Boundary trace of positive solutions of nonlinear elliptic inequalities
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 5, Volume 3 (2004) no. 3, pp. 481-533.

We develop a new method for proving the existence of a boundary trace, in the class of Borel measures, of nonnegative solutions of $-\Delta u+g\left(x,u\right)\ge 0$ in a smooth domain $\Omega$ under very general assumptions on $g$. This new definition which extends the previous notions of boundary trace is based upon a sweeping technique by solutions of Dirichlet problems with measure boundary data. We also prove a boundary pointwise blow-up estimate of any solution of such inequalities in terms of the Poisson kernel. If the nonlinearity is very degenerate near the boundary, for example if $g\left(x,u\right)\approx exp\left(-{\rho }_{\partial \Omega }^{-1}\left(x\right)\right){u}^{q}$, we exhibit a new full boundary blow-up phenomenon.

Classification: 35K60
Marcus, Moshe 1; Véron, Laurent 2

1 Department of Mathematics, Israel Institute of Technology Technion, Haifa 32000, Israel
2 Université François Rabelais Tours 37200, France
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Marcus, Moshe; Véron, Laurent. Boundary trace of positive solutions of nonlinear elliptic inequalities. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 5, Volume 3 (2004) no. 3, pp. 481-533. http://archive.numdam.org/item/ASNSP_2004_5_3_3_481_0/

[1] P. Baras - M. Pierre, Singularités éliminables pour des équations semi-linéaires, Ann. Inst. Fourier (Grenoble) 34 (1984), 185-206. | Numdam | MR | Zbl

[2] P. Baras - J. Goldstein, The heat equation with a singular potential, Trans. Amer. Math. Soc. 284 (1984), 121-139. | MR | Zbl

[3] Ph. Benilan - H. Brezis, Nonlinear problems related to the Thomas-Fermi equation, unpublished paper, see [6]. | MR

[4] M. F. Bidaut-Véron - L. Vivier, An elliptic semilinear equation with source term involving boundary measures: the subcritical case, Rev. Mat. Iberoamericana 16 (2000), 477-513. | MR | Zbl

[5] H. Brezis, Une équation semi-linéaire avec conditions aux limites dans ${L}^{1}$, unpublished paper. See also [32]-Chap. 4.

[6] H. Brezis, Some variational problems of the Thomas-Fermi type, in “Variational Inequalities”, R. W. Cottle, F. Giannessi and J.-L. Lions (eds.), Wiley, Chichester (1980), 53-73. | MR | Zbl

[7] X. Cabre, Extremal solutions and instantaneous complete blow-up for elliptic and parabolic problems, preprint. | MR

[8] R. Dautray - J. L. Lions, “Analyse Mathématique et Calcul Numérique”, Masson, Paris, 1987. | MR

[9] J. Doob, “Classical Potential Theory and its Probabilistic Counterpart”, Springer-Verlag, Berlin-New York, 1984. | MR | Zbl

[10] E. B. Dynkin - S. E. Kuznetsov, Trace on the boundary for solutions of nonlinear differential equations, Trans. Amer. Math. Soc. 350 (1998), 4499-4519. | MR | Zbl

[11] E. B. Dynkin - S. E. Kuznetsov, Solutions of nonlinear differential equtions on a Riemannian manifold and their trace on the Martin boundary, Trans. Amer. Math. Soc. 350 (1998), 4521-4552. | MR | Zbl

[12] E. B. Dynkin - S. E. Kuznetsov, Fine topology and fine trace on the boundary associated with a class of quasilinear differential equations, Comm. Pure Appl. Math. 51 (1998), 897-936. | MR | Zbl

[13] J. Fabbri - J. R. Licois, Behavior at boundary of solutions of a weakly superlinear elliptic equation, Adv. Nonlinear Stud. 2 (2002), 147-176. | MR | Zbl

[14] D. Gilbarg - N. S. Trudinger, “Partial Differential Equations of Second Order”, 2nd Ed. Springer-Verlag, Berlin-New York, 1983. | MR | Zbl

[15] A. Gmira - L. Véron, Boundary singularities of solutions of some nonlinear elliptic equations, Duke Math. J. 64 (1991), 271-324. | MR | Zbl

[16] M. Grillot - L. Véron, Boundary trace of solutions of the Prescribed Gaussian curvature equation, Proc. Roy. Soc. Edinburgh 130 A (2000), 1-34. | MR | Zbl

[17] I. Iscoe, On the support of measure-valued critical branching Brownian motion, Ann. Prob. 16 (1988), 200-221. | MR | Zbl

[18] J. B. Keller, On solutions of $\Delta u=f\left(u\right)$, Comm. Pure Appl. Math. 10 (1957), 503-510. | MR | Zbl

[19] J. F. Le Gall, Les solutions positives de $\Delta u={u}^{{}^{2}}$ dans le disque unité, C.R. Acad. Sci. Paris 317 Ser. I (1993), 873-878. | MR | Zbl

[20] J. F. Le Gall, The brownian snake and solutions of $\Delta u={u}^{{}^{2}}$ in a domain, Probab. Theory Related Fields 102 (1995), 393-432. | MR | Zbl

[21] M. Marcus - L. Véron, Uniqueness and asymptotic behaviour of solutions with boundary blow-up for a class of nonlinear elliptic equations, Ann. Inst. H. Poincaré 14 (1997), 237-274. | Numdam | MR | Zbl

[22] M. Marcus - L. Véron, Traces au bord des solutions positives d'équations elliptiques non-linéaires, C.R. Acad. Sci. Paris 321 Ser. I (1995), 179-184. | MR | Zbl

[23] M. Marcus - L. Véron, Traces au bord des solutions positives d'équations elliptiques et paraboliques non-linéaires: résultats d'existence et d'unicité, C.R. Acad. Sci. Paris 323 Ser. I (1996), 603-608. | MR | Zbl

[24] M. Marcus - L. Véron, The boundary trace of positive solutions of semilinear elliptic equations: the subcritical case, Arch. Ration. Mech. Anal. 144 (1998), 201-231. | MR | Zbl

[25] M. Marcus - L. Véron, The boundary trace of positive solutions of semilinear elliptic equations: the supercritical case, J. Math. Pures Appl. 77, 481-524 (1998). | MR | Zbl

[26] M. Marcus - L. Véron, Removable singularities and boundary traces, J. Math. Pures Appl. 80 (2001), 879-900. | MR | Zbl

[27] M. Marcus - L. Véron, The boundary trace and generalyzed boundary value problem for semilinear elliptic equations with coercive absorption, Comm. Pure Appl. Math. 56 (2003), 0689-0731. | MR | Zbl

[28] M. Marcus - L. Véron, Initial trace of positive solutions to semilinear parabolic inequalities, Adv. Nonlinear Studies 2 (2002), 395-436. | MR | Zbl

[29] A. Ratto - M. Rigoli - L. Véron, Scalar curvature and conformal deformation of hyperbolic space, J. Funct. Anal. 121 (1994), 15-77. | MR | Zbl

[30] Y. Richard - L. Véron, Isotropic singularities of nonlinear elliptic inequalities, Ann. Inst. H. Poincaré 6 (1989), 37-72. | Numdam | MR | Zbl

[31] J. L. Vazquez, An a priori interior estimate for the solution of a nonlinear problem representing weak diffusion, Nonlinear Anal. 5 (1981), 119-135. | MR | Zbl

[32] L. Véron, “Singularities of Solutions of Second Order Quasilinear Equations”, Pitman Research Notes in Math. 353, Addison Wesley Longman Inc., 1996. | MR | Zbl