Dirichlet problems with singular and gradient quadratic lower order terms
ESAIM: Control, Optimisation and Calculus of Variations, Tome 14 (2008) no. 3, pp. 411-426.

We present a revisited form of a result proved in [Boccardo, Murat and Puel, Portugaliae Math. 41 (1982) 507-534] and then we adapt the new proof in order to show the existence for solutions of quasilinear elliptic problems also if the lower order term has quadratic dependence on the gradient and singular dependence on the solution.

DOI : 10.1051/cocv:2008031
Classification : 35J20, 35J25, 35J65
Mots clés : quadratic gradient, singular lower order term
@article{COCV_2008__14_3_411_0,
     author = {Boccardo, Lucio},
     title = {Dirichlet problems with singular and gradient quadratic lower order terms},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     pages = {411--426},
     publisher = {EDP-Sciences},
     volume = {14},
     number = {3},
     year = {2008},
     doi = {10.1051/cocv:2008031},
     mrnumber = {2434059},
     zbl = {1147.35034},
     language = {en},
     url = {http://archive.numdam.org/articles/10.1051/cocv:2008031/}
}
TY  - JOUR
AU  - Boccardo, Lucio
TI  - Dirichlet problems with singular and gradient quadratic lower order terms
JO  - ESAIM: Control, Optimisation and Calculus of Variations
PY  - 2008
SP  - 411
EP  - 426
VL  - 14
IS  - 3
PB  - EDP-Sciences
UR  - http://archive.numdam.org/articles/10.1051/cocv:2008031/
DO  - 10.1051/cocv:2008031
LA  - en
ID  - COCV_2008__14_3_411_0
ER  - 
%0 Journal Article
%A Boccardo, Lucio
%T Dirichlet problems with singular and gradient quadratic lower order terms
%J ESAIM: Control, Optimisation and Calculus of Variations
%D 2008
%P 411-426
%V 14
%N 3
%I EDP-Sciences
%U http://archive.numdam.org/articles/10.1051/cocv:2008031/
%R 10.1051/cocv:2008031
%G en
%F COCV_2008__14_3_411_0
Boccardo, Lucio. Dirichlet problems with singular and gradient quadratic lower order terms. ESAIM: Control, Optimisation and Calculus of Variations, Tome 14 (2008) no. 3, pp. 411-426. doi : 10.1051/cocv:2008031. http://archive.numdam.org/articles/10.1051/cocv:2008031/

[1] D. Arcoya, S. Barile and P.J. Martinez-Aparicio, Singular quasilinear equations with quadratic growth in the gradient without sign condition. Preprint. | MR

[2] D. Arcoya and P.J. Martinez-Aparicio, Quasilinear equations with natural growth Rev. Mat. Iberoamericana (to appear). | MR | Zbl

[3] D. Arcoya, J. Carmona, T. Leonori, P.J. Martínez, L. Orsina and F. Petitta, Quadratic quasilinear equations with general singularities. Preprint.

[4] A. Bensoussan, L. Boccardo and F. Murat, On a nonlinear partial differential equation having natural growth terms and unbounded solution. Ann. Inst. H. Poincaré Anal. Non Linéaire 5 (1988) 347-364. | Numdam | MR | Zbl

[5] L. Boccardo, Some nonlinear Dirichlet problems in L 1 involving lower order terms in divergence form, in Progress in elliptic and parabolic partial differential equations (Capri, 1994), Pitman Res. Notes Math. Ser. 350, Longman, Harlow (1996) 43-57. | MR | Zbl

[6] L. Boccardo, Positive solutions for some quasilinear elliptic equations with natural growths. Atti Accad. Naz. Lincei 11 (2000) 31-39. | MR | Zbl

[7] L. Boccardo, Hardy potential and quasi-linear elliptic problems having natural growth terms, in Proceedings of the Conference held in Gaeta on the occasion of the 60th birthday of Haim Brezis, Progr. Nonlinear Differential Equations Appl. 63, Birkhauser, Basel (2005) 67-82. | MR | Zbl

[8] L. Boccardo and T. Gallouët, Nonlinear elliptic and parabolic equations involving measure data. J. Funct. Anal. 87 (1989) 149-169. | MR | Zbl

[9] L. Boccardo and T. Gallouët, Strongly nonlinear elliptic equations having natural growth terms and L 1 data. Nonlinear Anal. 19 (1992) 573-579. | MR | Zbl

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

[11] L. Boccardo and D. Giachetti, Existence results via regularity for some nonlinear elliptic problems. Comm. Partial Diff. Eq. 14 (1989) 663-680. | MR | Zbl

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

[13] L. Boccardo, F. Murat and J.-P. Puel, Existence de solutions non bornées pour certaines équations quasi-linéaires. Portugaliae Math. 41 (1982) 507-534. | MR | Zbl

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

[15] L. Boccardo, F. Murat and J.-P. Puel, Existence of bounded solutions for nonlinear elliptic unilateral problems. Ann. Mat. Pura Appl. 152 (1988) 183-196. | MR | Zbl

[16] L. Boccardo, F. Murat and J.-P. Puel, L -estimates for some nonlinear partial differential equations and application to an existence result. SIAM J. Math. Anal. 23 (1992) 326-333. | MR | Zbl

[17] H. Brezis and L. Nirenberg, Removable singularities for nonlinear elliptic equations. Topol. Methods Nonlinear Anal. 9 (1997) 201-219. | MR | Zbl

[18] M.G. Crandall, P.H. Rabinowitz and L. Tartar, On a Dirichlet problem with a singular nonlinearity. Comm. Partial Diff. Eq. 2 (1977) 193-222. | MR | Zbl

[19] A. Dall'Aglio, D. Giachetti and J.-P. Puel, Nonlinear elliptic equations with natural growth in general domains. Ann. Mat. Pura Appl. 181 (2002) 407-426. | MR | Zbl

[20] A. Dall'Aglio, V. De Cicco, D. Giachetti and J.-P. Puel, Existence of bounded solutions for nonlinear elliptic equations in unbounded domains. NoDEA 11 (2004) 431-450. | MR | Zbl

[21] T. Del Vecchio, Strongly nonlinear problems with Hamiltonian having natural growth. Houston J. Math. 16 (1990) 7-24. | MR | Zbl

[22] D. Giachetti and F. Murat, Personal communication.

[23] J.B. Keller, On solutions of Δu=f(u). Commun. Pure Appl. Math. 10 (1957) 503-510. | MR | Zbl

[24] A.C. Lazer and P.J. Mckenna, On a singular nonlinear elliptic boundary-value problem. Proc. Amer. Math. Soc. 111 (1991) 721-730. | MR | Zbl

[25] T. Leonori, Large solutions for a class of nonlinear elliptic equations with gradient terms. Adv. Nonlinear Stud. 7 (2007) 237-269. | MR | Zbl

[26] R. Osserman, On the inequality Δuf(u). Pacific J. Math. 7 (1957) 1641-1647. | MR | Zbl

[27] A. Porretta, Existence for elliptic equations in L 1 having lower order terms with natural growth. Portugaliae Math. 57 (2000) 179-190. | MR | Zbl

[28] A. Porretta, A local estimates and large solutions for some elliptic equations with absorption. Adv. Differential Equations 9 (2004) 329-351. | MR | Zbl

[29] A. Porretta and S. Segura De Leon, Nonlinear elliptic equations having a gradient term with natural growth. J. Math. Pures Appl. 85 (2006) 465-492. | MR | Zbl

[30] J.-P. Puel, Existence, comportement à l’infini et stabilité dans certains problèmes quasilinéaires elliptiques et paraboliques d’ordre 2. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 3 (1976) 89-119. | Numdam | MR | Zbl

[31] 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

[32] N.S. Trudinger, On Harnack type inequalities and their application to quasilinear elliptic equations. Comm. Pure Appl. Math. 20 (1967) 721-747. | MR | Zbl

[33] J.L. Vazquez, The Porous Medium Equation: Mathematical Theory, Oxford Mathematical Monographs. Oxford University Press, Oxford (2007). | MR | Zbl

Cité par Sources :