 Solution

Take a minimizing sequence for the variational regularization functionalIts coercivity makes the sequence bounded. Since $X$ is a reflexive Banach space, a subsequence converges weakly to some $u\in X$. A convex norm-lower-semicontinuous functional has weak lower semicontinuity, so this applies to both $J$ and the convex continuous map $u\mapsto \Vert Au-f{\Vert }_Y^2$. Thereforeand $u$ is a minimizer. This is the direct method in the calculus of variations.