On two open problems on the normality of bent functions

Non-normal Boolean bent functions are one of the least understood classes of bent functions, and only a few difficult-to-find examples of such functions are known. In this paper, we consider the following two open problems on the normality of bent functions:

1. Do non-normal bent functions in 8 variables and degree 4 exist?

2. Do non-normal bent functions in the ${\mathrm{PS}}^{-}\setminus {\mathrm{PS}}_{ap}$ class exist?

We solve both of these problems by finding among the known $\mathrm{PS}$ bent functions in $n=8$ variables a non-normal bent function in the ${\mathrm{PS}}^{-}\setminus {\mathrm{PS}}_{ap}$ class.