• Previous Article
    Local maximum principle for $L^p$-viscosity solutions of fully nonlinear elliptic PDEs with unbounded coefficients
  • CPAA Home
  • This Issue
  • Next Article
    On a singular Hamiltonian elliptic systems involving critical growth in dimension two
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, 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 $m> \frac{N}{2}$.
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

[1]

Li Yin, Jinghua Yao, Qihu Zhang, Chunshan Zhao. Multiple solutions with constant sign of a Dirichlet problem for a class of elliptic systems with variable exponent growth. Discrete & Continuous Dynamical Systems, 2017, 37 (4) : 2207-2226. doi: 10.3934/dcds.2017095

[2]

Juan Pablo Rincón-Zapatero. Hopf-Lax formula for variational problems with non-constant discount. Journal of Geometric Mechanics, 2009, 1 (3) : 357-367. doi: 10.3934/jgm.2009.1.357

[3]

Kais Hamza, Fima C. Klebaner. On nonexistence of non-constant volatility in the Black-Scholes formula. Discrete & Continuous Dynamical Systems - B, 2006, 6 (4) : 829-834. doi: 10.3934/dcdsb.2006.6.829

[4]

Steven M. Pederson. Non-turning Poincaré map and homoclinic tangencies in interval maps with non-constant topological entropy. Conference Publications, 2001, 2001 (Special) : 295-302. doi: 10.3934/proc.2001.2001.295

[5]

Ignacio Guerra. A semilinear problem with a gradient term in the nonlinearity. Discrete & Continuous Dynamical Systems, 2021  doi: 10.3934/dcds.2021110

[6]

R. P. Gupta, Shristi Tiwari, Shivam Saxena. The qualitative behavior of a plankton-fish interaction model with food limited growth rate and non-constant fish harvesting. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021160

[7]

Vanessa Barros, Carlos Nonato, Carlos Raposo. Global existence and energy decay of solutions for a wave equation with non-constant delay and nonlinear weights. Electronic Research Archive, 2020, 28 (1) : 205-220. doi: 10.3934/era.2020014

[8]

Isaac A. García, Claudia Valls. The three-dimensional center problem for the zero-Hopf singularity. Discrete & Continuous Dynamical Systems, 2016, 36 (4) : 2027-2046. doi: 10.3934/dcds.2016.36.2027

[9]

V. V. Motreanu. Multiplicity of solutions for variable exponent Dirichlet problem with concave term. Discrete & Continuous Dynamical Systems - S, 2012, 5 (4) : 845-855. doi: 10.3934/dcdss.2012.5.845

[10]

Craig Cowan, Abdolrahman Razani. Singular solutions of a Hénon equation involving a nonlinear gradient term. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021172

[11]

Gisella Croce. An elliptic problem with degenerate coercivity and a singular quadratic gradient lower order term. Discrete & Continuous Dynamical Systems - S, 2012, 5 (3) : 507-530. doi: 10.3934/dcdss.2012.5.507

[12]

Xin Zhong. Singularity formation to the two-dimensional non-barotropic non-resistive magnetohydrodynamic equations with zero heat conduction in a bounded domain. Discrete & Continuous Dynamical Systems - B, 2020, 25 (3) : 1083-1096. doi: 10.3934/dcdsb.2019209

[13]

Francis J. Chung. Partial data for the Neumann-Dirichlet magnetic Schrödinger inverse problem. Inverse Problems & Imaging, 2014, 8 (4) : 959-989. doi: 10.3934/ipi.2014.8.959

[14]

Harald Garcke, Kei Fong Lam. Analysis of a Cahn--Hilliard system with non-zero Dirichlet conditions modeling tumor growth with chemotaxis. Discrete & Continuous Dynamical Systems, 2017, 37 (8) : 4277-4308. doi: 10.3934/dcds.2017183

[15]

Huxiao Luo, Xianhua Tang, Zu Gao. Sign-changing solutions for non-local elliptic equations with asymptotically linear term. Communications on Pure & Applied Analysis, 2018, 17 (3) : 1147-1159. doi: 10.3934/cpaa.2018055

[16]

Freddy Dumortier, Santiago Ibáñez, Hiroshi Kokubu, Carles Simó. About the unfolding of a Hopf-zero singularity. Discrete & Continuous Dynamical Systems, 2013, 33 (10) : 4435-4471. doi: 10.3934/dcds.2013.33.4435

[17]

Martino Bardi, Paola Mannucci. On the Dirichlet problem for non-totally degenerate fully nonlinear elliptic equations. Communications on Pure & Applied Analysis, 2006, 5 (4) : 709-731. doi: 10.3934/cpaa.2006.5.709

[18]

Davide Guidetti. Partial reconstruction of the source term in a linear parabolic initial problem with Dirichlet boundary conditions. Discrete & Continuous Dynamical Systems, 2013, 33 (11&12) : 5107-5141. doi: 10.3934/dcds.2013.33.5107

[19]

Tommaso Leonori, Martina Magliocca. Comparison results for unbounded solutions for a parabolic Cauchy-Dirichlet problem with superlinear gradient growth. Communications on Pure & Applied Analysis, 2019, 18 (6) : 2923-2960. doi: 10.3934/cpaa.2019131

[20]

Bouthaina Abdelhedi, Hatem Zaag. Single point blow-up and final profile for a perturbed nonlinear heat equation with a gradient and a non-local term. Discrete & Continuous Dynamical Systems - S, 2021, 14 (8) : 2607-2623. doi: 10.3934/dcdss.2021032

2020 Impact Factor: 1.916

Metrics

  • PDF downloads (72)
  • HTML views (0)
  • Cited by (15)

[Back to Top]