Positive solutions of semilinear equations in cones

Abstract:

In this paper we consider the problem of finding a positive solution of the equation $\Delta u + |x{|^\nu }{u^{(n + 2 + 2\nu )/(n - 2)}} = 0$ in a cone $\mathcal {C}$, with zero boundary data. We are only interested in solutions that are regular at infinity (i.e. such that $u(x) = o(|x{|^{2 - n}})$, as $\mathcal {C} \ni x \to \infty$). We will always assume that $\nu > - 2$. We show that the existence of a solution depends on the sign of $\nu$ and also on the shape of the cone $\mathcal {C}$.