Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 154 / 2

Set $p=2+4/d$. For $u_{a,\lambda }(x)=au(\lambda x)$, the change of variables $y=\lambda x$ givesandThe factors cancel, so the Weinstein functional satisfies $J(u_{a,\lambda })=J(u)$.

The Gagliardo-Nirenberg interpolation inequality givesConsequently $J(u)\ge C_d^{-1}$ for every nonzero $u\in H^1$, and therefore $I>0$.

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.

WriteDifferentiating $\log J(u+th)$ at $t=0$ for a real test function $h$ givesAfter integration by parts,Rescaling the dependent and independent variables reduces this Euler-Lagrange equation to the ground-state equation for $Q$. The uniqueness of its positive radial solution and the equality cases in rearrangement show that all minimizers areThis is the classification of Weinstein-functional minimizers.

The Pohozaev identity for the mass-critical NLS ground state givesSince $Q$ is a minimizer,

The definition of the infimum $I$ givesSubstituting the value of $I$ from part 4 into the energy yieldsThis is coercivity below the NLS ground-state mass.