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May  2019, 39(5): 2637-2659. doi: 10.3934/dcds.2019110

Positive solution to extremal Pucci's equations with singular and gradient nonlinearity

Discipline of Mathematics, Indian Institute of Technology Gandhinagar, Palaj, Gandhinagar Gujarat, 382355, India

* Corresponding author: J. Tyagi

Received  May 2018 Revised  November 2018 Published  January 2019

Fund Project: The first author thanks DST/SERB for the financial support under the grant EMR/2015/001908

In this paper, we establish the existence of a positive solution to
$ \begin{equation*} \label{deff} \left\{ \begin{aligned}{} -\mathcal{M}^{+}_{\lambda,\Lambda}(D^{2}u)+H(x,Du)& = \frac{k(x)f(u)}{u^{\alpha}} \;\text{in}\; \Omega,\\ u&>0 \;\text{in}\; \Omega,\\ u& = 0 \;\text{on} \;\partial\Omega, \end{aligned} \right. \end{equation*} $
under certain conditions on
$ k,f $
and
$ H, $
using viscosity sub-and supersolution method. The main feature of this problem is that it has singularity as well as a superlinear growth in the gradient term. We use Hopf-Cole transformation to handle the superlinear gradient term and an approximation method combined with suitable stability result for viscosity solution to outfit the singular nonlinearity. This work extends and complements the recent works on elliptic equations involving singular as well as superlinear gradient nonlinearities.
Citation: Jagmohan Tyagi, Ram Baran Verma. Positive solution to extremal Pucci's equations with singular and gradient nonlinearity. Discrete & Continuous Dynamical Systems - A, 2019, 39 (5) : 2637-2659. doi: 10.3934/dcds.2019110
References:
[1]

M. E. Amendola, L. Rossi and A. Vitolo, Harnack inequalities and ABP estimates for nonlinear second-order elliptic equations in unbounded domains, Abstr. Appl. Anal., (2008), Art. ID 178534, 19 pp. doi: 10.1155/2008/178534. Google Scholar

[2]

D. ArcoyaJ. CarmonaT. LeonoriP. J. Martínez-AparicioL. 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

[3]

D. ArcoyaS. 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

[4]

S. N. Armstrong, Principal eigenvalues and an anti-maximum principle for homogeneous fully nonlinear elliptic equations, J. Differential Equations, 246 (2009), 2958-2987. doi: 10.1016/j.jde.2008.10.026. Google Scholar

[5]

A. L. Bertozzi and M. C. Pugh, Long-wave instabilities and saturation in thin film equations, Comm. Pure Appl. Math., 51 (1998), 625-661. doi: 10.1002/(SICI)1097-0312(199806)51:6<625::AID-CPA3>3.0.CO;2-9. Google Scholar

[6]

A. L. Bertozzi and M. C Pugh, Finite-time blow-up of solutions of some long-wave unstable thin film equations, Indiana Univ. Math. J., 49 (2000), 1323-1366. doi: 10.1512/iumj.2000.49.1887. Google Scholar

[7]

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

[8]

L. Boccardo and L. Orsina, Semilinear elliptic equations with singular nonlinearities, Calc. Var. Partial. Differ. Equ., 37 (2010), 363-380. doi: 10.1007/s00526-009-0266-x. Google Scholar

[9]

J. BuscaM. J. Esteban and A. Quaas, Nonlinear eigenvalues and bifurcation problems for Pucci's operator, Ann. Inst. H. Poincaré Anal. Non Linéaire, 22 (2005), 187-206. doi: 10.1016/j.anihpc.2004.05.004. Google Scholar

[10]

L. CaffarelliR. Hardt and L. Simon, Minimal surfaces with isolated singularities, Manuscripta Math., 48 (1984), 1-18. doi: 10.1007/BF01168999. Google Scholar

[11]

A. Callegari and A. Nachman, Some singular nonlinear equations arising in boundary layer theory, J. Math. Anal. Appl., 64 (1978), 96-105. doi: 10.1016/0022-247X(78)90022-7. Google Scholar

[12]

A. Callegari and A. Nachman, A nonlinear singular boundary value problem in the theory of pseudoplastic fluids, SIAM J. Appl. Math., 38 (1980), 275-281. doi: 10.1137/0138024. Google Scholar

[13]

I. Capuzzo DolcettaF. Leoni and A. Vitolo, On the inequality $F(x, D^2u) \geq f(u)+g(u)|Du|^q,$, Math. Ann., 365 (2016), 423-448. doi: 10.1007/s00208-015-1280-2. Google Scholar

[14]

M. Chhetri and S. Robinson, Existence and multiplicity of positive solutions for classes of singular elliptic PDEs, J. Math. Anal. Appl., 357 (2009), 176-182. doi: 10.1016/j.jmaa.2009.03.033. Google Scholar

[15]

M. M. Coclite, On a singular nonlinear Dirichlet problem Ⅲ, Nonlinear Anal. Theory Methods Appl., 21 (1993), 547-564. doi: 10.1016/0362-546X(93)90010-P. Google Scholar

[16]

M. M. Coclite and G. Palmieri, On a singular nonlinear dirichlet problem, Comm. Partial Differential Equations, 14 (1989), 1315-1327. doi: 10.1080/03605308908820656. Google Scholar

[17]

M. G. CrandallL. CaffarelliM. Kocan and A. Świech, On viscosity solutions of fully nonlinear equations with measurable ingredients, Comm. Pure Appl. Math., 49 (1996), 365-397. doi: 10.1002/(SICI)1097-0312(199604)49:4<365::AID-CPA3>3.0.CO;2-A. Google Scholar

[18]

M. G. CrandallH. Ishii and P.-L. Lions, User's guide to viscosity solutions of second order partial differential equations, Bull. Amer. Math. Soc., 27 (1992), 1-67. doi: 10.1090/S0273-0979-1992-00266-5. Google Scholar

[19]

M. G. CrandallP. H. Rabinowitz and L. Tartar, On a Dirichlet problem with a singular nonlinearity, Comm. Part. Diff. Eq., 2 (1977), 193-222. doi: 10.1080/03605307708820029. Google Scholar

[20]

L. DupaigneM. Ghergu and V. Rădulescu, Lane–Emden–Fowler equations with convection and singular potential, J. Math. Pures. Appl., 87 (2007), 563-581. doi: 10.1016/j.matpur.2007.03.002. Google Scholar

[21]

L. C. Evans, Partial Differential Equations, Amer. Math. Soc., Second Edition, Graduate Studies in Mathematics, 19. American Mathematical Society, Providence, RI, 2010. doi: 10.1090/gsm/019. Google Scholar

[22]

F. Faraci and D. Puglisi, A singular semilinear problem with dependence on the gradient, J. Differential Equations, 260 (2016), 3327-3349. doi: 10.1016/j.jde.2015.10.031. Google Scholar

[23]

P. FelmerA. Quaas and B. Sirakov, Existence and regularity results for fully nonlinear equations with singularities, Math. Ann., 354 (2012), 377-400. doi: 10.1007/s00208-011-0741-5. Google Scholar

[24]

P. Fok, Some Maximum Principles and Continuity Estimates for Fully Nonlinear Elliptic Equations of Second Order, Ph.D. Thesis, UCSB, 1996. Google Scholar

[25]

W. Fulks and J. S. Maybee, A singular non-linear equation, Osaka Math. J., 12 (1960), 1-19. Google Scholar

[26]

G. GaliseS. KoikeO. Ley and A. Vitolo, Entire solutions of fully nonlinear elliptic equations with a superlinear gradient term, J. Math. Anal. Appl., 441 (2016), 194-210. doi: 10.1016/j.jmaa.2016.03.083. Google Scholar

[27]

M. Ghergu and V. Rădulescu, Bifurcation for a class of singular elliptic problems with quadratic convection term, C. R. Math., 338 (2004), 831-836. doi: 10.1016/j.crma.2004.03.020. Google Scholar

[28]

M. Ghergu and V. Rădulescu, On a class of sublinear singular elliptic problems with convection term, J. Math. Anal. Appl., 311 (2005), 635-646. doi: 10.1016/j.jmaa.2005.03.012. Google Scholar

[29]

M. Ghergu and V. Rădulescu, Sublinear singular elliptic problems with two parameters, J. Differential Equations, 195 (2003), 520-536. doi: 10.1016/S0022-0396(03)00105-0. Google Scholar

[30]

E. Giarrusso, G. Porru, Problems for elliptic singular equations with a gradient term, Nonlinear Anal. Theory Methods Appl. 65 (2006), 107–128. doi: 10.1016/j.na.2005.08.007. Google Scholar

[31]

J. Hernández, F. J. Mancebo and J. M. Vega, Nonlinear singular elliptic problems: Recent results and open problems, Progr. Nonlinear Differential Equations Appl., Nonlinear elliptic and parabolic problems, Birkhuser, Basel, 64 (2005), 227–242. doi: 10.1007/3-7643-7385-7_12. Google Scholar

[32]

S. Koike and A. Świech, Existence of strong solutions of Pucci extremal equations with superlinear growth in $Du$, J. Fixed Point Theory Appl., 5 (2009), 291-304. doi: 10.1007/s11784-009-0106-9. Google Scholar

[33]

S. Koike and A. Świech, Weak Harnack inequality for fully nonlinear uniformly elliptic PDE with unbounded ingredients, J. Math. Soc. Japan, 61 (2009), 723-755. doi: 10.2969/jmsj/06130723. Google Scholar

[34]

S. Koike and A. Świech, Maximum principle and existence of $L^p$-viscosity solutions for fully nonlinear uniformly elliptic equations with measurable and quadratic term, Nonlinear Differ. Equ. Appl., 11 (2004), 491-509. doi: 10.1007/s00030-004-2001-9. Google Scholar

[35]

A. C. Lazer and P. J. McKenna, On a singular nonlinear elliptic boundary-value problem, Proc. Amer. Math. Soc., 111 (1991), 721-730. doi: 10.1090/S0002-9939-1991-1037213-9. Google Scholar

[36]

A. C. Lazer and P. J. McKenna, Asymptotic behavior of solutions of boundary blowup problems, Differ. Integral Equ., 7 (1994), 1001-1019. Google Scholar

[37]

P. J. McKenna and W. Reichel, Sign changing solutions to singular second order boundary value problems, Adv. Differential Equations, 6 (2001), 441-460. Google Scholar

[38]

M. Benrhouma, On a singular elliptic system with quadratic growth in the gradient, J. Math. Anal. Appl., 448 (2017), 1120-1146. doi: 10.1016/j.jmaa.2016.11.038. Google Scholar

[39]

J. A. Pelesko, Mathematical modeling of electrostatic MEMS with tailored dielectric properties, SIAM J. Appl. Math., 62 (2002), 888-908. doi: 10.1137/S0036139900381079. Google Scholar

[40]

G. Porru and A. Vitolo, Problems for elliptic singular equations with a quadratic gradient term, J. Math. Anal. Appl., 334 (2007), 467-486. doi: 10.1016/j.jmaa.2006.12.017. Google Scholar

[41]

A. Quaas and B. Sirakov, Principal eigenvalues and the Dirichlet problem for fully nonlinear elliptic operators, Adv. Math., 218 (2008), 105-135. doi: 10.1016/j.aim.2007.12.002. Google Scholar

[42]

W. Rudin, Real and Complex Analysis, Tata McGraw-Hill, Third Edition, McGraw-Hill Book Co., New York, 1987. Google Scholar

[43]

Sh ang Bin Cui, Positive solutions for Dirichlet problems associated to semilinear elliptic equations with singular nonlinearity, Nonlinear Anal. Theory Methods Appl., 21 (1993), 181-190. doi: 10.1016/0362-546X(93)90108-5. Google Scholar

[44]

B. Sirakov, Solvability of uniformly elliptic fully nonlinear PDE, Arch. Rational Mech. Anal., 195 (2010), 579-607. doi: 10.1007/s00205-009-0218-9. Google Scholar

[45]

J. Tyagi and R. B. Verma, A survey on the existence, uniqueness and regularity questions to fully nonlinear elliptic partial differential equations, Diff. Eq. Appl., 8 (2016), 135-205. doi: 10.7153/dea-08-09. Google Scholar

[46]

J. Tyagi and R. B. Verma, Positive solution of extremal Pucci's equations with singular and sublinear nonlinearity, Mediterr. J. Math., 14 (2017), Art. 148, 17 pp. doi: 10.1007/s00009-017-0950-6. Google Scholar

[47]

J. Tyagi and R. B. Verma, Lyapunov type inequality for extremal Pucci's equations, (submitted).Google Scholar

[48]

Z. Zhang and J. Yu, On a singular nonlinear Dirichlet problem with a convection term, SIAM J. Math. Anal., 32 (2000), 916-927. doi: 10.1137/S0036141097332165. Google Scholar

[49]

Z. Zhang, Nonexistence of positive classical solutions of a singular nonlinear Dirichlet problem with a convection term, Nonlinear Anal. Theory Methods Appl., 27 (1996), 957-961. doi: 10.1016/0362-546X(94)00367-Q. Google Scholar

show all references

References:
[1]

M. E. Amendola, L. Rossi and A. Vitolo, Harnack inequalities and ABP estimates for nonlinear second-order elliptic equations in unbounded domains, Abstr. Appl. Anal., (2008), Art. ID 178534, 19 pp. doi: 10.1155/2008/178534. Google Scholar

[2]

D. ArcoyaJ. CarmonaT. LeonoriP. J. Martínez-AparicioL. 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

[3]

D. ArcoyaS. 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

[4]

S. N. Armstrong, Principal eigenvalues and an anti-maximum principle for homogeneous fully nonlinear elliptic equations, J. Differential Equations, 246 (2009), 2958-2987. doi: 10.1016/j.jde.2008.10.026. Google Scholar

[5]

A. L. Bertozzi and M. C. Pugh, Long-wave instabilities and saturation in thin film equations, Comm. Pure Appl. Math., 51 (1998), 625-661. doi: 10.1002/(SICI)1097-0312(199806)51:6<625::AID-CPA3>3.0.CO;2-9. Google Scholar

[6]

A. L. Bertozzi and M. C Pugh, Finite-time blow-up of solutions of some long-wave unstable thin film equations, Indiana Univ. Math. J., 49 (2000), 1323-1366. doi: 10.1512/iumj.2000.49.1887. Google Scholar

[7]

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

[8]

L. Boccardo and L. Orsina, Semilinear elliptic equations with singular nonlinearities, Calc. Var. Partial. Differ. Equ., 37 (2010), 363-380. doi: 10.1007/s00526-009-0266-x. Google Scholar

[9]

J. BuscaM. J. Esteban and A. Quaas, Nonlinear eigenvalues and bifurcation problems for Pucci's operator, Ann. Inst. H. Poincaré Anal. Non Linéaire, 22 (2005), 187-206. doi: 10.1016/j.anihpc.2004.05.004. Google Scholar

[10]

L. CaffarelliR. Hardt and L. Simon, Minimal surfaces with isolated singularities, Manuscripta Math., 48 (1984), 1-18. doi: 10.1007/BF01168999. Google Scholar

[11]

A. Callegari and A. Nachman, Some singular nonlinear equations arising in boundary layer theory, J. Math. Anal. Appl., 64 (1978), 96-105. doi: 10.1016/0022-247X(78)90022-7. Google Scholar

[12]

A. Callegari and A. Nachman, A nonlinear singular boundary value problem in the theory of pseudoplastic fluids, SIAM J. Appl. Math., 38 (1980), 275-281. doi: 10.1137/0138024. Google Scholar

[13]

I. Capuzzo DolcettaF. Leoni and A. Vitolo, On the inequality $F(x, D^2u) \geq f(u)+g(u)|Du|^q,$, Math. Ann., 365 (2016), 423-448. doi: 10.1007/s00208-015-1280-2. Google Scholar

[14]

M. Chhetri and S. Robinson, Existence and multiplicity of positive solutions for classes of singular elliptic PDEs, J. Math. Anal. Appl., 357 (2009), 176-182. doi: 10.1016/j.jmaa.2009.03.033. Google Scholar

[15]

M. M. Coclite, On a singular nonlinear Dirichlet problem Ⅲ, Nonlinear Anal. Theory Methods Appl., 21 (1993), 547-564. doi: 10.1016/0362-546X(93)90010-P. Google Scholar

[16]

M. M. Coclite and G. Palmieri, On a singular nonlinear dirichlet problem, Comm. Partial Differential Equations, 14 (1989), 1315-1327. doi: 10.1080/03605308908820656. Google Scholar

[17]

M. G. CrandallL. CaffarelliM. Kocan and A. Świech, On viscosity solutions of fully nonlinear equations with measurable ingredients, Comm. Pure Appl. Math., 49 (1996), 365-397. doi: 10.1002/(SICI)1097-0312(199604)49:4<365::AID-CPA3>3.0.CO;2-A. Google Scholar

[18]

M. G. CrandallH. Ishii and P.-L. Lions, User's guide to viscosity solutions of second order partial differential equations, Bull. Amer. Math. Soc., 27 (1992), 1-67. doi: 10.1090/S0273-0979-1992-00266-5. Google Scholar

[19]

M. G. CrandallP. H. Rabinowitz and L. Tartar, On a Dirichlet problem with a singular nonlinearity, Comm. Part. Diff. Eq., 2 (1977), 193-222. doi: 10.1080/03605307708820029. Google Scholar

[20]

L. DupaigneM. Ghergu and V. Rădulescu, Lane–Emden–Fowler equations with convection and singular potential, J. Math. Pures. Appl., 87 (2007), 563-581. doi: 10.1016/j.matpur.2007.03.002. Google Scholar

[21]

L. C. Evans, Partial Differential Equations, Amer. Math. Soc., Second Edition, Graduate Studies in Mathematics, 19. American Mathematical Society, Providence, RI, 2010. doi: 10.1090/gsm/019. Google Scholar

[22]

F. Faraci and D. Puglisi, A singular semilinear problem with dependence on the gradient, J. Differential Equations, 260 (2016), 3327-3349. doi: 10.1016/j.jde.2015.10.031. Google Scholar

[23]

P. FelmerA. Quaas and B. Sirakov, Existence and regularity results for fully nonlinear equations with singularities, Math. Ann., 354 (2012), 377-400. doi: 10.1007/s00208-011-0741-5. Google Scholar

[24]

P. Fok, Some Maximum Principles and Continuity Estimates for Fully Nonlinear Elliptic Equations of Second Order, Ph.D. Thesis, UCSB, 1996. Google Scholar

[25]

W. Fulks and J. S. Maybee, A singular non-linear equation, Osaka Math. J., 12 (1960), 1-19. Google Scholar

[26]

G. GaliseS. KoikeO. Ley and A. Vitolo, Entire solutions of fully nonlinear elliptic equations with a superlinear gradient term, J. Math. Anal. Appl., 441 (2016), 194-210. doi: 10.1016/j.jmaa.2016.03.083. Google Scholar

[27]

M. Ghergu and V. Rădulescu, Bifurcation for a class of singular elliptic problems with quadratic convection term, C. R. Math., 338 (2004), 831-836. doi: 10.1016/j.crma.2004.03.020. Google Scholar

[28]

M. Ghergu and V. Rădulescu, On a class of sublinear singular elliptic problems with convection term, J. Math. Anal. Appl., 311 (2005), 635-646. doi: 10.1016/j.jmaa.2005.03.012. Google Scholar

[29]

M. Ghergu and V. Rădulescu, Sublinear singular elliptic problems with two parameters, J. Differential Equations, 195 (2003), 520-536. doi: 10.1016/S0022-0396(03)00105-0. Google Scholar

[30]

E. Giarrusso, G. Porru, Problems for elliptic singular equations with a gradient term, Nonlinear Anal. Theory Methods Appl. 65 (2006), 107–128. doi: 10.1016/j.na.2005.08.007. Google Scholar

[31]

J. Hernández, F. J. Mancebo and J. M. Vega, Nonlinear singular elliptic problems: Recent results and open problems, Progr. Nonlinear Differential Equations Appl., Nonlinear elliptic and parabolic problems, Birkhuser, Basel, 64 (2005), 227–242. doi: 10.1007/3-7643-7385-7_12. Google Scholar

[32]

S. Koike and A. Świech, Existence of strong solutions of Pucci extremal equations with superlinear growth in $Du$, J. Fixed Point Theory Appl., 5 (2009), 291-304. doi: 10.1007/s11784-009-0106-9. Google Scholar

[33]

S. Koike and A. Świech, Weak Harnack inequality for fully nonlinear uniformly elliptic PDE with unbounded ingredients, J. Math. Soc. Japan, 61 (2009), 723-755. doi: 10.2969/jmsj/06130723. Google Scholar

[34]

S. Koike and A. Świech, Maximum principle and existence of $L^p$-viscosity solutions for fully nonlinear uniformly elliptic equations with measurable and quadratic term, Nonlinear Differ. Equ. Appl., 11 (2004), 491-509. doi: 10.1007/s00030-004-2001-9. Google Scholar

[35]

A. C. Lazer and P. J. McKenna, On a singular nonlinear elliptic boundary-value problem, Proc. Amer. Math. Soc., 111 (1991), 721-730. doi: 10.1090/S0002-9939-1991-1037213-9. Google Scholar

[36]

A. C. Lazer and P. J. McKenna, Asymptotic behavior of solutions of boundary blowup problems, Differ. Integral Equ., 7 (1994), 1001-1019. Google Scholar

[37]

P. J. McKenna and W. Reichel, Sign changing solutions to singular second order boundary value problems, Adv. Differential Equations, 6 (2001), 441-460. Google Scholar

[38]

M. Benrhouma, On a singular elliptic system with quadratic growth in the gradient, J. Math. Anal. Appl., 448 (2017), 1120-1146. doi: 10.1016/j.jmaa.2016.11.038. Google Scholar

[39]

J. A. Pelesko, Mathematical modeling of electrostatic MEMS with tailored dielectric properties, SIAM J. Appl. Math., 62 (2002), 888-908. doi: 10.1137/S0036139900381079. Google Scholar

[40]

G. Porru and A. Vitolo, Problems for elliptic singular equations with a quadratic gradient term, J. Math. Anal. Appl., 334 (2007), 467-486. doi: 10.1016/j.jmaa.2006.12.017. Google Scholar

[41]

A. Quaas and B. Sirakov, Principal eigenvalues and the Dirichlet problem for fully nonlinear elliptic operators, Adv. Math., 218 (2008), 105-135. doi: 10.1016/j.aim.2007.12.002. Google Scholar

[42]

W. Rudin, Real and Complex Analysis, Tata McGraw-Hill, Third Edition, McGraw-Hill Book Co., New York, 1987. Google Scholar

[43]

Sh ang Bin Cui, Positive solutions for Dirichlet problems associated to semilinear elliptic equations with singular nonlinearity, Nonlinear Anal. Theory Methods Appl., 21 (1993), 181-190. doi: 10.1016/0362-546X(93)90108-5. Google Scholar

[44]

B. Sirakov, Solvability of uniformly elliptic fully nonlinear PDE, Arch. Rational Mech. Anal., 195 (2010), 579-607. doi: 10.1007/s00205-009-0218-9. Google Scholar

[45]

J. Tyagi and R. B. Verma, A survey on the existence, uniqueness and regularity questions to fully nonlinear elliptic partial differential equations, Diff. Eq. Appl., 8 (2016), 135-205. doi: 10.7153/dea-08-09. Google Scholar

[46]

J. Tyagi and R. B. Verma, Positive solution of extremal Pucci's equations with singular and sublinear nonlinearity, Mediterr. J. Math., 14 (2017), Art. 148, 17 pp. doi: 10.1007/s00009-017-0950-6. Google Scholar

[47]

J. Tyagi and R. B. Verma, Lyapunov type inequality for extremal Pucci's equations, (submitted).Google Scholar

[48]

Z. Zhang and J. Yu, On a singular nonlinear Dirichlet problem with a convection term, SIAM J. Math. Anal., 32 (2000), 916-927. doi: 10.1137/S0036141097332165. Google Scholar

[49]

Z. Zhang, Nonexistence of positive classical solutions of a singular nonlinear Dirichlet problem with a convection term, Nonlinear Anal. Theory Methods Appl., 27 (1996), 957-961. doi: 10.1016/0362-546X(94)00367-Q. Google Scholar

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