= Solution
A <classical field-theory soliton> is a smooth, spatially localized, finite-energy solution which retains its identity under time evolution and is stable against small perturbations, commonly because of a <topological charge> or a balance between energy terms with different scaling behavior.
For a static field write
$$
E=E_2+E_4+E_0,
$$
where
$$
E_2=\frac12\int|\nabla\phi|^2d^Dx,
\qquad
E_4=\kappa^2\int|\nabla\phi|^4d^Dx,
\qquad
E_0=\int U(\phi)d^Dx.
$$
Under the <Derrick scaling> $\phi_\lambda(\mathbf x)=\phi(\lambda\mathbf x)$, a <change of variables> gives
$$
E(\lambda)=\lambda^{2-D}E_2
+\lambda^{4-D}E_4+\lambda^{-D}E_0.
$$
A static solution must be stationary under this variation, so the <Derrick virial identity> is
$$
\boxed{(2-D)E_2+(4-D)E_4-DE_0=0}.
$$
All three energies are nonnegative. For $D\geq4$, every coefficient is nonpositive and the coefficient of the strictly positive $E_2$ of any nonconstant field is negative. The identity is impossible. Thus, when the quartic-gradient term is available,
$$
\boxed{D\geq4\quad\Longrightarrow\quad\text{no nontrivial static soliton}.}
$$
For $D=2$ or $3$, the $E_4$ term has the opposite sign to at least one other term and the <Derrick theorem> does not rule out a soliton. If $\kappa=0$ from the outset, the identity reduces to $(2-D)E_2-DE_0=0$, recovering the stronger standard obstruction for $D\geq2$.
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