= Solution
Use a <signed power test for a Laplacian eigenfunction>. For $\gamma\geq2$, test the <Dirichlet Laplacian eigenfunction> equation with $|u|^{\gamma-2}u$. Its derivative is $(\gamma-1)|u|^{\gamma-2}Du$; at $\gamma=2$ the test function is just $u$. <Integration by parts> gives the exact <energy estimate>
$$
(\gamma-1)\int_\Omega |u|^{\gamma-2}|Du|^2=\lambda\int_\Omega |u|^\gamma.
$$
Set $v=|u|^{\gamma/2}$. It belongs to the <zero-boundary Sobolev space> and obeys the <Sobolev chain rule>:
$$
\int_\Omega|Dv|^2=\frac{\gamma^2}{4}\int_\Omega|u|^{\gamma-2}|Du|^2
=\frac{\lambda\gamma^2}{4(\gamma-1)}\int_\Omega|u|^\gamma.
$$
For $\gamma>2$, the power map is $C^1$ with bounded derivative on the bounded range of $u$. For $\gamma=2$, the <absolute value> map is Lipschitz, and $D|u|=\operatorname{sgn}(u)Du$ <almost everywhere>; on the zero set, use the fact that a <gradient of a Sobolev function vanishes on a level set>. These facts justify the identity even when $u$ changes sign.
Apply the given <Sobolev inequality> with the <Sobolev conjugate exponent> $2\kappa=2n/(n-2)$:
$$
\boxed{\left(\int_\Omega|u|^{\gamma\kappa}\right)^{1/\kappa}
\leq\frac{C(n)\lambda\gamma^2}{4(\gamma-1)}\int_\Omega|u|^\gamma.}
$$
\b[Testing with a signed power raises the integrability exponent from $\gamma$ to $\gamma\kappa$.] Absorbing the factor $1/4$ into $C(n)$ gives precisely the requested inequality.
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