## A slight variant of the Lane-Emden equation.

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 $1 < p < \frac{N+2}{N-2}$ ?

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