc. 88 (1983) 486-490. | MR 699419 | Zbl 0526.46037

[13] Brezis H., Ponce A., Kato's inequality when Δu is a measure, C. R. Acad. Sci. Paris, Ser. I 338 (2004) 599-604. | MR 2056467 | Zbl 1101.35028

[14] Brezis H., Strauss W.A., Semilinear second order elliptic equations in $L^{1}$ , J. Math. Soc. Japan 25 (1973) 565-590. | MR 336050 | Zbl 0278.35041

[15] Dal Maso G., Murat F., Orsina L., Prignet A., Renormalized solutions of elliptic equations with general measure data, Ann. Sc. Norm. Super. Pisa Cl. Sci. 28 (4) (1999) 741-808. | Numdam | MR 1760541 | Zbl 0958.35045

[16] Bénilan P., Boccardo L., Gallouët T., Gariepy R., Pierre M., Vazquez J.L., An $L^{1}$ -theory of existence and uniqueness of solutions of nonlinear elliptic equations, Ann. Sc. Norm. Super. Pisa Cl. Sci. 22 (2) (1995) 241-273. | Numdam | MR 1354907 | Zbl 0866.35037

[17] Ferone V., Murat V.F., Quasilinear problems having quadratic growth in the gradient: an existence result when the source term is small, in: Equations aux dérivées partialles et applications, Gauthier-Villars, Ed. Sci. Méd. Elsevier, Paris, 1998, pp. 497-515. | MR 1648236 | Zbl 0917.35039

[18] Garcia Azorero J., Peral I., Hardy inequalities and some critical elliptic and parabolic problems, J. Differential Equations 144 (1998) 441-476. | MR 1616905 | Zbl 0918.35052

[19] Hansson K., Maz'Ya V.G., Verbitsky I.E., Criteria of solvability for multidimensional Riccati equations, Ark. Mat. 37 (1999) 87-120. | MR 1673427 | Zbl 1087.35513

[20] Kato T., Schrödinger operators with singular potentials, Israel J. Math. 13 (1972) 135-148. | MR 333833 | Zbl 0246.35025

[21] Lieb E.H., Loos M., Analysis, Graduate Studies in Mathematics, vol. 14, American Mathematical Society, Providence, RI, 2001. | MR 1817225 | Zbl 0966.26002

[22] Lions P.L., Résolution de problèmes elliptiques quasilinéaires, Arch. Rational Mech. Anal. 74 (4) (1980) 335-353. | MR 588033 | Zbl 0449.35036

[23] Pucci P., Serrin J., The strong maximum principle revisited, J. Differential Equations 196 (1) (2004) 1-66. | MR 2025185 | Zbl 1109.35022

[24] Stampacchia G., Le problème de Dirichlet pour les équations elliptiques du second ordre à coefficients discontinus, Ann. Inst. Fourier 15 (1965) 189-258. | Numdam | MR 192177 | Zbl 0151.15401