# American Institute of Mathematical Sciences

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 and Continuous Dynamical Systems, 2019, 39 (5) : 2731-2742. doi: 10.3934/dcds.2019114
##### 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 [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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##### 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 [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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