I need to evaluate the linearity (and even if they are PDEs) of the following:
$ i) \int_{\mathbb{R}^d} | \partial u |^2 dx + \int_{\mathbb{R}^d} u(x) f(x) dx = 0 $ with $ u : \mathbb{R}^d \mapsto \mathbb{R}$.
$ ii) | \nabla u(x) | = |x|^2 $
$ iii) \Delta u (x) = u(x+1) $
It should be specified whether these are linear, semilinear, quasilinear or fully non-linear.
I am confused about whether you can observe the integral as a simple function or not.
My guesses:
$i)$ is fully non-linear, since the integral is used on the highest degree.
$ii)$ Also fully non-linear, since $| \cdot | $ is being used on the the highest derivative and counts as a function $f(u_x)$
$iii)$ Not sure here. If you can rewrite $u(x+1)$ as $\lim_{x \to x+1} u(x)$ then perhaps I'd argue semilinear, with the limit being a function of $u$.