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Given a unbounded domain $\Omega$(a region or an open manifold) with $\dim \geq3$, consider the equation $$A u+Vu+|u|^ku=0,$$ here $A=\sum_{ij}\partial_ia^{ij}(x)\partial_j$ is an elliptic operator with smooth coefficients. Suppose that $A+V$ is the semi positive Laplacian, here $V$ is a (smooth) function with $V<0$, we may assume that $V$ is bounded. Assume there is the solution $u$ with $u\in L^2(\Omega)$.

Q I just want to ask if there are some work or results to study the exponential decay of $u$, i.e. $\int_\Omega |u|^2\exp(h)<\infty$, here $h$ is a distance function.

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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.)

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