June  2012, 5(3): 507-530. doi: 10.3934/dcdss.2012.5.507

An elliptic problem with degenerate coercivity and a singular quadratic gradient lower order term

1. 

Laboratoire de Mathématiques Appliquées du Havre, Université du Havre, 25, rue Philippe Lebon, 76063 Le Havre, France

Received  August 2010 Revised  September 2010 Published  October 2011

In this paper we study a Dirichlet problem for an elliptic equation with degenerate coercivity and a singular lower order term with natural growth with respect to the gradient. The model problem is $$ \begin{equation} \left\{\begin{array}{11} -div\left(\frac{\nabla u}{(1+|u|)^p}\right) + \frac{|\nabla u|^{2}}{|u|^{\theta}} = f & \mbox{in $\Omega$,} \\ u = 0 & \mbox{on $\partial\Omega$,} \end{array} \right. \end{equation} $$ where $\Omega$ is an open bounded set of $\mathbb{R}^N$, $N\geq 3$ and $p, \theta>0$. The source $f$ is a positive function belonging to some Lebesgue space. We will show that, even if the lower order term is singular, it has some regularizing effects on the solutions, when $p>\theta-1$ and $\theta<2$.
Citation: Gisella Croce. An elliptic problem with degenerate coercivity and a singular quadratic gradient lower order term. Discrete and Continuous Dynamical Systems - S, 2012, 5 (3) : 507-530. doi: 10.3934/dcdss.2012.5.507
References:
[1]

A. Alvino, L. Boccardo, V. Ferone, L. Orsina and G. Trombetti, Existence results for nonlinear elliptic equations with degenerate coercivity, Ann. Mat. Pura Appl. (4), 182 (2003), 53-79. doi: 10.1007/s10231-002-0056-y.

[2]

D. Arcoya, S. Barile, 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, 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.

[4]

D. Arcoya and P. J. Martínez-Aparicio, Quasilinear equations with natural growth, Rev. Mat. Iberoam., 24 (2008), 597-616.

[5]

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

[6]

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.

[7]

L. Boccardo, Quasilinear elliptic equations with natural growth terms: The regularizing effect of the lower order terms, J. Nonlin. Conv. Anal., 7 (2006), 355-365.

[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, A. Dall'Aglio and L. Orsina, Existence and regularity results for some elliptic equations with degenerate coercivity. Dedicated to Prof. C. Vinti, (Italian) (Perugia, 1996), Atti Sem. Mat. Fis. Univ. Modena, 46 (1998), 51-81.

[10]

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.

[11]

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

[12]

L. Boccardo, F. Murat and J.-P. Puel, $L^{\infty}$ estimate for some nonlinear elliptic partial differential equations and application to an existence result, SIAM J. Math. Anal., 23 (1992), 326-333. doi: 10.1137/0523016.

[13]

L. Boccardo, L. Orsina and M. M. Porzio, Existence results for quasilinear elliptic and parabolic problems with quadratic gradient terms and sources, preprint.

[14]

G. Croce, The regularizing effects of some lower order terms on the solutions in an elliptic equation with degenerate coercivity, Rendiconti di Matematica (7), 27 (2007), 299-314.

[15]

D. Giachetti and F. Murat, An elliptic problem with a lower order term having singular behaviour, Boll. Unione Mat. Ital. Sez. B, in press.

[16]

D. Giachetti and M. M. Porzio, Existence results fo some nonuniformly elliptic equations with irregular data, J. Math. Anal. Appl., 257 (2001), 100-130. doi: 10.1006/jmaa.2000.7324.

[17]

J. B. Keller, On the solutions of $\Delta u= f(u)$, Comm. Pure Appl. Math., 10 (1957), 503-510. doi: 10.1002/cpa.3160100402.

[18]

F. Leoni and B. Pellacci, Local estimates and global existence for strongly nonlinear parabolic equations with locally integrable data, J. Evol. Equ., 6 (2006), 113-144. doi: 10.1007/s00028-005-0234-7.

[19]

R. Osserman, On the inequality $\Delta u\geq f(u)$, Pacific J. Math., 7 (1957), 1641-1647.

[20]

A. Porretta, Uniqueness and homogeneization for a class of noncoercive operators in divergence form, Atti Sem. Mat. Fis. Univ. Modena, 46 (1998), 915-936.

[21]

A. Porretta, Existence for elliptic equations in $L^1$ having lower order terms with natural growth, Port. Math., 57 (2000), 179-190.

show all references

References:
[1]

A. Alvino, L. Boccardo, V. Ferone, L. Orsina and G. Trombetti, Existence results for nonlinear elliptic equations with degenerate coercivity, Ann. Mat. Pura Appl. (4), 182 (2003), 53-79. doi: 10.1007/s10231-002-0056-y.

[2]

D. Arcoya, S. Barile, 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, 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.

[4]

D. Arcoya and P. J. Martínez-Aparicio, Quasilinear equations with natural growth, Rev. Mat. Iberoam., 24 (2008), 597-616.

[5]

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

[6]

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.

[7]

L. Boccardo, Quasilinear elliptic equations with natural growth terms: The regularizing effect of the lower order terms, J. Nonlin. Conv. Anal., 7 (2006), 355-365.

[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, A. Dall'Aglio and L. Orsina, Existence and regularity results for some elliptic equations with degenerate coercivity. Dedicated to Prof. C. Vinti, (Italian) (Perugia, 1996), Atti Sem. Mat. Fis. Univ. Modena, 46 (1998), 51-81.

[10]

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.

[11]

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

[12]

L. Boccardo, F. Murat and J.-P. Puel, $L^{\infty}$ estimate for some nonlinear elliptic partial differential equations and application to an existence result, SIAM J. Math. Anal., 23 (1992), 326-333. doi: 10.1137/0523016.

[13]

L. Boccardo, L. Orsina and M. M. Porzio, Existence results for quasilinear elliptic and parabolic problems with quadratic gradient terms and sources, preprint.

[14]

G. Croce, The regularizing effects of some lower order terms on the solutions in an elliptic equation with degenerate coercivity, Rendiconti di Matematica (7), 27 (2007), 299-314.

[15]

D. Giachetti and F. Murat, An elliptic problem with a lower order term having singular behaviour, Boll. Unione Mat. Ital. Sez. B, in press.

[16]

D. Giachetti and M. M. Porzio, Existence results fo some nonuniformly elliptic equations with irregular data, J. Math. Anal. Appl., 257 (2001), 100-130. doi: 10.1006/jmaa.2000.7324.

[17]

J. B. Keller, On the solutions of $\Delta u= f(u)$, Comm. Pure Appl. Math., 10 (1957), 503-510. doi: 10.1002/cpa.3160100402.

[18]

F. Leoni and B. Pellacci, Local estimates and global existence for strongly nonlinear parabolic equations with locally integrable data, J. Evol. Equ., 6 (2006), 113-144. doi: 10.1007/s00028-005-0234-7.

[19]

R. Osserman, On the inequality $\Delta u\geq f(u)$, Pacific J. Math., 7 (1957), 1641-1647.

[20]

A. Porretta, Uniqueness and homogeneization for a class of noncoercive operators in divergence form, Atti Sem. Mat. Fis. Univ. Modena, 46 (1998), 915-936.

[21]

A. Porretta, Existence for elliptic equations in $L^1$ having lower order terms with natural growth, Port. Math., 57 (2000), 179-190.

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