February  2020, 40(2): 933-982. doi: 10.3934/dcds.2020067

Radial solutions of scaling invariant nonlinear elliptic equations with mixed reaction terms

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 de Chile

Received  March 2019 Revised  August 2019 Published  November 2019

We study global properties of positive radial solutions of $ -\Delta u = u^p+M\left |{\nabla u}\right |^{\frac{2p}{p+1}} $ in $ \mathbb R^N $ where $ p>1 $ and $ M $ is a real number. We prove the existence or the non-existence of ground states and of solutions with singularity at $ 0 $ according to the values of $ M $ and $ p $.

Citation: Marie-Françoise Bidaut-Véron, Marta Garcia-Huidobro, Laurent Véron. Radial solutions of scaling invariant nonlinear elliptic equations with mixed reaction terms. Discrete and Continuous Dynamical Systems, 2020, 40 (2) : 933-982. doi: 10.3934/dcds.2020067
References:
[1]

S. AlarcónJ. García-Melián 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.

[2]

L. R. Anderson and W. Leighton, Liapunov functions for autonomous systems of second order, J. Math. Anal. Appl., 23 (1968), 645-664.  doi: 10.1016/0022-247X(68)90145-5.

[3]

M. F. Bidaut-Véron, Local and global behaviour of solutions of quasilinear elliptic equations of Emden-Fowler type, Arch. Rat. Mech. Anal., 107 (1989), 293-324.  doi: 10.1007/BF00251552.

[4]

M. F. Bidaut-Véron, Self-similar solutions of the $p$ -Laplace heat equation: The fast diffusion case, Pacific Journal of Maths, 227 (2006), 201-269.  doi: 10.2140/pjm.2006.227.201.

[5]

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

[6]

M. F. Bidaut-VéronM. Garcia-Huidobro and L. Véron, Estimates of solutions of elliptic equations with a source reaction term involving the product of the function and its gradient, Duke Math. J., 168 (2019), 1487-1537.  doi: 10.1215/00127094-2018-0067.

[7]

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. Annalen, (2019), 1-44.  doi: 10.1007/s00208-019-01872-x.

[8]

M. F. Bidaut-Véron and P. Grillot, Asymptotic behaviour of elliptic systems with mixed absorption and source terms, Asymptotic Anal., 19 (1999), 117-147. 

[9]

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

[10]

M. F. Bidaut-Véron and S. Pohozaev, Local and global behavior of solutions of quasilinear equations of Emden-Fowler type, Journal d'Analyse Mathématique, 84 (2001), 1-49.  doi: 10.1007/BF02788105.

[11]

M. F. Bidaut-Véron and Th. Raoux, Asymptotic of solutions of some nonlinear elliptic systems, Comm. Part. Diff. Equ., 21 (1996), 1035-1086. 

[12]

C. Chicone and J. H. Tian, On general properties of quadratic systems, Amer. Math. Monthly, 89 (1982), 167-178.  doi: 10.1080/00029890.1982.11995405.

[13]

M. Chipot, On a class of nonlinear elliptic equations, Partial Differential Equations, Part 1, 2 (Warsaw, 1990), 27 (1992), 75-80. 

[14]

M. Chipot and F. Weissler, Some blow-up results for a nonlinear parabolic equation with a gradient term, S.I.A.M. J. of Num. Anal., 20 (1989), 886-907.  doi: 10.1137/0520060.

[15]

M. Fila, Remarks on blow-up for a nonlinear parabolic equation with a gradient term, Proc. A.M.S., 111 (1991), 795-801.  doi: 10.1090/S0002-9939-1991-1052569-9.

[16]

M. Fila and P. Quittner, Radial positive solutions for a semilinear elliptic equation with a gradient term, Adv. Math. Sci. Appl, 2 (1993), 39-45. 

[17]

J. H. Hubbard and B. H. West, Differential Equations: A Dynamical Systems Approach: Ordinary Differential Equations, Texts in Applied Mathematics, 5, Springer-Verlag Berlin Heidelberg, 1991.

[18]

Y. Kuznetsov, Elements of Applied Bifurcation Theory, Applied Math Sci., 110, Springer-Verlag, 2004. doi: 10.1007/978-1-4757-3978-7.

[19]

J. Serrin and H. Zou, Existence and non-existence results for ground states of quasilinear elliptic equations, Arch. Rat. Mech. Anal., 121 (1992), 101-130.  doi: 10.1007/BF00375415.

[20]

Ph. Souplet, Recent results and open problems on parabolic equations with gradient nonlinearities, Electron. J. Differential Equations, 20 (2001), 1-19. 

[21]

J. L. Vazquez and L. Véron, Singularities of elliptic equations with an exponential nonlinearity, Math. Ann., 269 (1984), 119-135.  doi: 10.1007/BF01456000.

[22]

F. X. Voirol, Thèse de Doctorat, Université de Metz, 1994.

[23]

F. X. Voirol, Coexistence of singular and regular solutions for the equation of Chipot and Weissler, Acta Math. Univ-Comenianae, 65 (1996), 53-64. 

show all references

References:
[1]

S. AlarcónJ. García-Melián 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.

[2]

L. R. Anderson and W. Leighton, Liapunov functions for autonomous systems of second order, J. Math. Anal. Appl., 23 (1968), 645-664.  doi: 10.1016/0022-247X(68)90145-5.

[3]

M. F. Bidaut-Véron, Local and global behaviour of solutions of quasilinear elliptic equations of Emden-Fowler type, Arch. Rat. Mech. Anal., 107 (1989), 293-324.  doi: 10.1007/BF00251552.

[4]

M. F. Bidaut-Véron, Self-similar solutions of the $p$ -Laplace heat equation: The fast diffusion case, Pacific Journal of Maths, 227 (2006), 201-269.  doi: 10.2140/pjm.2006.227.201.

[5]

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

[6]

M. F. Bidaut-VéronM. Garcia-Huidobro and L. Véron, Estimates of solutions of elliptic equations with a source reaction term involving the product of the function and its gradient, Duke Math. J., 168 (2019), 1487-1537.  doi: 10.1215/00127094-2018-0067.

[7]

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. Annalen, (2019), 1-44.  doi: 10.1007/s00208-019-01872-x.

[8]

M. F. Bidaut-Véron and P. Grillot, Asymptotic behaviour of elliptic systems with mixed absorption and source terms, Asymptotic Anal., 19 (1999), 117-147. 

[9]

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

[10]

M. F. Bidaut-Véron and S. Pohozaev, Local and global behavior of solutions of quasilinear equations of Emden-Fowler type, Journal d'Analyse Mathématique, 84 (2001), 1-49.  doi: 10.1007/BF02788105.

[11]

M. F. Bidaut-Véron and Th. Raoux, Asymptotic of solutions of some nonlinear elliptic systems, Comm. Part. Diff. Equ., 21 (1996), 1035-1086. 

[12]

C. Chicone and J. H. Tian, On general properties of quadratic systems, Amer. Math. Monthly, 89 (1982), 167-178.  doi: 10.1080/00029890.1982.11995405.

[13]

M. Chipot, On a class of nonlinear elliptic equations, Partial Differential Equations, Part 1, 2 (Warsaw, 1990), 27 (1992), 75-80. 

[14]

M. Chipot and F. Weissler, Some blow-up results for a nonlinear parabolic equation with a gradient term, S.I.A.M. J. of Num. Anal., 20 (1989), 886-907.  doi: 10.1137/0520060.

[15]

M. Fila, Remarks on blow-up for a nonlinear parabolic equation with a gradient term, Proc. A.M.S., 111 (1991), 795-801.  doi: 10.1090/S0002-9939-1991-1052569-9.

[16]

M. Fila and P. Quittner, Radial positive solutions for a semilinear elliptic equation with a gradient term, Adv. Math. Sci. Appl, 2 (1993), 39-45. 

[17]

J. H. Hubbard and B. H. West, Differential Equations: A Dynamical Systems Approach: Ordinary Differential Equations, Texts in Applied Mathematics, 5, Springer-Verlag Berlin Heidelberg, 1991.

[18]

Y. Kuznetsov, Elements of Applied Bifurcation Theory, Applied Math Sci., 110, Springer-Verlag, 2004. doi: 10.1007/978-1-4757-3978-7.

[19]

J. Serrin and H. Zou, Existence and non-existence results for ground states of quasilinear elliptic equations, Arch. Rat. Mech. Anal., 121 (1992), 101-130.  doi: 10.1007/BF00375415.

[20]

Ph. Souplet, Recent results and open problems on parabolic equations with gradient nonlinearities, Electron. J. Differential Equations, 20 (2001), 1-19. 

[21]

J. L. Vazquez and L. Véron, Singularities of elliptic equations with an exponential nonlinearity, Math. Ann., 269 (1984), 119-135.  doi: 10.1007/BF01456000.

[22]

F. X. Voirol, Thèse de Doctorat, Université de Metz, 1994.

[23]

F. X. Voirol, Coexistence of singular and regular solutions for the equation of Chipot and Weissler, Acta Math. Univ-Comenianae, 65 (1996), 53-64. 

Figure 1.  $ M>0 $, $ K>0 $
Figure 2.  $ M<0 $, $ K\geq 0 $
Figure 3.  $ -\mu^*<M<0 $, $ K<0 $
Figure 4.  $ M = -\mu^* $, $ K<0 $
Figure 5.  $ M<-\mu^* $, $ K<0 $
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