May  2019, 39(5): 2731-2742. doi: 10.3934/dcds.2019114

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

Received  May 2018 Revised  September 2018 Published  January 2019

Fund Project: The research of the second author was supported in part by NSERC.

In this paper we consider positive supersolutions of the nonlinear elliptic equation
$ - \Delta u = \rho(x) f(u)|\nabla u|^p, ~~~~ {\rm{ in }}~~~~ \Omega, $
where
$ 0\le p<1 $
,
$ \Omega $
is an arbitrary domain (bounded or unbounded) in
$ {\mathbb{R}}^N $
(
$ N\ge 2 $
),
$ f: [0, a_{f}) \rightarrow {\mathbb{R}}_{+} $
$ (0 < a_{f} \leq +\infty) $
is a non-decreasing continuous function and
$ \rho: \Omega \rightarrow \mathbb{R} $
is a positive function. Using the maximum principle we give explicit estimates on positive supersolutions
$ u $
at each point
$ x\in\Omega $
where
$ \nabla u\not\equiv0 $
in a neighborhood of
$ x $
. As applications, we discuss the dead core set of supersolutions on bounded domains, and also obtain Liouville type results in unbounded domains
$ \Omega $
with the property that
$ \sup_{x\in\Omega}dist (x, \partial\Omega) = \infty $
. In particular when
$ \rho(x) = |x|^\beta $
(
$ \beta\in {\mathbb{R}} $
) and
$ f(u) = u^q $
with
$ q+p>1 $
then every positive supersolution in an exterior domain is eventually constant if
$ (N-2)q+p(N-1)< N+\beta. $
Citation: Asadollah Aghajani, Craig Cowan. Explicit estimates on positive supersolutions of nonlinear elliptic equations and applications. Discrete & Continuous Dynamical Systems - A, 2019, 39 (5) : 2731-2742. doi: 10.3934/dcds.2019114
References:
[1]

S. AlarconJ. 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.  Google Scholar

[2]

S. AlarconJ. 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.  Google Scholar

[3]

S. AlarconJ. 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.  Google Scholar

[4]

S. AlarconJ. 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.  Google Scholar

[5]

D. ArcoyaC. De CosterL. 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.  Google Scholar

[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.  Google Scholar

[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].  Google Scholar

[8]

H. BerestyckiF. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

M. F. Bidaut-VeronM. 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.  Google Scholar

[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.  Google Scholar

[14]

M. A. Burgos-PerezJ. 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.  Google Scholar

[15]

G. Caristi and E. Mitidieri, Nonexistence of positive solutions of quasilinear equations, Adv. Diff. Equ., 2 (1997), 317-359.   Google Scholar

[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.  Google Scholar

[17]

P. FelmerA. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

show all references

References:
[1]

S. AlarconJ. 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.  Google Scholar

[2]

S. AlarconJ. 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.  Google Scholar

[3]

S. AlarconJ. 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.  Google Scholar

[4]

S. AlarconJ. 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.  Google Scholar

[5]

D. ArcoyaC. De CosterL. 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.  Google Scholar

[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.  Google Scholar

[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].  Google Scholar

[8]

H. BerestyckiF. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

M. F. Bidaut-VeronM. 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.  Google Scholar

[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.  Google Scholar

[14]

M. A. Burgos-PerezJ. 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.  Google Scholar

[15]

G. Caristi and E. Mitidieri, Nonexistence of positive solutions of quasilinear equations, Adv. Diff. Equ., 2 (1997), 317-359.   Google Scholar

[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.  Google Scholar

[17]

P. FelmerA. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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