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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

 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, Università 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 $m> \frac{N}{2}$.
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 and 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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [15] F. E. Browder, Existence theorems for nonlinear partial differential equations, "Global Analysis" (Proc. Sympos. Pre Math., vol XVI, Berkeley, California, 1968), [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. [17] D. Giachetti and S. Segura de León, Quasilinear stationary problems with a quadratic gradient term having singularities,, J. London Math. Soc., (). [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. [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.

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##### 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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [15] F. E. Browder, Existence theorems for nonlinear partial differential equations, "Global Analysis" (Proc. Sympos. Pre Math., vol XVI, Berkeley, California, 1968), [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. [17] D. Giachetti and S. Segura de León, Quasilinear stationary problems with a quadratic gradient term having singularities,, J. London Math. Soc., (). [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. [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.
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