Some one-dimensional elliptic problems with constraints

TL;DR

Problem: find normalized solutions to a one-dimensional higher-order elliptic equation $(-\frac{d^2}{dx^2})^m u + \lambda G'(u) = F'(u)$ under the constraint $\int_{\mathbb{R}} K(u)dx=\rho$. Methods: a global-branch/bifurcation strategy for $m=1$, a variational approach in the special case $2G=K=s^2$, and, for the poly-harmonic setting $m\ge2$, Pohožaev identities plus Nehari-manifold/ constrained minimization. Results: existence of normalized solutions with prescribed mass in multiple regimes for $m=1$, and existence results in subcritical/critical and supercritical growth for poly-harmonic problems, together with asymptotics and constraints-driven minimization frameworks. Significance: extends standing-wave analysis for Schrödinger-type equations to higher-order one-dimensional problems, clarifies mass-subcritical/critical/supercritical geometry, and provides variational constructions for higher-order constrained elliptic equations.

Abstract

Given $m \in \mathbb{N} \setminus \{0\}$ and $ρ> 0$, we find solutions $(λ,u)$ 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.

Theorems & Definitions (32)

• Theorem 1.1
• Corollary 1.2
• Remark 1.3
• Theorem 1.4
• Theorem 1.5
• Remark 2.1
• Lemma 2.2
• proof
• Lemma 2.3
• proof