Suppose I have the variational problem
$$ E(y) = \frac{1}{2}\int_{a}^{b} y^2 + \alpha y'^2dx $$
Variational derivative will provide
$$ \frac{\delta E}{ \delta y} = y -\alpha y'', $$
Is there a variational problem known in literature that can give me something like as a variational derivative the following
$$ \frac{\delta E}{\delta y} = \left(1-\alpha( y\right)) y - \alpha(y)y'' $$
?
The reason is the following, if I write down a gradient descent I can get something like
$$ \partial_t y = - (1 - \alpha(y))y + \alpha(y)y'' $$
And what I want to describe with $\alpha = \alpha(y) \in [0,1]$ is some measure of some feature of $y$, and I want the update to weight more one term of the gradient rather than the other, based on such function.
Is there anything like this known in literature? The closest think I could come up with was the EM algorithm, but it doesn't seem to me there's a variational framework for this.
More specifically $\alpha(y)$ could be a sigmoid or some gaussian depending on $y$.