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.
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://www.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://www.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://www.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://www.numdam.org/articles/10.1051/cocv:2008031/
[1] Singular quasilinear equations with quadratic growth in the gradient without sign condition. Preprint. | MR
, and ,[2] Quasilinear equations with natural growth Rev. Mat. Iberoamericana (to appear). | MR | Zbl
and ,[3] Quadratic quasilinear equations with general singularities. Preprint.
, , , , and ,[4] 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
, and ,
[5] Some nonlinear Dirichlet problems in
[6] Positive solutions for some quasilinear elliptic equations with natural growths. Atti Accad. Naz. Lincei 11 (2000) 31-39. | MR | Zbl
,[7] 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] Nonlinear elliptic and parabolic equations involving measure data. J. Funct. Anal. 87 (1989) 149-169. | MR | Zbl
and ,
[9] Strongly nonlinear elliptic equations having natural growth terms and
[10] Existence and nonexistence of solutions for some nonlinear elliptic equations. J. Anal. Math. 73 (1997) 203-223. | MR | Zbl
, and ,[11] Existence results via regularity for some nonlinear elliptic problems. Comm. Partial Diff. Eq. 14 (1989) 663-680. | MR | Zbl
and ,[12] Almost everywhere convergence of the gradients of solutions to elliptic and parabolic equations. Nonlinear Anal. TMA 19 (1992) 581-597. | MR | Zbl
and ,[13] Existence de solutions non bornées pour certaines équations quasi-linéaires. Portugaliae Math. 41 (1982) 507-534. | MR | Zbl
, and ,[14] Résultats d'existence pour certains problèmes elliptiques quasi linéaires. Ann. Sc. Norm. Sup. Pisa 11 (1984) 213-235. | Numdam | MR | Zbl
, and ,[15] Existence of bounded solutions for nonlinear elliptic unilateral problems. Ann. Mat. Pura Appl. 152 (1988) 183-196. | MR | Zbl
, and ,
[16]
[17] Removable singularities for nonlinear elliptic equations. Topol. Methods Nonlinear Anal. 9 (1997) 201-219. | MR | Zbl
and ,[18] On a Dirichlet problem with a singular nonlinearity. Comm. Partial Diff. Eq. 2 (1977) 193-222. | MR | Zbl
, and ,[19] Nonlinear elliptic equations with natural growth in general domains. Ann. Mat. Pura Appl. 181 (2002) 407-426. | MR | Zbl
, and ,[20] Existence of bounded solutions for nonlinear elliptic equations in unbounded domains. NoDEA 11 (2004) 431-450. | MR | Zbl
, , and ,[21] Strongly nonlinear problems with Hamiltonian having natural growth. Houston J. Math. 16 (1990) 7-24. | MR | Zbl
,[22] Personal communication.
and ,
[23] On solutions of
[24] On a singular nonlinear elliptic boundary-value problem. Proc. Amer. Math. Soc. 111 (1991) 721-730. | MR | Zbl
and ,[25] Large solutions for a class of nonlinear elliptic equations with gradient terms. Adv. Nonlinear Stud. 7 (2007) 237-269. | MR | Zbl
,
[26] On the inequality
[27] Existence for elliptic equations in
[28] A local estimates and large solutions for some elliptic equations with absorption. Adv. Differential Equations 9 (2004) 329-351. | MR | Zbl
,[29] Nonlinear elliptic equations having a gradient term with natural growth. J. Math. Pures Appl. 85 (2006) 465-492. | MR | Zbl
and ,
[30] Existence, comportement à l’infini et stabilité dans certains problèmes quasilinéaires elliptiques et paraboliques d’ordre
[31] 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] On Harnack type inequalities and their application to quasilinear elliptic equations. Comm. Pure Appl. Math. 20 (1967) 721-747. | MR | Zbl
,[33] The Porous Medium Equation: Mathematical Theory, Oxford Mathematical Monographs. Oxford University Press, Oxford (2007). | MR | Zbl
,Cité par Sources :