Some one-dimensional elliptic problems with constraints

Given $m\in \mathbb{N}\setminus \{0\}$ and $\rho >0$, we find solutions $(\lambda ,u)$ to the problem \begin{equation*} \begin{cases} \bigl(-\frac{$\mathrm{d}$^2}{$\mathrm{d}$ x^2}\bigr)^m u + λG’(u) = F’(u)\ \int_{$\mathbb{R}$} K(u) , $\mathrm{d}$x = ρ\end{cases} \end{equation*} in the following cases: $m=1$ or $2G(s)=K(s)=s^2$. In the former, we follow a bifurcation argument; in the latter, we use variational methods.