For the solution of semi-linear elliptic equation, for example I'm considering the 2D cubic nonlinear Schroedinger equation, the correspongding elliptic equation is $\Delta u+u^3=u$, with $u>0$. By variational method, we can prove the existence of the solution(the minimizer of some functional), and moreover the uniqueness is also proved by MK Kwong, "Uniqueness of positive solutions of ...".

Now I want to ask if there's any result about the integrability of the solution, like if the solution lies in $H^s$ or even Schwartz? because what we know about this kind of solution is radially decay with respect to some given point.

For the 1D case, the solution can be written explicitely, $u(x)=\frac{2^{-1/2}}{cosh(x)}$, which is Schwartz function, maybe this could make my question more reasonable?


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