Abstrakt
Let $\kappa < \lambda$ be regular cardinals. We say that an embedding $j: V\to M$ with critical point $\kappa$ is \emph{$\lambda$-tall} if $\lambda<j(\kappa)$ and $M$ is closed under $\kappa$-sequences in $V$.
Silver showed that GCH can fail at a measurable cardinal $\kappa$, starting with $\kappa$ being $\kappa^{++}$-supercompact. Later, Woodin improved this result, starting from the optimal hypothesis of a $\kappa^{++}$-tall measurable cardinal $\kappa$.
Now more generally, suppose that $\kappa \le \lambda$ are regular and one wishes the GCH to fail at $\lambda$ with $\kappa$ being $\lambda$-supercompact. Silver's methods show that this can be done starting with $\kappa$ being $\lambda^{++}$-supercompact (note that Silver's result above is the special case when $\kappa = \lambda$).
One can ask if there is an analogue of Woodin's result for $\lambda$-supercompactness. We answer this question in the following strong sense: starting with the GCH and $\kappa$ being $\lambda$-supercompact and $\lambda^{++}$-tall, we preserve $\lambda$-supercompactness of $\kappa$ and kill the GCH at $\lambda$ by directly manipulating the size of $2^\lambda$ (i.e.\ we do not force the failure of GCH at $\lambda$ as a consequence of having $2^\kappa$ large enough).
The direct manipulation of $2^\lambda$, where $\lambda$ can be a successor cardinal, is the first step toward understanding which Easton functions can be realized as the continuum function on regular cardinals while preserving instances of $\lambda$-supercompactness.