Recall one can write the Laplacian in $N$ dimensions as
$ \Delta u = u_{rr} + \frac{N-1}{r} u_r + \frac{1}{r^2} \Delta_\theta u$ where $ \Delta_\theta$ is the Laplace Beltrami operator on $ S^{N-1}$.
I am interested in the existence of positive solutions of the following equation:
$L(u) = u^p$ in $ \Omega$ with $ u=0$ on $ \partial \Omega$ where
$L(u) = - u_{rr} - \frac{N-1}{r} u_{r} - \frac{C}{r^2} \Delta_\theta u$. Here $ \Omega$ is a bounded domain in $ R^N$.
Here $C$ is a constant strictly between zero and one.
QUESTION
Is it obvious that for that there is a positive solution of the above PDE for $ 1 < p < \frac{N+2}{N-2}$ ?
