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 & Continuous Dynamical Systems - A, 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[11]

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

[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.  Google Scholar

[13]

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

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[18]

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

[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.  Google Scholar

[20]

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

[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.  Google Scholar

[22]

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

[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.   Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[11]

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

[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.  Google Scholar

[13]

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

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[18]

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

[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.  Google Scholar

[20]

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

[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.  Google Scholar

[22]

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

[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.   Google Scholar

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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