Consider first passage percolation with identical and independent weight distributions and first passage time ${\rm T}$. In this paper, we study the upper tail large deviations $\mathbb{P}({\rm T}(0,nx)>n(μ+ξ))$, for $ξ>0$ and $x\neq 0$ with a time constant $μ$ and a dimension $d$, for weights that satisfy a tail assumption $ β_1\exp{(-αt^r)}\leq \mathbb P(τ_e>t)\leq β_2\exp{(-αt^r)}.$ When $r\leq 1$ (this includes the well-known Eden growth model), we show that the upper tail large deviation decays as $\exp{(-(2dξ+o(1))n)}$. When $1< r\leq d$, we find that the rate function can be naturally described by a variational formula, called the discrete p-Capacity, and we study its asymptotics. For $r<d$, we show that the large deviation event ${\rm T}(0,nx)>n(μ+ξ)$ is described by a localization of high weights around the origin. The picture changes for $r\geq d$ where the configuration is not anymore localized.