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In this talk, we discuss a Lane-Emden type equation of higher order with a supercritical polynomial non-linearity
\begin{equation*}
(-1)^m \Delta^m u = |u|^{p-2} u
\end{equation*}
\(m \in \mathbb N\) with \(n > 2m\) and \(p> p^\ast = \frac{2n} {n-2m}\).
For \(m=1\) this equation was proposed by Lane to study the interior of a star.
We will discuss some new and well-known results for these equations. In the center of our attention will be a new monotonicity formula for the triharmonic case \(m=3\) for certain combinations of \(p\) and \(n\). This formula will allow us to bound the Hausdorff-dimension of the singular set of stationary solutions.
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