Consider the following non linear pde in the unknown $v(x,y)$: 
$$ \frac{\partial v(x,y)}{\partial x} + 
\Big(\frac{\partial v(x,y)}{\partial x} \Big)^2 = e^{2 ty}-1 $$
where $t$ is some fixed small non zero real number. Now define the 
following sequence of functions: 
$$ v\_{0}(x,y) : \equiv 0 $$
$$ v\_{n+1}(x,y):= \int\_{0}^{x} (e^{2 ty}-1)- \Big(\frac{\partial v\_n(x^{\prime},y)}{\partial x^{\prime}} \Big)^2 d x^{\prime} $$ 


My question is the following: is there any hope of saying that this sequence 
"converges" in any reasonable sense, for small $t$? More precisely, 
the idea is as follows: Is there some appropriate "weighted Sobolev space" 
in which these functions $v\_{n}(x,y)$ live; and hopefully with respect 
to that weighted norm, the sequence is a Cauchy sequence? 


Note that it can not be the usual Sobolev space, because the function 
$e^{2t y}$ does not have finite $L^p$ norm. 


For the time being I am not asking whether the "limit" $v$, if it exists 
is smooth or not. Right now all I am looking for is a "weak" solution. 
Eventually of course I would like to know about the regularity of $v$. 
But that is a meaningful question only if $v$ exists. 


Every thing is over the real numbers, the functions $v(x,y)$ and 
$v\_{n}(x,y)$ are supposed to be defined on the whole of $\mathbb{R}^2$ (at least they should be measurable with respect to the right weight). I am not imposing any boundary conditions; 
I am simply looking for some solution of the pde. 



Does a particular iteration produce a weak solution to a non linear pde?
This is 
$$
(\partial\_x\nu+\frac12)^2=\frac14+e^{2ty}-1=e^{2ty}-\frac34.
$$
The parameter $ty$ should be chosen so that $2ty\ge \ln 3-\ln 4$, and then a solution is given by
$$
\partial\_x\nu+\frac12=\alpha,\quad \alpha^2=e^{2ty}-\frac34,
$$
e.g.
$
\nu=(\alpha -\frac12)x+\phi(y).
$


