The authors prove a lower semicontinuity for the \(BV\) extension of the functional defined in \(W^{1,1}(\Omega)\) \[ \int_\Omega f(x,u(x),\nabla u(x))dx, \] where the energy density \(f\) is not coercive. Here \(\Omega\) denotes an open subset of \({\mathbb R}^N\) and \(f\) is a Caratheodory function with \(f(\cdot,u,\xi)\) weakly differentiable in \(\Omega\) for every \((u,\xi)\in {\mathbb R}\times {\mathbb R}^N\); moreover, the authors assume that there exists \(Z\subset \Omega\) with null Lebesgue measure such that \(f(x,u,\cdot)\) is convex on \({\mathbb R}^N\) for every \((x,u)\in (\Omega\setminus Z)\times {\mathbb R}\) and \(f(x,\cdot,\xi)\) continuous in \({\mathbb R}\) for every \((x,\xi)\in (\Omega\setminus Z)\times {\mathbb R}^N\) and for any open set \(\Omega^\prime\subset\subset \Omega\) and any bounded set \(B\subset {\mathbb R}\times{\mathbb R}^N\) the following estimate holds: \[ \int_{\Omega^\prime}| \nabla _x f(x,u,\xi)| dx \leq L(\Omega^\prime,B) \] for every \((u,\xi)\in B\). Under these assumptions the authors give a lower semicontinuity result for the functional extended to the space \(BV(\Omega)\) \[ \int_\Omega f(x,u,\nabla u)dx +\int_\Omega \bar f^\infty\left( x,\tilde u, \frac{D^c u}{| D^c u| }\right) d| D^c u| +\int_{J_u} d{\mathbb H}^{N-1}\int_{u^-}^{u^+} \bar f^\infty (x,t,\nu_u)dt \] where \(\bar f^\infty\) is the recession function of \(\bar f\), \(\bar f\) is a suitable Borel function, continuous in \(u\), convex in \(\xi\) and such that for any \((u,\xi)\in {\mathbb R}\times {\mathbb R}^N\) \(\bar f(\cdot,u,\xi)\) coincides \({\mathbb H}^{N-1}\) a.e. with the precise representative of \(f(\cdot,u,\xi)\).