Some one-dimensional elliptic problems with constraints

Given m ∈ N \0\ and ρ > 0, we find solutions (λ,u) to the problem {equation*} {cases} (-d^2/d x^2)^m u + λ G'(u) = F'(u),\\ {5pt} ∫R K(u) dx = ρ, {cases} {equation*} in the following cases: $m=1$ or 2G(s) = K(s) = s². In the former, we follow a bifurcation argument; in the latter, we use variational methods.