A sup+inf inequality near R=0

• Autores: Wenxiong Chen, Congming Li
• Localización: Advances in mathematics, ISSN 0001-8708, Vol. 220, Nº 1, 2009, págs. 219-245
• Idioma: inglés
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• Resumen

  • Let O be a domain in R2, not necessarily bounded. Consider the semi-linear elliptic equation We prove that, for any compact subset K of O, there is a constant C, such that the inequality (1)holds for all solutions u.

    This type of inequality was first established by Brezis, Li, and Shafrir [H. Brezis, Y.Y. Li, I. Shafrir, A sup+inf inequality for some nonlinear elliptic equations involving exponential nonlinearities, J. Funct. Anal. 115 (1993) 344�358] under the assumption that R(x) is positive and bounded away from zero.

    It has become a useful tool in estimating the solutions of semi-linear elliptic equations either in Euclidean spaces or on Riemannian manifolds (see [H. Brezis, Y.Y. Li, I. Shafrir, A sup+inf inequality for some nonlinear elliptic equations involving exponential nonlinearities, J. Funct. Anal. 115 (1993) 344�358; C.-C. Chen, C.-S. Lin, A sharp sup+inf inequality for a nonlinear elliptic equation in R2, Comm. Anal. Geom. 6 (1998) 1�19; W. Chen, C. Li, Gaussian curvature in the negative case, Proc. Amer. Math. Soc. 131 (2003) 741�744; W. Chen, C. Li, Indefinite elliptic problems with critical exponent, in: Advances in Non-linear PDE and Related Areas, World Scientific, 1998, pp. 67�79; Y.Y. Li, I. Shafrir, Blow up analysis for solutions of -?u=Veu in dimension two, Indiana Univ. Math. J. 43 (1994) 1255�1270])