Positive solutions to the Neumann problem for nonlinear elliptic equations in conic domains (Q1280335)

The author establishes conditions for the absence of solutions to the Neumann problem \[ \Delta u + | x| ^\sigma u^p = 0 \quad\text{in }K,\qquad\dfrac{\partial u}{\partial \nu} = 0 \quad\text{on }\partial K\backslash \{0\} \] in the cone \(K = \{x\in \mathbb{R}^n\: 0<r=| x| <+\infty, \theta\in \Omega\}\), where \(\sigma\in \mathbb{R}\), \(p>1\), \(\nu\) is the outer unit normal to \(\partial K\), \((r,\theta) = (r,\theta_1,\dots,\theta_{n-1})\) are polar coordinates in \(\mathbb{R}^n\), \(\Omega\) is a domain on the unit sphere \(S^{n-1}\) with a boundary of the class \(C^1\).