This is an extended comment.

Your question really needs to be stated more precisely.

First, presumably there are conditions on the dimension $d$, on the geometry of $\Omega$, and on behavior of $V$. Absent these conditions certainly exponential decay cannot be expected.

(You probably also want to give a boundary condition.)

For example, in dimension 1, the equation
$$ - u'' + |u|^2 u - \frac{3x^2}{(1+x^2)^2} u = 0 $$
is of the type you are looking for, by admits the $L^2(\mathbb{R})$ solution
$$ u = (1 + x^2)^{-1/2} $$
that does not decay exponentially.

Related to this point is the fact that the exponential decay of solutions to *linear* Schrodinger operators are usually proven for the eigenvalue problem where the eigenvalue has a non-zero real part. There are some classical works of Agmon (this, this, and this for example) in this case. Is it the case that you want $V$ to be non-zero at infinity?

*[Following paragraph is struck since the OP edited to fix it]* ~~Secondly, your definition of "exponential decay" is odd. Taking for example $h(x) = \ln (1 + \ln(1 + |x|))$, this function grows to infinity as $x$ goes to infinity, but I would not really call functions for which $\int |u|^2 \exp(h) < \infty$ as "exponentially decaying". ~~

Finally, if you are willing to take the existence of a solution for granted, and if you are willing to guess that the expected solution has *some* decay, then can you not just look at the linear theory, where you look at estimates for decay of the linear equation
$$ - \triangle u + \tilde{V} u = 0 $$
where $\tilde{V} = V + |u|^k$? You may be able to upgrade slower decay to exponential decay in this case. (For example, if you know $V$ gives a potential well and limits to $-1$ near infinity sufficiently fast, then Agmon's results cited above would imply that any sufficiently fast polynomially decaying solution is in fact decaying exponentially.)