content_image/365634.jpg content_image/365635.jpg content_image/365636.jpg content_image/365637.jpg content_image/365638.jpg content_image/365639.jpg content_image/365640.jpg
## D. Euler-Lagrange equations in semi-infinite system 

Our goal is to determine the shape of the ion number and charge density profiles near the charged wall, therefore we need to consider the Euler-Lagrange (EL) equations corresponding to the minimum of the functional (26). 

From δ L /δφ ( z ) = 0 we obtain 

where 

<img src='content_image/365634.jpg'>

<img src='content_image/365635.jpg'>

Note that − 1 < g ( z ) < 1. From the above we can obtain ψ as a Taylor series in φ and η , of odd orders in φ . The charge density can be eliminated by using (39), and after some algebra we obtain three EL equations for ϑ , η and g : 

<img src='content_image/365636.jpg'>

<img src='content_image/365637.jpg'>

<img src='content_image/365638.jpg'>

where 

<img src='content_image/365639.jpg'>

and 

<img src='content_image/365640.jpg'>

