In this paper, we study the minimization of the integral functionals of the type $$ \int\_{\Omega}\frac{|\nabla v|^2}{\big\[1+B|v|\big]^{2}} + \frac12\int\_{\Omega} a(x)|v|^2 - \frac1q\int\_{\Omega} \rho(x)|v|^q, $$ where $0\<B$, $0\<a\_0\leq a(x)\in L^1\_{\mathrm{loc}}(\Omega)$, $0\not\equiv\rho^+(x)\in L^{{\frac{2}{2-q}}}(\Omega)$, and $0\<q<2$. The degeneracy of the principal part of the functional implies that it is not coercive in $W\_{0}^{1,2}(\Omega)$ and pushes to set and solve the minimization problem in $W\_{0}^{1,1}(\Omega)$ instead of the space $W\_{0}^{1,2}(\Omega)$. In addition, when the coefficients $a(x)$ and $\rho(x)$ are merely in $L^1(\Omega)$, but satisfy for some $Q>0$ the condition $|\rho(x)|\leq Q a(x)$, we show the existence of a minimum of the functional which belongs to $W\_{0}^{1,2}(\Omega){\cap L^\infty(\Omega)}\setminus{0}$.