= Solution
For an admissible nonzero real trial function, <integration by parts> and the zero-<flux> <boundary conditions> reduce the <Rayleigh quotient> to
$$
\frac{\int_{-L}^Ly\mathcal Ly\,dx}{\int_{-L}^Lwy^2\,dx}
=\frac{\int_{-L}^Lp(y')^2\,dx}{\int_{-L}^Lwy^2\,dx}\geq0.
$$
A nonzero constant trial function attains zero. Positivity of $p$ also shows that equality requires $y'=0$. Hence
$$
\boxed{\lambda_0=0,\qquad f_0(x)=C,\qquad c_{\rm eq}(x)=C e^{-\beta V(x)}.}
$$
If the conserved total amount is $M=\int_{-L}^Lc\,dx$, the normalized <stationary solution> is
$$
\boxed{c_{\rm eq}(x)=\frac{M e^{-\beta V(x)}}{\int_{-L}^Le^{-\beta V(s)}\,ds}.}
$$
Its <diffusive flux> vanishes everywhere. The <Boltzmann distribution> fixes the spatial profile, while the <diffusivity> affects the relaxation toward it. Boundedness of $V$ makes the normalization finite on this finite interval.
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