given the 2 PDE
$$ \Delta u-au_{tt}+u_{t}=0$$
and $$ \Delta u + Du*Df=0 $$
here $ \Delta u $ is the Laplacian $ Du= grad(u) $ is the gradient and * means scalar product $u_{t} = \frac{\partial u}{\partial t}$
my doubt is what term should i include to get the linear part of the equations $ u_{t} $ and $Du*Df $ from the Euler Lagrange equations in $ R^{n} $ thanks
Well, for the second case, the Lagrangian you want is $$ \Lambda_D(u) = \tfrac12 \int_D\ e^{f(x)}\ |\nabla u|^2\ \mathrm{d}x $$ where $D$ is an appropriate domain in $\mathbb{R}^n$. The first case is similar (if you are assuming that $a$ is a constant), for then, it will just be $$ \Lambda_D(u) = \tfrac12 \int_{D'}\ e^{t}\ \bigl(|\nabla u|^2-a|u_t|^2\bigr)\ \mathrm{d}x\mathrm{d}t $$ where $D'$ is an appropriate domain in $\mathbb{R}^n\times \mathbb{R}$. If $a$ is not a constant, there are conditions.