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Global well-posedness for the 2D Boussinesq equations with a velocity damping term
Explicit estimates on positive supersolutions of nonlinear elliptic equations and applications
1. | School of Mathematics, Iran University of Science and Technology, Narmak, Tehran, Iran |
2. | Department of Mathematics, University of Manitoba, Winnipeg, Manitoba, R3T 2N2, Canada |
$ - \Delta u = \rho(x) f(u)|\nabla u|^p, ~~~~ {\rm{ in }}~~~~ \Omega, $ |
$ 0\le p<1 $ |
$ \Omega $ |
$ {\mathbb{R}}^N $ |
$ N\ge 2 $ |
$ f: [0, a_{f}) \rightarrow {\mathbb{R}}_{+} $ |
$ (0 < a_{f} \leq +\infty) $ |
$ \rho: \Omega \rightarrow \mathbb{R} $ |
$ u $ |
$ x\in\Omega $ |
$ \nabla u\not\equiv0 $ |
$ x $ |
$ \Omega $ |
$ \sup_{x\in\Omega}dist (x, \partial\Omega) = \infty $ |
$ \rho(x) = |x|^\beta $ |
$ \beta\in {\mathbb{R}} $ |
$ f(u) = u^q $ |
$ q+p>1 $ |
$ (N-2)q+p(N-1)< N+\beta. $ |
References:
[1] |
S. Alarcon, J. Garcia-Melian and A. Quaas,
Liouville type theorems for elliptic equations with gradient terms, Milan J. Math., 81 (2013), 171-185.
doi: 10.1007/s00032-013-0197-z. |
[2] |
S. Alarcon, J. Garcia-Melian and A. Quaas,
Nonexistence of positive supersolutions to some nonlinear elliptic problems, J. Math. Pures Appl., 99 (2013), 618-634.
doi: 10.1016/j.matpur.2012.10.001. |
[3] |
S. Alarcon, J. Garcia-Melian and A. Quaas,
Existence and non-existence of solutions to elliptic equations with a general convection term, Proc. Roy. Soc. Edinburgh Sect. A, 144 (2014), 225-239.
doi: 10.1017/S030821051200100X. |
[4] |
S. Alarcon, J. Garcia-Melian and A. Quaas,
Keller-Osserman type conditions for some elliptic problems with gradient terms, J. Differential Equations, 252 (2012), 886-914.
doi: 10.1016/j.jde.2011.09.033. |
[5] |
D. Arcoya, C. De Coster, L. Jeanjean and K. Tanaka,
Continuum of solutions for an elliptic problem with critical growth in the gradient, J. Funct. Anal., 268 (2015), 2298-2335.
doi: 10.1016/j.jfa.2015.01.014. |
[6] |
S. N. Armstrong and B. Sirakov,
Nonexistence of positive supersolutions of elliptic equations via the maximum principle, Comm. Part. Diff. Eqns., 36 (2011), 2011-2047.
doi: 10.1080/03605302.2010.534523. |
[7] |
S. N. Armstrong and B. Sirakov, Liouville results for fully nonlinear elliptic equations with
power growth nonlinearities, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), 10 (2011), 711–728,
arXiv: 1001.4489. [math.AP]. |
[8] |
H. Berestycki, F. Hamel and N. Nadirashvili,
The speed of propagation for KPP type problems Ⅰ. Periodic framework, J. Europ. Math. Soc., 7 (2005), 173-213.
doi: 10.4171/JEMS/26. |
[9] |
H. Berestycki, F. Hamel and L. Rossi, Liouville type results for semilinear elliptic equations in unbounded domains, Ann. Mat. Pura Appl., (4) 186 (2007), 469–507.
doi: 10.1007/s10231-006-0015-0. |
[10] |
M. F. Bidaut-Veron, M. Garcia-Huidobro and L. Veron, Estimates of solutions of elliptic equations with a source reaction term involving the product of the function and its gradient, available at https://arXiv.org/pdf/1711.11489.pdf Google Scholar |
[11] |
M. F. Bidaut-Veron,
Local and global behavior of solutions of quasilinear equations of EmdenFowler type, Arch. Rat. Mech. Anal., 107 (1989), 293-324.
doi: 10.1007/BF00251552. |
[12] |
M. F. Bidaut-Veron, M. Garcia-Huidobro and L. Veron,
Local and global behavior of solutions of quasilinear equations of Emden-Fowler type, Calc. Var. Part. Diff. Equ., 54 (2015), 3471-3515.
doi: 10.1007/s00526-015-0911-5. |
[13] |
I. Birindelli and F. Demengel,
Comparison principle and Liouville type results for singular fully nonlinear operators, Ann. Fac. Sci. Toulouse, 13 (2004), 261-287.
doi: 10.5802/afst.1070. |
[14] |
M. A. Burgos-Perez, J. Garcia Melian and A. Quaas,
Classification of supersolutions and Liouville theorems for some nonlinear elliptic problems, Discrete Contin. Dyn. Syst., 36 (2016), 4703-4721.
doi: 10.3934/dcds.2016004. |
[15] |
G. Caristi and E. Mitidieri,
Nonexistence of positive solutions of quasilinear equations, Adv. Diff. Equ., 2 (1997), 317-359.
|
[16] |
H. Chen and P. Felmer,
On Liouville type theorems for fully nonlinear elliptic equations with gradient term, J. Differential Equations, 255 (2013), 2167-2195.
doi: 10.1016/j.jde.2013.06.009. |
[17] |
P. Felmer, A. Quaas and B. Sirakov,
Solvability of nonlinear elliptic equations with gradient terms, J. Diff. Eq., 254 (2013), 4327-4346.
doi: 10.1016/j.jde.2013.03.003. |
[18] |
R. Filippucci,
Nonexistence of positive weak solutions of elliptic inequalities, Nonlinear Anal., 70 (2009), 2903-2916.
doi: 10.1016/j.na.2008.12.018. |
[19] |
L. Jeanjean and B. Sirakov,
Existence and multiplicity for elliptic problems with quadratic growth in the gradient, Comm. Part. Diff. Eq., 38 (2013), 244-264.
doi: 10.1080/03605302.2012.738754. |
[20] |
L. Rossi,
Non-existence of positive solutions of fully nonlinear elliptic bounded domains, Commun. Pure Appl. Anal., 7 (2008), 125-141.
doi: 10.3934/cpaa.2008.7.125. |
[21] |
J. Serrin and H. Zou,
CauchyLiouville and universal boundedness theorems for quasilinear elliptic equations and inequalities, Acta. Math., 189 (2002), 79-142.
doi: 10.1007/BF02392645. |
[22] |
L. Veron, Local and Global Aspects of Quasilinear Degenerate Elliptic Equations, Quasilinear elliptic singular problems. World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2017.
doi: 10.1142/9850. |
show all references
References:
[1] |
S. Alarcon, J. Garcia-Melian and A. Quaas,
Liouville type theorems for elliptic equations with gradient terms, Milan J. Math., 81 (2013), 171-185.
doi: 10.1007/s00032-013-0197-z. |
[2] |
S. Alarcon, J. Garcia-Melian and A. Quaas,
Nonexistence of positive supersolutions to some nonlinear elliptic problems, J. Math. Pures Appl., 99 (2013), 618-634.
doi: 10.1016/j.matpur.2012.10.001. |
[3] |
S. Alarcon, J. Garcia-Melian and A. Quaas,
Existence and non-existence of solutions to elliptic equations with a general convection term, Proc. Roy. Soc. Edinburgh Sect. A, 144 (2014), 225-239.
doi: 10.1017/S030821051200100X. |
[4] |
S. Alarcon, J. Garcia-Melian and A. Quaas,
Keller-Osserman type conditions for some elliptic problems with gradient terms, J. Differential Equations, 252 (2012), 886-914.
doi: 10.1016/j.jde.2011.09.033. |
[5] |
D. Arcoya, C. De Coster, L. Jeanjean and K. Tanaka,
Continuum of solutions for an elliptic problem with critical growth in the gradient, J. Funct. Anal., 268 (2015), 2298-2335.
doi: 10.1016/j.jfa.2015.01.014. |
[6] |
S. N. Armstrong and B. Sirakov,
Nonexistence of positive supersolutions of elliptic equations via the maximum principle, Comm. Part. Diff. Eqns., 36 (2011), 2011-2047.
doi: 10.1080/03605302.2010.534523. |
[7] |
S. N. Armstrong and B. Sirakov, Liouville results for fully nonlinear elliptic equations with
power growth nonlinearities, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), 10 (2011), 711–728,
arXiv: 1001.4489. [math.AP]. |
[8] |
H. Berestycki, F. Hamel and N. Nadirashvili,
The speed of propagation for KPP type problems Ⅰ. Periodic framework, J. Europ. Math. Soc., 7 (2005), 173-213.
doi: 10.4171/JEMS/26. |
[9] |
H. Berestycki, F. Hamel and L. Rossi, Liouville type results for semilinear elliptic equations in unbounded domains, Ann. Mat. Pura Appl., (4) 186 (2007), 469–507.
doi: 10.1007/s10231-006-0015-0. |
[10] |
M. F. Bidaut-Veron, M. Garcia-Huidobro and L. Veron, Estimates of solutions of elliptic equations with a source reaction term involving the product of the function and its gradient, available at https://arXiv.org/pdf/1711.11489.pdf Google Scholar |
[11] |
M. F. Bidaut-Veron,
Local and global behavior of solutions of quasilinear equations of EmdenFowler type, Arch. Rat. Mech. Anal., 107 (1989), 293-324.
doi: 10.1007/BF00251552. |
[12] |
M. F. Bidaut-Veron, M. Garcia-Huidobro and L. Veron,
Local and global behavior of solutions of quasilinear equations of Emden-Fowler type, Calc. Var. Part. Diff. Equ., 54 (2015), 3471-3515.
doi: 10.1007/s00526-015-0911-5. |
[13] |
I. Birindelli and F. Demengel,
Comparison principle and Liouville type results for singular fully nonlinear operators, Ann. Fac. Sci. Toulouse, 13 (2004), 261-287.
doi: 10.5802/afst.1070. |
[14] |
M. A. Burgos-Perez, J. Garcia Melian and A. Quaas,
Classification of supersolutions and Liouville theorems for some nonlinear elliptic problems, Discrete Contin. Dyn. Syst., 36 (2016), 4703-4721.
doi: 10.3934/dcds.2016004. |
[15] |
G. Caristi and E. Mitidieri,
Nonexistence of positive solutions of quasilinear equations, Adv. Diff. Equ., 2 (1997), 317-359.
|
[16] |
H. Chen and P. Felmer,
On Liouville type theorems for fully nonlinear elliptic equations with gradient term, J. Differential Equations, 255 (2013), 2167-2195.
doi: 10.1016/j.jde.2013.06.009. |
[17] |
P. Felmer, A. Quaas and B. Sirakov,
Solvability of nonlinear elliptic equations with gradient terms, J. Diff. Eq., 254 (2013), 4327-4346.
doi: 10.1016/j.jde.2013.03.003. |
[18] |
R. Filippucci,
Nonexistence of positive weak solutions of elliptic inequalities, Nonlinear Anal., 70 (2009), 2903-2916.
doi: 10.1016/j.na.2008.12.018. |
[19] |
L. Jeanjean and B. Sirakov,
Existence and multiplicity for elliptic problems with quadratic growth in the gradient, Comm. Part. Diff. Eq., 38 (2013), 244-264.
doi: 10.1080/03605302.2012.738754. |
[20] |
L. Rossi,
Non-existence of positive solutions of fully nonlinear elliptic bounded domains, Commun. Pure Appl. Anal., 7 (2008), 125-141.
doi: 10.3934/cpaa.2008.7.125. |
[21] |
J. Serrin and H. Zou,
CauchyLiouville and universal boundedness theorems for quasilinear elliptic equations and inequalities, Acta. Math., 189 (2002), 79-142.
doi: 10.1007/BF02392645. |
[22] |
L. Veron, Local and Global Aspects of Quasilinear Degenerate Elliptic Equations, Quasilinear elliptic singular problems. World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2017.
doi: 10.1142/9850. |
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