I encountered a sentence which says it is well known that problem $$ \begin{cases} -\Delta u =|u|^{p-1} u & in \,\, \Omega \\ u=0 & on \,\, \partial \Omega \end{cases} $$ have a solution for $1<p<\frac{N+2}{N-2}$ and doesn't have any solution for $p>\frac{N+2}{N-2}$.

The existence is okay by mountain pass theorem. But how about the non-existence case? Can some one give a reference or hint?

Thanks alot indeed.