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Uniqueness and asymptotic behavior of solutions of a biharmonic equation with supercritical exponent
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 |
$ \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*} $ |
$ k,f $ |
$ H, $ |
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. |
[2] |
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. |
[3] |
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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[9] |
J. Busca, M. 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. |
[10] |
L. Caffarelli, R. Hardt and L. Simon,
Minimal surfaces with isolated singularities, Manuscripta Math., 48 (1984), 1-18.
doi: 10.1007/BF01168999. |
[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. |
[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. |
[13] |
I. Capuzzo Dolcetta, F. 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. |
[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. |
[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. |
[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. |
[17] |
M. G. Crandall, L. Caffarelli, M. 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. |
[18] |
M. G. Crandall, H. 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. |
[19] |
M. G. Crandall, P. 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. |
[20] |
L. Dupaigne, M. 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. |
[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. |
[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. |
[23] |
P. Felmer, A. 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. |
[24] |
P. Fok, Some Maximum Principles and Continuity Estimates for Fully Nonlinear Elliptic Equations of Second Order, Ph.D. Thesis, UCSB, 1996. |
[25] |
W. Fulks and J. S. Maybee,
A singular non-linear equation, Osaka Math. J., 12 (1960), 1-19.
|
[26] |
G. Galise, S. Koike, O. 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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[36] |
A. C. Lazer and P. J. McKenna,
Asymptotic behavior of solutions of boundary blowup problems, Differ. Integral Equ., 7 (1994), 1001-1019.
|
[37] |
P. J. McKenna and W. Reichel,
Sign changing solutions to singular second order boundary value problems, Adv. Differential Equations, 6 (2001), 441-460.
|
[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. |
[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. |
[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. |
[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. |
[42] |
W. Rudin, Real and Complex Analysis, Tata McGraw-Hill, Third Edition, McGraw-Hill Book Co., New York, 1987. |
[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. |
[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. |
[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. |
[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. |
[47] |
J. Tyagi and R. B. Verma, Lyapunov type inequality for extremal Pucci's equations, (submitted). |
[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. |
[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. |
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. |
[2] |
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. |
[3] |
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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[9] |
J. Busca, M. 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. |
[10] |
L. Caffarelli, R. Hardt and L. Simon,
Minimal surfaces with isolated singularities, Manuscripta Math., 48 (1984), 1-18.
doi: 10.1007/BF01168999. |
[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. |
[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. |
[13] |
I. Capuzzo Dolcetta, F. 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. |
[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. |
[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. |
[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. |
[17] |
M. G. Crandall, L. Caffarelli, M. 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. |
[18] |
M. G. Crandall, H. 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. |
[19] |
M. G. Crandall, P. 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. |
[20] |
L. Dupaigne, M. 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. |
[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. |
[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. |
[23] |
P. Felmer, A. 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. |
[24] |
P. Fok, Some Maximum Principles and Continuity Estimates for Fully Nonlinear Elliptic Equations of Second Order, Ph.D. Thesis, UCSB, 1996. |
[25] |
W. Fulks and J. S. Maybee,
A singular non-linear equation, Osaka Math. J., 12 (1960), 1-19.
|
[26] |
G. Galise, S. Koike, O. 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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[36] |
A. C. Lazer and P. J. McKenna,
Asymptotic behavior of solutions of boundary blowup problems, Differ. Integral Equ., 7 (1994), 1001-1019.
|
[37] |
P. J. McKenna and W. Reichel,
Sign changing solutions to singular second order boundary value problems, Adv. Differential Equations, 6 (2001), 441-460.
|
[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. |
[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. |
[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. |
[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. |
[42] |
W. Rudin, Real and Complex Analysis, Tata McGraw-Hill, Third Edition, McGraw-Hill Book Co., New York, 1987. |
[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. |
[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. |
[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. |
[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. |
[47] |
J. Tyagi and R. B. Verma, Lyapunov type inequality for extremal Pucci's equations, (submitted). |
[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. |
[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. |
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