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A. Porretta, L. Véron / Journal of Functional Analysis 236 (2006) 581–591 591<br />

which extends a simi<strong>la</strong>r result in [9].<br />

Corol<strong>la</strong>ry 1. Assume that g satisfies (2.2)–(2.4). Then any solution of (2.29) satisfies (2.17) and<br />

verifies ∂ru>0 on ΓR,(R+r)/2.<br />

References<br />

[1] A. Aftalion, M. <strong>de</strong>l Pino, R. Letelier, Multip<strong>le</strong> boundary blow-up solutions for nonlinear elliptic equations, Proc.<br />

Roy. Soc. Edinburgh Sect. A 133 (2) (2003) 225–235.<br />

[2] C. Band<strong>le</strong>, M. Essen, On the solutions of quasilinear elliptic prob<strong>le</strong>ms with boundary blow-up, in: Sympos. Math.,<br />

vol. 35, Cambridge Univ. Press, 1994, pp. 93–111.<br />

[3] C. Band<strong>le</strong>, M. Marcus, Large solutions of <strong>semi</strong>linear elliptic equations: Existence, uniqueness and asymptotic behaviour,<br />

J. Anal. Math. 58 (1992) 9–24.<br />

[4] C. Band<strong>le</strong>, M. Marcus, Asymptotic behaviour of solutions and their <strong>de</strong>rivatives, for <strong>semi</strong>linear elliptic prob<strong>le</strong>ms<br />

with blowup on the boundary, Ann. Inst. H. Poincaré Anal. Non Linéaire 12 (2) (1995) 155–171.<br />

[5] H. Brezis, personal communication, January 2005.<br />

[6] Y. Du, Z. Guo, Boundary blow-up solutions and their applications in quasilinear elliptic equations, J. Anal. Math. 89<br />

(2003) 277–302.<br />

[7] Y. Du, Z. Guo, Uniqueness and <strong>la</strong>yer analysis for boundary blow-up solutions, J. Math. Pures Appl. 83 (6) (2004)<br />

739–763.<br />

[8] Y. Du, S. Yan, Boundary blow-up solutions with a spike <strong>la</strong>yer, J. Differential Equations 205 (1) (2004) 156–184.<br />

[9] B. Gidas, W.M. Ni, L. Nirenberg, Symmetry and re<strong>la</strong>ted properties via the maximum princip<strong>le</strong>, Comm. Math.<br />

Phys. 68 (1979) 209–243.<br />

[10] J.B. Kel<strong>le</strong>r, On solutions of u = f(u), Comm. Pure Appl. Math. 10 (1957) 503–510.<br />

[11] M. Marcus, L. Véron, Uniqueness and asymptotic behaviour of solutions with boundary blow-up for a c<strong>la</strong>ss of<br />

nonlinear elliptic equations, Ann. Inst. H. Poincaré 14 (1997) 237–274.<br />

[12] M. Marcus, L. Véron, Existence and uniqueness results for <strong>la</strong>rge solutions of general nonlinear elliptic equations,<br />

J. Evol. Equ. 3 (2003) 637–652.<br />

[13] P.J. McKenna, W. Reichel, W. Walter, Symmetry and multiplicity for nonlinear elliptic differential equations with<br />

boundary blow-up, Nonlinear Anal. 28 (7) (1997) 1213–1225.<br />

[14] R. Osserman, On the inequality u f(u), Pacific J. Math. 7 (1957) 1641–1647.<br />

[15] S.I. Pohožaev, The boundary value prob<strong>le</strong>m for equation U = U 2 , Dokl. Akad. Nauk SSSR 138 (1961) 305–308<br />

(in Russian).<br />

[16] A. Porretta, L. Véron, Asymptotic behaviour for the <strong>gradient</strong> of <strong>la</strong>rge solutions to some nonlinear elliptic equations,<br />

Adv. Nonlinear Stud., in press.<br />

[17] J. Serrin, A symmetry prob<strong>le</strong>m in potential theory, Arch. Rat. Mech. Anal. 43 (1971) 304–318.<br />

[18] L. Véron, Semilinear elliptic equations with uniform blow-up on the boundary, J. Anal. Math. 59 (1992) 231–250.

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