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Letbe the harmonic-oscillator energy space, and denote the minimized quadratic functional by $J(u)$. A minimizing sequence $(u_n)\subset \mathcal{A}(M)$ is bounded in $\Sigma$, because $J$ is its squared Hilbert norm with positive coefficients. After taking a subsequence, $u_n\rightharpoonup u$ weakly in $\Sigma$.

The embedding $\Sigma \hookrightarrow L^4({\mathbb{R}}^2)$ is compact. On any fixed ball this follows from the Rellich-Kondrachov compactness theorem. Outside a large ball, the moment bound makes the $L^2$ tail uniformly small, and interpolation with the uniform $H^1\hookrightarrow L^q$ bounds for some $q>4$ makes the $L^4$ tail uniformly small. Hence $u_n\to u$ strongly in $L^4$.

The constraint passes to the limit, so $\int |u{|}^4=M$. Weak lower semicontinuity gives $J(u)\le \mathrm{liminf}J(u_n)=I(M)$. Therefore $u$ attains the infimum. This is Fixed-L4 minimization in the harmonic-oscillator energy space.

Let $(v_n)$ be a minimizing sequence. The coercive lower bound from part 2 makes it bounded in the harmonic-oscillator energy space and in $L^4$. After taking a subsequence, $v_n$ converges weakly in both spaces. The compact embedding of the harmonic-oscillator energy space gives strong convergence in $L^2$, while weak lower semicontinuity of the gradient, moment, and $L^4$ terms yieldsThus $v$ attains the infimum. Replacing $v$ by $|v|$ does not increase the gradient norm, so a minimizer may be chosen nonnegative. It is nonzero because the infimum is negative whereas $J(0)=0$.

Taking the first variation against a smooth compactly supported function gives the Euler-Lagrange equationwhich is the Schrödinger trapped defocusing stationary equation.