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My question is about non-uniqueness of solutions of an elliptic PDE in $\mathbb R^n$ with source term in a scaling-subcritical space (regular, but with too slow decay at infinity), and with some nice ...

I want to prove the uniqueness of the solution of the following problem: $$\eqalign{ & - d\,\Delta u + u = {u^p} \text{ in } \Omega \cr & u > 0 \text{ in } \Omega \cr & \...