Existence of Positive Solutions of a Semilinear Nondivergence Form Elliptic Equation in a Conical Domain

摘要

We consider the uniformly elliptic equation ∑from n to _(i,j=1)a_(ij)(x)(a~2u)/(ax_jax_i)+∑from n to (i=1) a_i(x) (au)/(ax_i)+|u|~σu=0, σ=const ＞0, (1) in the cone K={r, ω ∈R~n|ω∈Ω,r＞0,} where (r,ω) are the polar coordinates in the space R~n and Ω is a domain of the class C~(1,1) on the unit sphere in R~n. We study the existence or nonexistence of a positive solution of Eq. (1) in the cone K_R=K ∩ {|x| > R} such that u|aK = 0.