$0^{\sharp}$ and elementary end extensions of $V_{\kappa}$

In this paper we prove that if $\kappa$ is a cardinal in $L[0^{\sharp}]$, then there is an inner model $M$ such that $M \models (V_{\kappa},\in)$ has no elementary end extension. In particular if $0^{\sharp}$ exists, then weak compactness is never downwards absolute. We complement the result with a lemma stating that any cardinal greater than $\aleph_1$ of uncountable cofinality in $L[0^{\sharp}]$ is Mahlo in every strict inner model of $L[0^{\sharp}]$.