Normalized solutions of Nehari-Pankov type to mass-supercritical indefinite variational problems

TL;DR

This paper develops a new variational approach to find solutions with prescribed norm for nonlinear equations, extending to mass-supercritical regimes and applying to Schrödinger equations on graphs and tori.

Contribution

It introduces a novel method to detect prescribed norm solutions without mass-subcriticality assumptions, applicable to various higher-order and bounded domain equations.

Findings

Established existence of solutions with large prescribed mass on graphs.

Proved multiple small-mass solutions for higher-order equations in bounded domains.

Extended solution existence results to mass-supercritical regimes.

Abstract

We consider abstract nonlinear equations of the form $A u = \lambda u + I'(u)$, where $A$ is a self-adjoint operator with compact resolvent on a Hilbert space $H$, $\lambda \in \mathbb{R}$ is a parameter, and $u \mapsto I'(u)$ is a superlinear term of variational nature. In this abstract setting, we develop a new approach to detect prescribed norm solutions in $H$ which does not rely on any mass-subcriticality assumptions. We then consider various applications of this approach. First, we obtain, under general assumptions including the full mass-supercritical parameter regime, the existence of (infinitely many) solutions to a class of nonlinear Schr\"odinger equations on a compact graph $\mathcal{G}$ with prescribed arbitrarily large mass, thereby improving previous results which only cover small masses. Moreover, we derive a similar result for a biharmonic Schr\"odinger equation in the $2$-torus. For a larger class of second order and higher order equations in a bounded domain with Dirichlet boundary conditions, we also show the existence of multiple solutions with prescribed small mass. The solutions we obtain are detected as ground states of Nehari-Pankov type for the associated $\lambda$-dependent action functional, where $\lambda$ varies in a spectral gap between sufficiently large eigenvalues of $A$. The key new observation in this abstract framework is the fact that the $H$-norms of these $\lambda$-dependent solution families form connected sets even though the solution families themselves may be disconnected. To estimate the size of these connected sets in specific settings, we use Weyl type estimates for the length of spectral gaps, variational characterizations of eigenvalues, bounds for associated eigenfunctions and a bound from analytic number theory.