  It has been recently presented some local versions of the
Bishop-Phelps-Bollob\'as type property for operators. In the present article,
we continue studying these properties for multilinear mappings. We show some
differences between the local and uniform versions of the
Bishop-Phelps-Bollob\'as type results for multilinear mappings, and also
provide some interesting examples which shows that this study is not just a
mere generalization of the linear case. We study those properties for bilinear
forms on $\ell_p \times \ell_q$ using the strong subdifferentiability of the
norm of the Banach space $\ell_p \hat{\otimes}_{\pi} \ell_{q}$. Moreover, we
present necessary and sufficient conditions for the norm of a Banach space $Y$
to be strongly subdifferentiable through the study of these properties for
bilinear mappings on $\ell_1^N \times Y$.
