Some one-dimensional elliptic problems with constraints

Abstract

Given $m\in \mathbb{N}\setminus \{0\}$ and $\ddot{I}>0$, we find solutions $(\hat{I}»,u)$ 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.

17 pages, to appear in Topol. Methods Nonlinear Anal