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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.

The Gagliardo-Nirenberg interpolation inequalities bound an intermediate derivative norm by powers of a higher derivative norm and a lower norm. In two-dimensional whole space, one useful case for $f\in H^2({\mathbb{R}}^2)$ isApply Ladyzhenskaya's inequality to the components of $\nabla f$, then combine their estimates by the Cauchy-Schwarz inequality. Density of smooth functions in a Sobolev space extends the result from smooth compactly supported functions to the stated Sobolev space.

The domain and boundary assumptions matter. On a bounded domain, the affine function $f(x)=x_1$ has nonzero gradient but $D^{2}f=0$, so this homogeneous inequality cannot hold for every $H^2$ function there. A bounded-domain version needs appropriate boundary restrictions or an additional lower-order term.

For the integer powers $p=2,3,4$, the Generalized Korteweg–De Vries equation is mass-subcritical. Its Gagliardo-Nirenberg interpolation inequality isThe exponent $\alpha =(p-1)/2$ is strictly below two. Since KdV mass and energy conservation fixes $\Vert u{\Vert }_2$ and $E(u)$, the gradient size $X=\Vert u_x{\Vert }_2$ obeys $X^2/2\le |E(u_0)|+C_{M(u_0)}{X}^{\alpha }$. Young's inequality for products absorbs the lower power into $X^2/4$ and leaves a uniform bound for $X$.

Under the standard local $H^1(\mathbb{R})$ solution theory with a continuation time controlled by the H1 norm, this bound extends the solution globally. At $p=5$, the nonlinear estimate has gradient exponent two, so this absorption argument requires additional information such as a small-mass threshold. The scaling identifies this as an L2 norm threshold.

In three dimensions, the Hardy–Littlewood–Sobolev inequality and Gagliardo-Nirenberg interpolation inequality giveConserved mass and energy therefore bound the $H^1$ norm by Young inequality, and the blowup alternative gives global existence.

Let $0<\lambda <d$ and $1<p,q<\infty$ satisfy $1/p+1/q+\lambda /d=2$. The Hardy–Littlewood–Sobolev inequality states thatHere the right side uses Lp norms, and the constant is independent of $f,g$. The exponents are linked by the displayed dimensional relation; they are not a conjugate exponents pair. This is a bound for a singular convolution rather than a pointwise bound on the kernel.^[ref]

For $d=3$, $\lambda =1$ and $p=q=6/5$, applying it to $f=g=|u{|}^2$ givesThis is the potential-energy estimate used for the Gravitational Hartree equation. A Gagliardo-Nirenberg interpolation inequality then relates that intermediate Lp norm to mass and the gradient L2 norm.

The one-dimensional Gagliardo-Nirenberg interpolation inequality yieldsMass conservation fixes $\Vert u{\Vert }_2$. If $X=\Vert u_x{\Vert }_2$, energy conservation therefore givesFor $p<5$, the second exponent is strictly smaller than two. The right side tends to infinity with $X$, so this inequality bounds $X$ uniformly throughout the lifespan. The conserved $L^2$ norm then bounds $\Vert u(t){\Vert }_{H^1}$. The stated blowup criterion rules out a finite endpoint, proving Global existence for the energy-subcritical generalized KdV equation.

In three dimensions, the Hardy–Littlewood–Sobolev inequality applied to the Newtonian kernel givesThe Gagliardo-Nirenberg interpolation inequality then givesWriting $X=\Vert \nabla u{\Vert }_2$ and using the conserved mass, conservation of energy impliesThus $X$ remains bounded. Together with the conserved $L^2$ norm this bounds $\Vert u(t){\Vert }_{H^1}$, and the supplied blowup criterion proves Global H1 solutions of the three-dimensional gravitational Hartree equation.

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 $I>0$.