= Solution
Let $T=\tfrac12\int|\nabla\phi|^2d^nx$ and $W=\int U(\phi)d^nx$. For a sufficiently regular <finite-energy field configuration> with vacuum asymptotics, the scaled field $\phi_\lambda(x)=\phi(\lambda x)$ stays in the same boundary sector. Changing variables gives the <Derrick scaling> formula
$$
V(\phi_\lambda)=\lambda^{2-n}T+\lambda^{-n}W.
$$
A static solution must be stationary under this variation, so the <Derrick virial identity> is
$$
\boxed{(n-2)T+nW=0.}
$$
For $n>2$, both terms are nonnegative, so both vanish and the field is constant at a vacuum. Hence \b[there are no nonconstant smooth finite-energy static solitons of this scalar type for $n>2$]. For $n=1$ the identity is $T=W$, allowing <kinks>. For $n=2$ it requires $W=0$: a nonzero potential contribution excludes a static soliton, but the scaling identity alone does not exclude a purely derivative energy.
There is a useful refinement specific to the unconstrained flat target $\mathbb R^l$. If $U$ is differentiable and nonnegative, $W=0$ implies $U(\phi(x))=0$ and $\nabla_\phi U=0$ on the image. The static <Euler-Lagrange field equations> then give $\Delta\phi=0$. Each first derivative is an entire harmonic function in <L2 space>. Its mean-value bound over a disk of radius $R$ is at most its total squared norm divided by $\pi R^2$; letting $R\to\infty$ shows that derivative is zero. Thus even the exceptional two-dimensional case has only constant smooth finite-energy solutions into this flat target. This <two-dimensional flat-target Derrick obstruction> uses the field equations in addition to the scale identity.
The <Yang-Mills theory> with a <Higgs field> escapes the original scaling because the gauge curvature is quadratic in spatial scaling. In three dimensions use $A_i^\lambda(x)=\lambda A_i(\lambda x)$ and $\Phi^\lambda(x)=\Phi(\lambda x)$. Then $F^\lambda_{ij}=\lambda^2F_{ij}(\lambda x)$ and $D_i^\lambda\Phi^\lambda=\lambda D_i\Phi(\lambda x)$. Writing the magnetic, Higgs-gradient and Higgs-potential energies as $V_B,V_D,V_U$, the <Derrick scaling of Yang-Mills-Higgs energy> gives
$$
V_\lambda=\lambda V_B+\lambda^{-1}V_D+\lambda^{-3}V_U,
\qquad V_B=V_D+3V_U
$$
at stationarity. These terms can balance. In particular, the zero-potential <Bogomolny-Prasad-Sommerfield monopole> has $V_B=V_D\ne0$.
For a <nonlinear sigma model> in two dimensions, the <harmonic map> energy is purely quadratic in first derivatives and is scale invariant: $V_\lambda=V$. Its target is a curved manifold, not the unconstrained Euclidean target of the refinement above. Its equation is the nonlinear harmonic-map equation, not componentwise $\Delta\phi=0$. The <Derrick theorem> therefore gives no obstruction. Nonconstant finite-energy maps into $S^2$, as in the <O3 nonlinear sigma model>, provide examples with nonzero degree.
For the complex theory, regard $\phi$ and $\bar\phi$ as independent variables in the variation, and write $s=|\phi|^2$, $U=U(s)$. Variation in $\bar\phi$ gives
$$
\frac{\partial\mathcal L}{\partial\bar\phi}=-U'(s)\phi,\qquad
\frac{\partial\mathcal L}{\partial(\partial_\mu\bar\phi)}=\frac12\partial^\mu\phi.
$$
Thus
$$
\boxed{\phi_{tt}-\Delta\phi+2U'(|\phi|^2)\phi=0.}
$$
The factor 2 follows from the printed $1/2$ normalization of the complex kinetic term; equivalently vary its two real components.
A <non-topological soliton> is a localized finite-energy persistent field solution with no protecting winding or other nontrivial topological sector. In this complex theory a <Q-ball> has a stationary spatial profile and a rotating phase,
$$
\phi(t,x)=e^{i\omega t}f(x),\qquad f(x)\to0\text{ at infinity},
$$
with real $f$. It is not static as a field even though its energy density is time-independent. The global phase symmetry gives the <Noether charge>
$$
Q=\int\operatorname{Im}(\bar\phi\phi_t)d^nx=\omega I,
\qquad I=\int f^2d^nx.
$$
The profile equation is $-\Delta f+2U'(f^2)f-\omega^2f=0$. Its energy is $T+W+\omega^2I/2$. To vary the size at fixed charge one must also vary the frequency, because scaling $f$ at fixed $\omega$ would change $Q$. Eliminating $\omega=Q/I$ gives <charge-constrained scalar-field scaling>:
$$
E_Q[f_\lambda]=\lambda^{2-n}T+\lambda^{-n}W+\lambda^n\frac{Q^2}{2I}.
$$
The new positive-power term can balance the other terms for $n>1$, including $n>2$, so the static <Derrick theorem> does not rule out these solutions. Equivalently, at fixed frequency the profile extremizes $T+\int[U(f^2)-\omega^2f^2/2]$, whose effective potential need not be nonnegative. This explains both the time dependence and the charge constraint behind the evasion; a complex field by itself is not enough.
For an explicit example take the <sextic unique-vacuum Q-ball potential>
$$
\boxed{U(|\phi|^2)=|\phi|^2-|\phi|^4+|\phi|^6.}
$$
It is nonnegative, since $U(s)=s[(s-1/2)^2+3/4]$, and its only vacuum is zero. Its small-field mass squared is $2U'(0)=2$, while $\min_{s>0}2U(s)/s=3/2$. It supports localized <Q-balls>, for example radial ones in three dimensions with $3/2<\omega^2<2$. In that range the profile potential is positive quadratic near zero but negative somewhere away from zero, permitting a localized nontrivial profile. This example uses a conserved phase charge and no vacuum winding.
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