科研之友
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摘要
We study unified asymptotic behavior of boundary blow-up solutions to semilinear elliptic equations of the form
{Delta u = b(x)f(u), x is an element of Omega
u(x) = infinity, x is an element of partial derivative Omega
where Omega subset of R-N is a bounded smooth domain, b(x) = 0 on partial derivative Omega is a non-negative function on Omega, f is non-negative on [0, infinity), and f(u)/u is increasing on (0, infinity), f (L(u)) = u(rho+r) L'(u) as u -> infinity with rho > 1 - r and 0 <= r <= 1, where L is an element of C-3([A, infinity)) satisfying lim(u ->infinity) L(u) = infinity, L' is an element of N RV-r. In our previous results, we obtained an explicit unified expression of boundary blow-up solutions when f is normalized regularly varying at infinity with index rho/(1-r) > 1 or grows at infinity faster than any power function. The effect of the mean curvature of the boundary in the second-order approximation was also discussed. In this paper, we will establish the second-order approximation of boundary blow-up solutions which depends on the distance of x from the boundary partial derivative Omega. Our analysis is based on the Karamata regular variation theory.
