= Existence of a Weinstein-functional minimizer
For nonzero $u\in H^1(\mathbb R^d)$, the <Weinstein functional> is invariant under amplitude multiplication and spatial dilation. Apply <symmetric decreasing rearrangement> to a minimizing sequence, then use those two invariances to arrange
$$
\|u_n\|_2=\|\nabla u_n\|_2=1
$$
with nonnegative radial, radially nonincreasing $u_n$. Rearrangement preserves the relevant <Lp norms> and does not increase the gradient <L2 norm>.
These monotone radial profiles have uniform pointwise bounds $|u_n(x)|\le C_d|x|^{-d/2}$, since their value on a ball is bounded by its average squared mass. For every $p>2$, this gives
$$
\int_{|x|>R}|u_n|^p\le C_{d,p}R^{-d(p-2)/2}.
$$
Local <Rellich-Kondrachov compactness theorem> and this tail bound yield strong convergence in $L^{2+4/d}$ after a subsequence; this exponent is below $2^*=2d/(d-2)$ for $d\ge3$, the upper range in the <Sobolev embedding theorem>; put $2^*=\infty$ for $d=1,2$. The argument also covers $d=1$ because monotone rearrangement, rather than radial symmetry alone, controls the tails.
The <Gagliardo-Nirenberg interpolation inequality> bounds the infimum of the <Weinstein functional> away from zero. The strong nonlinear-norm limit is therefore nonzero. Weak lower semicontinuity of the two normalized quadratic norms shows that the limit attains the infimum, giving a nonnegative minimizer.
Back to article page
