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Normalized solutions of Nehari-Pankov type to indefinite variational problems

Damien Galant, Tobias Weth

TL;DR

This work develops a novel variational framework for finding normalized solutions to indefinite problems of the form $A u = \lambda u + I'(u)$ by exploiting Nehari-Pankov type constraints in spectral gaps of $A$, and by revealing that the masses of ground-state branches form connected sets across gaps. The authors establish existence and multiplicity of normalized ground states via a λ-dependent action $\mathcal{J}_\lambda$ and show that prescribed-mass solutions exist for arbitrarily large mass in both graph-based and biharmonic settings, without requiring mass-critical growth. Central to the approach are (i) a detailed analysis of the λ-dependence of the minimization problem, (ii) action estimates controlling $c_\lambda$ in terms of spectral gap lengths, and (iii) Weyl-law and edge-ODE arguments that convert spectral-gap information into $L^\infty$ and $L^2$ norm bounds for solutions. The results yield infinitely many normalized nonlinear Schrödinger solutions on compact metric graphs and, in the torus and biharmonic settings, analogous multiplicity for prescribed mass, highlighting the method’s robustness beyond the mass-critical regime and its utility for higher-order problems.

Abstract

We consider abstract nonlinear equations of the form $A u = λu + I'(u)$, where $A$ is a self-adjoint operator with compact resolvent on a Hilbert space $H$, $λ\in \mathbf{R}$ 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 existence and multiplicity of solutions with prescribed norm in $H$. We then consider various applications of this approach. First, we obtain, under fairly general assumptions including the mass-supercritical case, the existence of infinitely many solutions to a class of nonlinear (time-independent) Schrödinger equations on a compact graph $\mathcal{G}$ with prescribed (arbitrarily large) mass. Moreover, we derive a similar existence and multiplicity result for a biharmonic semilinear 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 $λ$-dependent action functional, where $λ$ varies in a spectral gap between sufficiently large eigenvalues of $A$. The key observation in this abstract framework is the fact that the $H$-norms of these $λ$-dependent solution families form connected sets. 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 classical bound from analytic number theory.

Normalized solutions of Nehari-Pankov type to indefinite variational problems

TL;DR

This work develops a novel variational framework for finding normalized solutions to indefinite problems of the form by exploiting Nehari-Pankov type constraints in spectral gaps of , and by revealing that the masses of ground-state branches form connected sets across gaps. The authors establish existence and multiplicity of normalized ground states via a λ-dependent action and show that prescribed-mass solutions exist for arbitrarily large mass in both graph-based and biharmonic settings, without requiring mass-critical growth. Central to the approach are (i) a detailed analysis of the λ-dependence of the minimization problem, (ii) action estimates controlling in terms of spectral gap lengths, and (iii) Weyl-law and edge-ODE arguments that convert spectral-gap information into and norm bounds for solutions. The results yield infinitely many normalized nonlinear Schrödinger solutions on compact metric graphs and, in the torus and biharmonic settings, analogous multiplicity for prescribed mass, highlighting the method’s robustness beyond the mass-critical regime and its utility for higher-order problems.

Abstract

We consider abstract nonlinear equations of the form , where is a self-adjoint operator with compact resolvent on a Hilbert space , is a parameter, and is a superlinear term of variational nature. In this abstract setting, we develop a new approach to existence and multiplicity of solutions with prescribed norm in . We then consider various applications of this approach. First, we obtain, under fairly general assumptions including the mass-supercritical case, the existence of infinitely many solutions to a class of nonlinear (time-independent) Schrödinger equations on a compact graph with prescribed (arbitrarily large) mass. Moreover, we derive a similar existence and multiplicity result for a biharmonic semilinear equation in the -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 -dependent action functional, where varies in a spectral gap between sufficiently large eigenvalues of . The key observation in this abstract framework is the fact that the -norms of these -dependent solution families form connected sets. 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 classical bound from analytic number theory.
Paper Structure (14 sections, 27 theorems, 189 equations)

This paper contains 14 sections, 27 theorems, 189 equations.

Table of Contents

  1. Introduction
  2. Existence of Nehari-Pankov solutions for fixed $\lambda$
  3. The $\lambda$-dependence of the minimization problem and normalized solutions
  4. Norms of Nehari-Pankov solutions
  5. General action estimates
  6. An additional action estimate
  7. Further estimates in the case of a metric measure space
  8. Nonlinear Schrödinger equations on compact metric graphs
  9. Setup
  10. Weyl's law, large spectral gaps and large $L^\infty$ norms
  11. ODE arguments and the $L^\infty$ to $L^2$ estimate
  12. Sign of solutions in the presence of a constant-sign eigenfunction
  13. Proof of Theorem \ref{['graph_mult']}
  14. The nonlinear biharmonic equation on the $2$-torus

Key Result

Theorem 1.1

Let $\mathcal{G}$ be a compact metric graph and $(D(A), A)$ be a self-adjoint realisation of the Laplacian on $\mathcal{G}$ (acting as the second derivative on every edge). Let $f \in \mathcal{C}(\mathbb{R}, \mathbb{R})$ satisfy the following two assumptions: Then, for any $\mu > 0$, the nonlinear Schrödinger equation $-u" = \lambda u + f(u)$ coupled with vertex conditions from $D(A)$ has infinit

Theorems & Definitions (53)

  • Theorem 1.1
  • Remark 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Remark 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Theorem 1.8
  • Lemma 2.1
  • proof
  • ...and 43 more