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Choose a minimizing sequence $(u_n)$ withSincethe sequence is bounded in $H_0^1(\Omega )$. Passing to a subsequence, weak compactness givesin $H_0^1(\Omega )$. Rellich compactness gives $u_n\to w$ strongly in $L^2$, and hencePart (i) now yieldsThe definition of $\lambda$ gives the reverse inequality because $w$ is admissible. ThereforeThis is the constrained ground-state minimizer on a bounded domain obtained by the direct method.