Some one-dimensional elliptic problems with constraints

Abstract

Given $m \in \mathbb{N} \setminus \{0\}$ and $\rho > 0$, we find solutions $(\lambda,u)$ to the problem \begin{equation*} \begin{cases} \biggl(-\dfrac{d^2}{d x^2}\biggr)^m u + \lambda G'(u) = F'(u),\\ \noalign{\vskip5pt} \displaystyle \int_{\mathbb{R}} K(u) dx = \rho, \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.