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September  2012, 11(5): 1875-1895. doi: 10.3934/cpaa.2012.11.1875

Elliptic equations having a singular quadratic gradient term and a changing sign datum

Daniela Giachetti 1, , Francesco Petitta 2, and  Sergio Segura de León 3,

1. 

Dip. Metodi e Modelli Matematici per le Scienze Applicate, Univ. Roma 1, Via Antonio Scarpa 16, 00161 Roma
2. 

Dipartimento di Scienze di Base e Applicate per l'Ingegneria, Sapienza, Università di Roma, Via Scarpa 16, 00161 Roma, Italy
3. 

Departament d'An, Spain

Received  March 2011 Revised  February 2012 Published  March 2012

In this paper we study a singular elliptic problem whose model is \begin{eqnarray*} - \Delta u= \frac{|\nabla u|^2}{|u|^\theta}+f(x), in \Omega\\ u = 0, on \partial \Omega; \end{eqnarray*} where $\theta\in (0,1)$ and $f \in L^m (\Omega)$, with $m\geq \frac{N}{2}$. We do not assume any sign condition on the lower order term, nor assume the datum $f$ has a constant sign. We carefully define the meaning of solution to this problem giving sense to the gradient term where $u=0$, and prove the existence of such a solution. We also discuss related questions as the existence of solutions when the datum $f$ is less regular or the boundedness of the solutions when the datum $f \in L^m (\Omega)$ with $m> \frac{N}{2}$.
Keywords: gradient term, Dirichlet problem, singularity at zero, data with non-constant sign..
Mathematics Subject Classification: 35J75, 35J2.
Citation: Daniela Giachetti, Francesco Petitta, Sergio Segura de León. Elliptic equations having a singular quadratic gradient term and a changing sign datum. Communications on Pure & Applied Analysis, 2012, 11 (5) : 1875-1895. doi: 10.3934/cpaa.2012.11.1875
References:
[1]

B. Abdellaoui, D. Giachetti, I. Peral and M. Walias, Elliptic problems with nonlinear terms depending on the gradient and singular on the boundary, Nonlinear Analysis, 74 (2011), 1355-1371.  Google Scholar

[2]

D. Arcoya, S. Barile and P. J. Martínez-Aparicio, Singular quasilinear equations with quadratic growth in the gradient without sign condition, J. Math. Anal. Appl., 350 (2009), 401-408. doi: 10.1016/j.jmaa.2008.09.073.  Google Scholar

[3]

D. Arcoya, L. Boccardo, T. Leonori and A. Porretta, Some elliptic problems with singular natural growth lower order terms, J. Differential Equations, 249 (2010), 2771-2795. doi: 10.1016/j.jde.2010.05.009.  Google Scholar

[4]

D. Arcoya, J. Carmona, T. Leonori, P. J. Martínez-Aparicio, L. Orsina and F. Petitta, Existence and nonexistence of solutions for singular quadratic quasilinear equations, J. Differential Equations, 246 (2009), 4006-4042. doi: 10.1016/j.jde.2009.01.016.  Google Scholar

[5]

D. Arcoya, J. Carmona and P. J. Martínez-Aparicio, Elliptic obstacle problems with natural growth on the gradient and singular nonlinear terms, Adv. Nonlinear Stud., 7 (2007), 299-317.  Google Scholar

[6]

D. Arcoya and P. J. Martínez-Aparicio, Quasilinear equations with natural growth, Rev. Mat. Iberoamericana, 24 (2008), 597-616. doi: 10.4171/RMI/548.  Google Scholar

[7]

D. Arcoya and S. Segura de León, Uniqueness of solutions for some elliptic equations with a quadratic gradient term, ESAIM: Control, Optimization and the Calculus of Variations, 16 (2010), 327-336. doi: 10.1051/cocv:2008072.  Google Scholar

[8]

L. Boccardo, Dirichlet problems with singular and gradient quadratic lower order terms, ESAIM Control Optim. Calc. Var., 14 (2008), 411-426. doi: 10.1051/cocv:2008031.  Google Scholar

[9]

L. Boccardo and T. Gallouët, Strongly nonlinear elliptic equations having natural growth terms and $L^1$ data, Nonlinear Anal., 19 (1992), 573-579. doi: 10.1016/0362-546X(92)90022-7.  Google Scholar

[10]

L. Boccardo, T. Leonori, L. Orsina and F. Petitta, Quasilinear elliptic equations with singular quadratic growth terms, Comm. Contemp. Math., 13 (2011), 607-642. doi: 10.1142/S0219199711004300.  Google Scholar

[11]

L. Boccardo, F. Murat and J. P. Puel, Existence de solutions non bornées pour certaines équations quasi-linéaires, Portugal. Math., 41 (1982), 507-534.  Google Scholar

[12]

L. Boccardo, F. Murat and J. P. Puel, Résultats d'existence pour certains problèmes elliptiques quasilinéaires, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 11 (1984), 213-235.  Google Scholar

[13]

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. doi: 10.1007/BF01766148.  Google Scholar

[14]

L. Boccardo, S. Segura de León and C. Trombetti, Bounded and unbounded solutions for a class of quasi-linear elliptic problems with a quadratic gradient term, J. Math. Pures Appl., 80 (2001), 919-940. doi: 10.1016/S0021-7824(01)01211-9.  Google Scholar

[15]

F. E. Browder, Existence theorems for nonlinear partial differential equations, "Global Analysis" (Proc. Sympos. Pre Math., vol XVI, Berkeley, California, 1968),  Google Scholar

[16]

D. Giachetti and F. Murat, An elliptic problem with a lower order term having singular behaviour, Boll. Unione Mat. Ital., 2 (2009), 349-370.  Google Scholar

[17]

D. Giachetti and S. Segura de León, Quasilinear stationary problems with a quadratic gradient term having singularities,, J. London Math. Soc., ().   Google Scholar

[18]

A. Porretta and S. Segura de León, Nonlinear elliptic equations having a gradient term with natural growth, J. Math. Pures Appl., 85 (2006), 465-492. doi: 10.1016/j.matpur.2005.10.009.  Google Scholar

[19]

S. Segura de León, Existence and uniqueness for $L^1$ data of some elliptic equations with natural growth, Adv. Diff. Eq., 8 (2003), 1377-1408.  Google Scholar

show all references

References:
[1]

B. Abdellaoui, D. Giachetti, I. Peral and M. Walias, Elliptic problems with nonlinear terms depending on the gradient and singular on the boundary, Nonlinear Analysis, 74 (2011), 1355-1371.  Google Scholar

[2]

D. Arcoya, S. Barile and P. J. Martínez-Aparicio, Singular quasilinear equations with quadratic growth in the gradient without sign condition, J. Math. Anal. Appl., 350 (2009), 401-408. doi: 10.1016/j.jmaa.2008.09.073.  Google Scholar

[3]

D. Arcoya, L. Boccardo, T. Leonori and A. Porretta, Some elliptic problems with singular natural growth lower order terms, J. Differential Equations, 249 (2010), 2771-2795. doi: 10.1016/j.jde.2010.05.009.  Google Scholar

[4]

D. Arcoya, J. Carmona, T. Leonori, P. J. Martínez-Aparicio, L. Orsina and F. Petitta, Existence and nonexistence of solutions for singular quadratic quasilinear equations, J. Differential Equations, 246 (2009), 4006-4042. doi: 10.1016/j.jde.2009.01.016.  Google Scholar

[5]

D. Arcoya, J. Carmona and P. J. Martínez-Aparicio, Elliptic obstacle problems with natural growth on the gradient and singular nonlinear terms, Adv. Nonlinear Stud., 7 (2007), 299-317.  Google Scholar

[6]

D. Arcoya and P. J. Martínez-Aparicio, Quasilinear equations with natural growth, Rev. Mat. Iberoamericana, 24 (2008), 597-616. doi: 10.4171/RMI/548.  Google Scholar

[7]

D. Arcoya and S. Segura de León, Uniqueness of solutions for some elliptic equations with a quadratic gradient term, ESAIM: Control, Optimization and the Calculus of Variations, 16 (2010), 327-336. doi: 10.1051/cocv:2008072.  Google Scholar

[8]

L. Boccardo, Dirichlet problems with singular and gradient quadratic lower order terms, ESAIM Control Optim. Calc. Var., 14 (2008), 411-426. doi: 10.1051/cocv:2008031.  Google Scholar

[9]

L. Boccardo and T. Gallouët, Strongly nonlinear elliptic equations having natural growth terms and $L^1$ data, Nonlinear Anal., 19 (1992), 573-579. doi: 10.1016/0362-546X(92)90022-7.  Google Scholar

[10]

L. Boccardo, T. Leonori, L. Orsina and F. Petitta, Quasilinear elliptic equations with singular quadratic growth terms, Comm. Contemp. Math., 13 (2011), 607-642. doi: 10.1142/S0219199711004300.  Google Scholar

[11]

L. Boccardo, F. Murat and J. P. Puel, Existence de solutions non bornées pour certaines équations quasi-linéaires, Portugal. Math., 41 (1982), 507-534.  Google Scholar

[12]

L. Boccardo, F. Murat and J. P. Puel, Résultats d'existence pour certains problèmes elliptiques quasilinéaires, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 11 (1984), 213-235.  Google Scholar

[13]

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. doi: 10.1007/BF01766148.  Google Scholar

[14]

L. Boccardo, S. Segura de León and C. Trombetti, Bounded and unbounded solutions for a class of quasi-linear elliptic problems with a quadratic gradient term, J. Math. Pures Appl., 80 (2001), 919-940. doi: 10.1016/S0021-7824(01)01211-9.  Google Scholar

[15]

F. E. Browder, Existence theorems for nonlinear partial differential equations, "Global Analysis" (Proc. Sympos. Pre Math., vol XVI, Berkeley, California, 1968),  Google Scholar

[16]

D. Giachetti and F. Murat, An elliptic problem with a lower order term having singular behaviour, Boll. Unione Mat. Ital., 2 (2009), 349-370.  Google Scholar

[17]

D. Giachetti and S. Segura de León, Quasilinear stationary problems with a quadratic gradient term having singularities,, J. London Math. Soc., ().   Google Scholar

[18]

A. Porretta and S. Segura de León, Nonlinear elliptic equations having a gradient term with natural growth, J. Math. Pures Appl., 85 (2006), 465-492. doi: 10.1016/j.matpur.2005.10.009.  Google Scholar

[19]

S. Segura de León, Existence and uniqueness for $L^1$ data of some elliptic equations with natural growth, Adv. Diff. Eq., 8 (2003), 1377-1408.  Google Scholar

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