NLS ground state ($Q$)

For the mass-critical focusing nonlinear Schrödinger equation in dimension $d$, the NLS ground state is the positive radial solution ofIts symmetry orbit consists of the optimizers of the sharp Gagliardo-Nirenberg inequality.

For $p=2+4/d$ and nonzero $u\in H^1({\mathbb{R}}^d)$, the mass-critical Weinstein functional isIt is invariant under multiplication by a nonzero scalar, spatial dilation, translation, and constant phase. Its positive minimizers are rescalings and translations of the NLS ground state.

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.

Every minimizer of the Weinstein functional has the formwhere $Q$ is the positive radial NLS ground state. The Euler-Lagrange equation first reduces a minimizer to a rescaled ground-state equation; the equality cases in the diamagnetic and rearrangement inequalities give the constant phase, translation, and radial profile.

In dimension $d$, the mass-critical sharp inequality isFor nonzero $u$, equality holds exactly on the amplitude, phase, translation, and spatial-scaling orbit of the NLS ground state $Q$; $u=0$ also gives equality. For $d=2$ this becomes $\Vert u{\Vert }_4^4\le 2\Vert Q{\Vert }_2^{-2}\Vert u{\Vert }_2^2\Vert \nabla u{\Vert }_2^2$.

The sharp inequality impliesThus energy controls the gradient whenever the mass is strictly below the mass of the NLS ground state.

Suppose $\Vert u_n{\Vert }_2=\Vert Q{\Vert }_2$, $\Vert \nabla u_n{\Vert }_2=\Vert \nabla Q{\Vert }_2$, and $E(u_n)\to 0$. The profile decomposition modulo translations and the sharp inequality force exactly one nonzero profile: splitting the mass would make the sharp inequality strict. Hence, after passage to a subsequence, some translations $u_n(\cdot +x_n)$ converge strongly in $H^1$, and therefore in $L^{2+4/d}$, to an optimizer.