Nonradial solutions of a Neumann Hénon equation on a ball (Q6583767)

The author investigates the existence of positive classical solutions to the Neumann Hénon problem. Precisely, one looks for a solution \(u>0\) of \(-\Delta u+u=|x|^\alpha u^{p-1}\) within the \(N\)-dimensional unit ball centered at the origin with even \(N\geq 4\). The parameters satisfy \(p>1\) and \(\alpha>0\). On the surface of the unit ball a homogeneous Neumann boundary condition is prescribed. The author shows that positive classical nonradial solutions to the aforementioned problem exist under a suitable condition on \(p\) depending on \(\alpha\) and \(N\) using a variational approach on convex cones.