Advanced Search
Article Contents
Article Contents

Trace and boundary singularities of positive solutions of a class of quasilinear equations

  • *Corresponding author: Laurent Véron

    *Corresponding author: Laurent Véron

A Juan-Luis por su 75 cumpleaños. Cuarenta y seis años de amistad, respeto y admiración

Abstract Full Text(HTML) Related Papers Cited by
  • We study positive functions satisfying (E)$ \;-\Delta u+m|\nabla u|^q-u^p = 0 $ in a domain $ {\Omega} $ or in $ {\mathbb R}^{_N}_+ $ when $ p>1 $ and $ 1<q<2 $. We give sufficient conditions for the existence of a solution to (E) with a nonnegative measure $ \mu $ as boundary data; these conditions are expressed in terms of Bessel capacities on the boundary. We also study removability of boundary singular sets, and solutions with an isolated singularity on $ \partial\Omega $. The different results depend on two critical exponents for $ p = p_c: = \frac{N+1}{N-1} $ and for $ q = q_c: = \frac{N+1}{N} $, and on the sign of $ q-\frac{2p}{p+1} $

    Mathematics Subject Classification: Primary: 35J62, 35J66, 35J75; Secondary: 31C15.


    \begin{equation} \\ \end{equation}
  • 加载中
  • [1] D. R. Adams and L. I. Hedberg, Function Spaces and Potential Theory, Theory, Springer-Verlag, London-Berlin-Heidelberg-New York (1996). doi: 10.1007/978-3-662-03282-4.
    [2] 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.
    [3] 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, 378 (2020), 13-58.  doi: 10.1007/s00208-019-01872-x.
    [4] 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 (2020), 261-280.  doi: 10.1515/ans-2021-2124.
    [5] M.-F. Bidaut-Véron, M. Garcia-Huidobro and L. Véron, Boundary singular solutions of a class of equations with mixed absorption-reaction, Calc. Var. part. Diff. Equ., 61 (2022), Paper No. 113, 46 pp. doi: 10.1007/s00526-022-02200-z.
    [6] M.-F. Bidaut-Véron, M. Garcia-Huidobro and L. Véron, New results on the Chipot-Weissler quasilinear equation, In preparation.
    [7] M.-F. Bidaut-VéronG. HoangQ.-H. Nguyen and L. Véron, An elliptic semilinear equation with source term and boundary measure data: The supercritical case, J. Funct. Anal., 269 (2015), 1995-2017.  doi: 10.1016/j.jfa.2015.06.020.
    [8] M.-F. Bidaut-VéronA. C. Ponce and L. Véron, Isolated boundary singularities of semilinear elliptic equations, Calc. Var. Part. Diff. Equ., 40 (2011), 183-221.  doi: 10.1007/s00526-010-0337-z.
    [9] M.-F. Bidaut-Véron and L. Vivier, An elliptic semilinear equation with source term involving boundary measures: The subcritical case, Rev. Mat. Iberoamericana, 16 (2000), 477-513.  doi: 10.4171/RMI/281.
    [10] L. BoccardoF. Murat and J. P. Puel, Résultats d'existence pour certains problèmes elliptiques quasilinéaires, Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, 11 (1984), 213-235. 
    [11] M. Chipot and F. B. Weissler, Some blowup results for a nonlinear parabolic equation with a gradient term, SIAM J. Math. Anal., 20 (1989), 886-907.  doi: 10.1137/0520060.
    [12] J. L. Doob, Classical Potential Theory and Its Probabilistic Counterpart, Springer-Verlag, London-Berlin-Heidelberg-New York, 1984. doi: 10.1007/978-1-4612-5208-5.
    [13] B. Gidas and J. Spruck, Global and local behaviour of positive solutions of nonlinear elliptic equations, Comm. Pure Appl. Math., 34 (1981), 525–598. doi: 10.1002/cpa.3160340406.
    [14] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, London-Berlin-Heidelberg-New York, 1983. doi: 10.1007/978-3-642-61798-0.
    [15] A. Gmira and L. Véron, Boundary singularities of solutions of some nonlinear elliptic equations, Duke Math. J., 64 (1991), 271-324.  doi: 10.1215/S0012-7094-91-06414-8.
    [16] M. Marcus and L. Véron, The boundary trace of positive solutions of semilinear elliptic equations: The subcritical case, Arch. Rat. Mech. Anal., 144 (1998), 200-231.  doi: 10.1007/s002050050116.
    [17] M. Marcus and L. Véron, Removable singularities and boundary traces, J. Math. Pures Appl., 80 (2001), 879-900.  doi: 10.1016/S0021-7824(01)01209-0.
    [18] M. Marcus and L. Véron, The boundary trace and generalized boundary value problem for semilinear elliptic equations with coercive absorption, Comm. Pure Appl. Math., 56 (2003), 689-731.  doi: 10.1002/cpa.3037.
    [19] M. Marcus and L. Véron, Nonlinear Elliptic Equations Involving Measures, De Gruyter Series in Nonlinear Analysis and Applications, 21, De Gruyter, Berlin, 2014
    [20] L. Montoro, Harnack inequalities and qualitative properties for some quasilinear elliptic equations, Nonlinear Diff. Equ. Appl., 26 (2019), 33 pp. doi: 10.1007/s00030-019-0591-5.
    [21] P. T. Nguyen and L. Véron, Boundary singularities of solutions to elliptic viscous Hamilton-Jacobi equations, J. Funct. Anal., 263 (2012), 1487-1538.  doi: 10.1016/j.jfa.2012.05.019.
    [22] B. Opic and A. Kufner, Hardy-Type Inequalities, Pitman Research Notes in Mathematics Series, 219, Longman Scientific Technical, Harlow, 1990.
    [23] P. PolacikP. Quittner and Ph. Souplet, Singularity and decay estimates in superlinear problems via Liouville-type theorems. I. Elliptic equations and systems, Duke Math. J., 139 (2007), 555-579.  doi: 10.1215/S0012-7094-07-13935-8.
    [24] D. Ruiz, A priori estimates and existence of positive solutions for strongly nonlinear problems, J. Diff. Equ., 199 (2004), 96–114. doi: 10.1016/j.jde.2003.10.021.
    [25] J. Serrin, Y. Yan and H. Zou, A numerical study of existence and non-existence of the ground states and bifurcation for the equation of Chipot and Weissler, AHPCRC Preprint, 93-056, Univesrity of Minnesota, (1993).
    [26] J. Serrin and H. Zou, Existence and non-existence results for ground states of quasilinear elliptic equations, Arch. Rational Mech. Anal., 121 (1992), 101-130.  doi: 10.1007/BF00375415.
    [27] J. Serrin and H. Zou, Classification of positive solutions of quasilinear elliptic equations, Top. Meth. in Nonlinear Anal., 3 (1994), 1-25.  doi: 10.12775/TMNA.1994.001.
    [28] 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.
    [29] L. Véron, Local and Global Aspects of Quasilinear Degenerate Elliptic Equations, World Scientific, 2017. doi: 10.1142/9850.
    [30] F. X. Voirol, Étude de Quelques Équations Elliptiques Fortement Non Linéaires, PhD Thesis, University of Metz, 1994.
  • 加载中

Article Metrics

HTML views(1405) PDF downloads(216) Cited by(0)

Access History

Other Articles By Authors



    DownLoad:  Full-Size Img  PowerPoint