Is the Bilateral Filter a Solution of Some Variational Method?
Yes indeed.
You may have a look on work (Paper) by Michael Elad which is called On the Origin of the Bilateral Filter and Ways to Improve It or Analysis of the Bilateral Filter.
You may have a look on the way he derives the Bilateral Filter from a Least Squares problem with weighted regularization on the Gradient of the image in the form of:
$$ \arg \min_{x} \frac{1}{2} {\left\| x - y \right\|}_{2}^{2} + \frac{\lambda}{2} \sum_{n = 1}^{N} { \left( x - {D}^{n} x \right) }^{T} W \left( y, n \right) \left( x - {D}^{n} x \right) $$
Where $ {D}^{n} $ is Image Gradient Operator (Raising by $ n $ simply shifts it) and $ W $ is the weights matrix of the Regularized Least Squares.