August  2022, 42(8): 3861-3930. doi: 10.3934/dcds.2022036

Singular solutions of some elliptic equations involving mixed absorption-reaction

1. 

Laboratoire de Mathématiques et Physique Théorique, Université de Tours, 37200 Tours, France

2. 

Departamento de Matematicas, Pontifica Universidad Catolica de Chile, Casilla 307, Correo 2, Santiago, Chile

* Corresponding author: Laurent Véron

This article has been prepared with the support of the FONDECYT grant 1210241 for the three authors

Received  August 2021 Published  August 2022 Early access  April 2022

We study properties of nonnegative functions satisfying (E)$ \;-{\Delta} u+u^p-M|\nabla u|^q = 0 $ in a domain of $ {\mathbb R}^N $ when $ p>1 $, $ M>0 $ and $ 1<q<p $. We concentrate our analysis on the solutions of (E) with an isolated singularity, or in an exterior domain, or in the whole space. The existence of such solutions and their behaviours depend strongly on the values of the exponents $ p $ and $ q $ and in particular according to the sign of $ q-\frac{2p}{p+1} $, and when $ q = \frac{2p}{p+1} $, also on the value of the parameter $ M $ which becomes a key element. The description of the different behaviours is made possible by a sharp analysis of the radial solutions of (E).

Citation: Marie-Françoise Bidaut-Véron, Marta Garcia-Huidobro, Laurent Véron. Singular solutions of some elliptic equations involving mixed absorption-reaction. Discrete and Continuous Dynamical Systems, 2022, 42 (8) : 3861-3930. doi: 10.3934/dcds.2022036
References:
[1]

C. Bandle and E. Giarrusso, Boundary blow-up for semilinear elliptic equation with nonlinear gradient term, Adv. Diff. Equ., 1 (1996), 133-150. 

[2]

C. Bandle and M. Marcus, Sur les solutions maximales de problèmes elliptiques non linéaires: Bornes isopérimétriques et comportement asymptotique, C. R. Acad. Sci. Paris, 311 (1990), 91-93. 

[3]

R. Bellman, Stability Theory of Differential Equations, McGraw-Hill Book Co., Inc., New York-Toronto-London, 1953.

[4]

M. F. Bidaut-VéronM. Garcia-Huidobro and L. Véron, Local and global properties of solutions of quasilinear HamiltonJacobi equations, J. Funct. Anal., 267 (2014), 3294-3331.  doi: 10.1016/j.jfa.2014.07.003.

[5]

M. F. Bidaut-VéronM. Garcia-Huidobro and L. Véron, A priori estimates for elliptic equations with reaction terms involving the function and its gradient, Math. Ann., 378 (2020), 13-56.  doi: 10.1007/s00208-019-01872-x.

[6]

M. F. Bidaut-VéronM. Garcia-Huidobro and L. Véron, Radial solutions of scaling invariant nonlinear elliptic equations with mixed reaction terms, Discrete Cont. Dyn. Systems, 40 (2020), 933-982.  doi: 10.3934/dcds.2020067.

[7]

M. F. Bidaut-VéronM. Garcia-Huidobro and L. Véron, Measure data problems for a class of elliptic equations with mixed absorption-reaction, Adv. Nonlinear Stud., 21 (2021), 261-280.  doi: 10.1515/ans-2021-2124.

[8]

M. F. Bidaut-Véron, M. Garcia-Huidobro and L. Véron, Boundary singular solutions of a class of equations with mixed absorption-reaction, to appear, Calculus of Variations and Part. Diff. Equ., arXiv: 2007.16097v2.

[9]

M. F. Bidaut-Véron and H. Giacomini, A new dynamical approach of Emden-Fowler equations and systems, Adv. Diff. Equ., 15 (2010), 1033-1082. 

[10]

M. F. Bidaut-Véron and T. Raoux, Asymptotic of solutions of some nonlinear elliptic system, Comm. Part. Diff. Equ., 21 (1996), 1035-1086.  doi: 10.1080/03605309608821217.

[11]

L. BoccardoF. Murat and J. P. Puel, Résultats d'existence pour certains problèmes quasilinéaires, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 11 (2015), 213-255. 

[12]

P. FelmerA. Quass and B. Sirakov, Solvability of nonlinear elliptic equations with gradient terms, J. Diff. Equ., 254 (2013), 4327-4346.  doi: 10.1016/j.jde.2013.03.003.

[13]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Reprint of the 1998 edition, Classics in Mathematics, Springer-Verlag, Berlin, 2001.

[14]

J. B. Keller, On the solutions of ${\Delta} u = f(u)$, Comm. Pure Appl. Math., 10 (1957), 503-510.  doi: 10.1002/cpa.3160100402.

[15]

S. Kichenassamy and L. Véron, Singular solutions of the $p$-Laplace equation, Math. Ann, 275 (1986), 599-615.  doi: 10.1007/BF01459140.

[16]

M. Marcus and L. Véron, Maximal solutions of semilinear elliptic equations with locally integrable forcing term, Israel J. Math., 152 (2006), 333-348.  doi: 10.1007/BF02771990.

[17]

P. T. Nguyen, Isolated singularities of positive solutions with weighted gradient term, Anal. PDE, 9 (2016), 1671-1692.  doi: 10.2140/apde.2016.9.1671.

[18]

R. Osserman, On the inequality ${\Delta} u\geq f(u)$, Pacific J. Math., 7 (1957), 1641-1647. 

[19]

Y. Richard and L. Véron, Isotropic singularities of solutions of nonlinear inequalities, Ann. Inst. H. Poincaré Anal. Non Linéaire, 6 (1989), 37-72.  doi: 10.1016/s0294-1449(16)30331-6.

[20]

J. Serrin, Isolated singularities of solutions of quasilinear equations, Acta Math., 113 (1965), 219-240.  doi: 10.1007/BF02391778.

[21]

L. Véron, Comportement asymptotique des solutions d'équations elliptiques semi-linéaires dans ${\mathbb R}^N$, Ann. Mat. Pura Appl., 127 (1981), 25-50.  doi: 10.1007/BF01811717.

[22]

L. Véron, Singular solutions of some nonlinear elliptic equations, Nonlinear Anal. Theory, Methods Appl., 5 (1981), 225-242.  doi: 10.1016/0362-546X(81)90028-6.

[23]

L. Véron, Singular solutions of some nonlinear elliptic equations, Ann. Fac. Sci. Toulouse, 6 (1984), 1-31. 

[24]

L. Véron, Semilinear elliptic equations with uniform blow-up on the boundary, J. Anal. Math., 59 (1992), 231-250.  doi: 10.1007/BF02790229.

[25]

L. Véron, Local and Global Aspects of Quasilinear Degenerate Elliptic Equations, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2017. doi: 10.1142/9850.

show all references

References:
[1]

C. Bandle and E. Giarrusso, Boundary blow-up for semilinear elliptic equation with nonlinear gradient term, Adv. Diff. Equ., 1 (1996), 133-150. 

[2]

C. Bandle and M. Marcus, Sur les solutions maximales de problèmes elliptiques non linéaires: Bornes isopérimétriques et comportement asymptotique, C. R. Acad. Sci. Paris, 311 (1990), 91-93. 

[3]

R. Bellman, Stability Theory of Differential Equations, McGraw-Hill Book Co., Inc., New York-Toronto-London, 1953.

[4]

M. F. Bidaut-VéronM. Garcia-Huidobro and L. Véron, Local and global properties of solutions of quasilinear HamiltonJacobi equations, J. Funct. Anal., 267 (2014), 3294-3331.  doi: 10.1016/j.jfa.2014.07.003.

[5]

M. F. Bidaut-VéronM. Garcia-Huidobro and L. Véron, A priori estimates for elliptic equations with reaction terms involving the function and its gradient, Math. Ann., 378 (2020), 13-56.  doi: 10.1007/s00208-019-01872-x.

[6]

M. F. Bidaut-VéronM. Garcia-Huidobro and L. Véron, Radial solutions of scaling invariant nonlinear elliptic equations with mixed reaction terms, Discrete Cont. Dyn. Systems, 40 (2020), 933-982.  doi: 10.3934/dcds.2020067.

[7]

M. F. Bidaut-VéronM. Garcia-Huidobro and L. Véron, Measure data problems for a class of elliptic equations with mixed absorption-reaction, Adv. Nonlinear Stud., 21 (2021), 261-280.  doi: 10.1515/ans-2021-2124.

[8]

M. F. Bidaut-Véron, M. Garcia-Huidobro and L. Véron, Boundary singular solutions of a class of equations with mixed absorption-reaction, to appear, Calculus of Variations and Part. Diff. Equ., arXiv: 2007.16097v2.

[9]

M. F. Bidaut-Véron and H. Giacomini, A new dynamical approach of Emden-Fowler equations and systems, Adv. Diff. Equ., 15 (2010), 1033-1082. 

[10]

M. F. Bidaut-Véron and T. Raoux, Asymptotic of solutions of some nonlinear elliptic system, Comm. Part. Diff. Equ., 21 (1996), 1035-1086.  doi: 10.1080/03605309608821217.

[11]

L. BoccardoF. Murat and J. P. Puel, Résultats d'existence pour certains problèmes quasilinéaires, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 11 (2015), 213-255. 

[12]

P. FelmerA. Quass and B. Sirakov, Solvability of nonlinear elliptic equations with gradient terms, J. Diff. Equ., 254 (2013), 4327-4346.  doi: 10.1016/j.jde.2013.03.003.

[13]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Reprint of the 1998 edition, Classics in Mathematics, Springer-Verlag, Berlin, 2001.

[14]

J. B. Keller, On the solutions of ${\Delta} u = f(u)$, Comm. Pure Appl. Math., 10 (1957), 503-510.  doi: 10.1002/cpa.3160100402.

[15]

S. Kichenassamy and L. Véron, Singular solutions of the $p$-Laplace equation, Math. Ann, 275 (1986), 599-615.  doi: 10.1007/BF01459140.

[16]

M. Marcus and L. Véron, Maximal solutions of semilinear elliptic equations with locally integrable forcing term, Israel J. Math., 152 (2006), 333-348.  doi: 10.1007/BF02771990.

[17]

P. T. Nguyen, Isolated singularities of positive solutions with weighted gradient term, Anal. PDE, 9 (2016), 1671-1692.  doi: 10.2140/apde.2016.9.1671.

[18]

R. Osserman, On the inequality ${\Delta} u\geq f(u)$, Pacific J. Math., 7 (1957), 1641-1647. 

[19]

Y. Richard and L. Véron, Isotropic singularities of solutions of nonlinear inequalities, Ann. Inst. H. Poincaré Anal. Non Linéaire, 6 (1989), 37-72.  doi: 10.1016/s0294-1449(16)30331-6.

[20]

J. Serrin, Isolated singularities of solutions of quasilinear equations, Acta Math., 113 (1965), 219-240.  doi: 10.1007/BF02391778.

[21]

L. Véron, Comportement asymptotique des solutions d'équations elliptiques semi-linéaires dans ${\mathbb R}^N$, Ann. Mat. Pura Appl., 127 (1981), 25-50.  doi: 10.1007/BF01811717.

[22]

L. Véron, Singular solutions of some nonlinear elliptic equations, Nonlinear Anal. Theory, Methods Appl., 5 (1981), 225-242.  doi: 10.1016/0362-546X(81)90028-6.

[23]

L. Véron, Singular solutions of some nonlinear elliptic equations, Ann. Fac. Sci. Toulouse, 6 (1984), 1-31. 

[24]

L. Véron, Semilinear elliptic equations with uniform blow-up on the boundary, J. Anal. Math., 59 (1992), 231-250.  doi: 10.1007/BF02790229.

[25]

L. Véron, Local and Global Aspects of Quasilinear Degenerate Elliptic Equations, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2017. doi: 10.1142/9850.

Figure 1.  $ M>0 $, $ K<0\Longleftrightarrow p<\frac{N}{N-2} $
Figure 2.  $ M>0 $, $ K = 0\Longleftrightarrow p = \frac{N}{N-2} $
Figure 3.  $ M>m^* $, $ K>0\Longleftrightarrow p>\frac{N}{N-2} $
Figure 4.  $ M = m^* $, $ K>0\Longleftrightarrow p>\frac{N}{N-2} $
Figure 5.  $ 0<M<m^* $, $ K>0\Longleftrightarrow p>\frac{N}{N-2} $
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