A sup plus inf inequality near R=0

Let Omega be a domain in R-2, not necessarily bounded. Consider the semi-linear elliptic equation -Delta u = R(x)e(u), x is an element of Omega. We prove that, for any compact subset K of Omega, there is a constant C, such that the inequality (sup)(K) (u) + (inf u)(Omega) <= C holds for all solutions u. This type of inequality was first established by Brezis, Li, and Shaftir [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 R-2, 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 -Delta u = V e(u) in dimension two, Indiana Univ. Math. J. 43 (1994) 1255-1270]). In Brezis, Li, and Shafrir's result, the constant C depends on the lower bound of the function R(x). In this paper, we remove this restriction and extend the inequality to the case where R(x) is allowed to have zeros, so that it can be applied to obtain a priori estimates for a broader class of equations, as we will illustrate in the last section. The key to prove this inequality is the analysis of asymptotic decay near infinity of solutions for a corresponding limiting equation in R-2, which is interesting and useful in its own right. (c) 2008 Elsevier Inc. All rights reserved.