It's known that we can have global failures of GCH---for example, where $\forall \lambda(2^\lambda = \lambda^{++})$---given suitable large cardinal axioms.


My question is whether we can have global failures of GCH where there is a weakly inaccessible cardinal between $\lambda$ and $2^\lambda$ for each $\lambda$. Similarly, whether we can have a cardinal fixed point between $\lambda$ and $2^\lambda$. I'd also be interested in whether $2^\lambda$ can be weakly inaccessible/a cardinal fixed point, for every $\lambda$.



How badly can the GCH fail globally?
In the Foreman-Woodin model [The generalized continuum hypothesis can fail everywhere.](https://www.jstor.org/stable/2944324?origin=crossref&seq=1) for each infinite cardinal $\kappa, 2^\kappa$ is weakly inaccessible.


This answers your last question. The answer to the first two questions can be yes as well. In the case of Foreman-Woodin model, they start with a supercompact $\kappa=\kappa\_0$ and infinitely many inaccessibles $\kappa\_n, n<\omega,$ above it. They first force to get $2^{\kappa\_n}=\kappa\_{n+1}$ preserving $\kappa$ supercompact, and this is reflected below for all cardinals. So if for example each $\kappa\_n$ is measurable, then what you get in the final model is that for each infinite cardinal $\lambda, 2^\lambda$ has been measurable in $V$, in particular there are both weakly inaccessible and cardinal fixed points between $\lambda$ and $2^\lambda.$


See also the paper [A model in which every Boolean algebra has many subalgebras](https://www.cambridge.org/core/journals/journal-of-symbolic-logic/article/model-in-which-every-boolean-algebra-has-many-subalgebras/2B6A5E860D17D730DEEE23C37A7CFC6D) by Cummings and Shelah, where they build a model in which for each infinite cardinal $\kappa, 2^\kappa$ is weakly inaccessible and $Pr(2^\kappa)$ holds. Here $Pr(\lambda)$ is in some sense a large cardinal property (for example it holds if $\lambda$ is a Ramsey cardinal). For its definition see the paper.


