Singular semilinear elliptic equations in the half-space

Abstract

We show that equation $x_{N}^{q}\Delta u+u^{p-1}=0$ on the half-space $Y=\mathbf{R}^{N-1}\times\left(0,\infty\right)$ and on some of its subsets has a ground state solution for $q=N-\frac{p\left(N-2\right)}{2},\; p\;\epsilon\left(2,2*\right)$. For N $\geq$ 3 the end point cases p=2 and p=2{*} correspond to eh Hardy inequality and the limit exponent Sobolev inequality respectively. For N=2 the problem can be interpreted in terms of Laplace-Beltrami operator on the hyperbolic half-plane.