\phi)^2 -(\nabla V_0)^2 - (\nabla b_0)^2 - (\nabla A_0)^2
\right] + V_{ef}\right)$$ with $$V_{ef}= \frac{1}{2} \left[
m_s^2 \phi^2 -m_v^2 V_0^2 -m_\rho^2 b_0^2  \right] - \mu_p \rho_p-\mu_n \rho_n$$ $$+ \frac{\gamma}{2 \pi^2} \sum_{i=p,n} \int_0^{k_{Fi}} \d p p^2 \epsilon +
\Gamma_v V_0 \rho + \Gamma_{\rho} \frac{b_0}{2} \rho_3
+ e A_0 \rho_p 
\label{vef}$$

Minimization of $\Omega$ with respect to $ k_{Fi}(r),\, i=p,n$, gives rise to the following conditions $$k_{Fp}^2\left(\mu_p-\sqrt{k_{Fp}^2+{M^*}^2}-
\Gamma_v V_0- \frac{\Gamma_\rho}{2} b_0 -e A_0 - \Sigma_0^R  \right)=0$$ and $$k_{Fn}^2\left(\mu_n-\sqrt{k_{Fn}^2+{M^*}^2}-
\Gamma_v V_0+ \frac{\Gamma_\rho}{2} b_0 - \Sigma_0^R \right)=0,$$ where the rearrangement term is $$\Sigma_0^R=\frac{\partial\, \Gamma_v}{\partial \rho}\, \rho\, V_0+\frac{\partial\, \Gamma_\rho}{\partial \rho}\, \rho_3\, \frac{b_0}{2}-
\frac{\partial\, \Gamma_s}{\partial \rho}\, \rho_s\, \phi .$$ From the above equations we obtain $k_{Fp}=0$ and $k_{Fn}=0$ or, for $k_{Fp}$ or $k_{Fn}$ different from zero, $$\mu_p=\sqrt{k_{Fp}^2+{M^*}^2}+ \Gamma_v V_0  + \frac{\Gamma_\rho}{2}  b_0 +
e A_0  + \Sigma_0^R, \label{mup}$$ $$\mu_n=\sqrt{k_{Fn}^2+{M^*}^2}+ \Gamma_v V_0  - \frac{\Gamma_\rho}{2}  b_0
+\Sigma_0^R.
\label{mun}$$ The values of $k_{Fp}$ and $k_{Fn}$ are obtained inverting these two last equations.

Such density dependences in the coupling parameters do not affect the energy functional but of course affect its derivative such as the pressure density and the chemical potentials. As already discussed in the literature [@inst04; @inst06; @inst062; @ddpeos], the rearrangement term is crucial in obtaining different behaviors 