 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 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.