Some one-dimensional elliptic problems with constraints

TL;DR

This paper investigates solutions to one-dimensional elliptic problems with constraints, employing bifurcation and variational methods depending on the specific case of the problem.

Contribution

It introduces new solution existence results for constrained elliptic problems using bifurcation and variational techniques for different cases.

Findings

Existence of solutions for m=1 using bifurcation methods.

Existence of solutions for m=2 with quadratic constraints via variational methods.

Characterization of solutions under specific constraints and problem settings.

Abstract

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 + \lambda G'(u) = F'(u)\\ \int_{\mathbb{R}} K(u) \, \mathrm{d}x = \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.