= Solution
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 yields
$$
J(v)\leq\liminf_{n\to\infty}J(v_n).
$$
Thus $v$ 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 equation>
$$
-\Delta v+|x|^2v-\omega v+v^3=0,
$$
which is the <Schrödinger trapped defocusing stationary equation>.
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