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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 arrangewith 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 givesLocal Rellich-Kondrachov compactness theorem and this tail bound yield strong convergence in $L^{2+4/d}$ after a subsequence; this exponent is below $2^{\ast }=2d/(d-2)$ for $d\ge 3$, the upper range in the Sobolev embedding theorem; put $2^{\ast }=\infty$ 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.

By the two invariances from part 1, normalize a minimizing sequence $(u_n)$ so thatReplacing $u_n$ by $|u_n|$ and then by its symmetric decreasing rearrangement preserves its $L^2$ and $L^p$ norms and does not increase the gradient norm. A harmless dilation restores the normalization, so we may take the sequence nonnegative, radial, and radially decreasing.

The sequence has a weakly convergent subsequence in $H^1$. Radial compactness and the fixed scale give strong convergence in $L^p$; in particular the limit is nonzero because $\Vert u_n{\Vert }_p^p\to I^{-1}$. Weak lower semicontinuity of the $L^2$ and gradient norms then shows that the limit attains the infimum. This is the existence of a Weinstein-functional minimizer.

For $u\in W^{1,p}({\mathbb{R}}^d)$, the symmetric decreasing rearrangement satisfies