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let $\{f_n\}_n \subset C^\infty \cap L^2(\mathbb R^N)$ be a sequence of functions that solves a linear differential equation $F_n(f_n, \nabla f_n) = 0$. Suppose that there exists a subsequence $n_k$ such that $f_{n_k} \to f$ weakly in $L^2$ and that is the weak solution (not necessarily unique) of the limit problem $F(f,\nabla f) =0$.

If we had a uniqueness result for the limit problem, it would be trivial that the whole sequence $f_n \to f$. What strategy can one use to show this kind of result without relying on the uniqueness for the limit problem?

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