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Ground state solutions to generalized nonlinear wave equations with infinite-dimensional kernel

Rainer Mandel, Tobias Weth

TL;DR

The paper addresses the existence of time-periodic solutions to generalized nonlinear wave equations on closed Riemannian manifolds in the doubly degenerate regime where the operator $L_{\mathcal{A}}=\mathcal{A}+\partial_{tt}$ has an infinite-dimensional kernel, and the weight $q(x,t)$ may vanish on open sets.A direct variational strategy is developed via a nonlinear saddle point reduction on the Nehari-Pankov set, yielding a minimax characterization of the ground-state energy $c=\inf_{\mathcal{N}_{\mathcal{P}}}\Phi=\inf_{w\in E^+\setminus\{0\}}\sup_{E_w}\Phi$ and guaranteeing the existence of ground states.The authors formulate an abstract framework $(E=E^+\oplus E^0\oplus E^-)$ with a functional $\Phi(u)=\tfrac{1}{2}(\|u^+\|^2-\|u^-\|^2)-I(u)$ and verify conditions $(I0)-(I3)$, $(S1)-(S2)$ to ensure a saddle-point reduction and minimization, then apply it to generalized nonlinear wave equations by checking $(CE)_p$ and $(CC)_q$ for several geometries.Key contributions include (i) an abstract existence theorem for ground states with a verifiable reduction framework, (ii) compact embedding criteria for $E^\pm$ into $L^p$ on tori and spheres, and (iii) a detailed control-condition analysis enabling ground-state results even when $q$ vanishes on open sets; collectively, these extend variational methods to settings with infinite-dimensional kernels and degeneracies.

Abstract

The present paper is devoted to existence results for time-periodic solutions of generalized nonlinear wave equations in a closed Riemannian manifold M. Our main focus lies on the doubly degenerate setting where the associated generalized wave operator has an infinite dimensional kernel and the nonlinearity may vanish on open subsets of M. To deal with this setting, we apply a direct variational approach based on a new variant of the nonlinear saddle point reduction to the associated Nehari-Pankov set. This allows us to find ground state solutions and to characterize the associated ground state energy by a fairly simple minimax principle.

Ground state solutions to generalized nonlinear wave equations with infinite-dimensional kernel

TL;DR

The paper addresses the existence of time-periodic solutions to generalized nonlinear wave equations on closed Riemannian manifolds in the doubly degenerate regime where the operator $L_{\mathcal{A}}=\mathcal{A}+\partial_{tt}$ has an infinite-dimensional kernel, and the weight $q(x,t)$ may vanish on open sets.A direct variational strategy is developed via a nonlinear saddle point reduction on the Nehari-Pankov set, yielding a minimax characterization of the ground-state energy $c=\inf_{\mathcal{N}_{\mathcal{P}}}\Phi=\inf_{w\in E^+\setminus\{0\}}\sup_{E_w}\Phi$ and guaranteeing the existence of ground states.The authors formulate an abstract framework $(E=E^+\oplus E^0\oplus E^-)$ with a functional $\Phi(u)=\tfrac{1}{2}(\|u^+\|^2-\|u^-\|^2)-I(u)$ and verify conditions $(I0)-(I3)$, $(S1)-(S2)$ to ensure a saddle-point reduction and minimization, then apply it to generalized nonlinear wave equations by checking $(CE)_p$ and $(CC)_q$ for several geometries.Key contributions include (i) an abstract existence theorem for ground states with a verifiable reduction framework, (ii) compact embedding criteria for $E^\pm$ into $L^p$ on tori and spheres, and (iii) a detailed control-condition analysis enabling ground-state results even when $q$ vanishes on open sets; collectively, these extend variational methods to settings with infinite-dimensional kernels and degeneracies.

Abstract

The present paper is devoted to existence results for time-periodic solutions of generalized nonlinear wave equations in a closed Riemannian manifold M. Our main focus lies on the doubly degenerate setting where the associated generalized wave operator has an infinite dimensional kernel and the nonlinearity may vanish on open subsets of M. To deal with this setting, we apply a direct variational approach based on a new variant of the nonlinear saddle point reduction to the associated Nehari-Pankov set. This allows us to find ground state solutions and to characterize the associated ground state energy by a fairly simple minimax principle.
Paper Structure (19 sections, 33 theorems, 207 equations, 1 table)

This paper contains 19 sections, 33 theorems, 207 equations, 1 table.

Table of Contents

  1. Introduction
  2. Organization of the paper
  3. An abstract existence result for ground state solutions
  4. Preliminary observations
  5. The saddle point reduction and ground states
  6. Completion of the proof of Theorem \ref{['main-abstract-result']}
  7. Some useful criteria
  8. Existence of ground state solutions to generalized nonlinear wave equations
  9. Verification of condition $(I0)$.
  10. Verification of condition $(I1)$.
  11. Verification of condition $(I2)$.
  12. Verification of condition $(I3)$.
  13. Verification of condition (S1).
  14. Verification of condition (S2).
  15. Compact Embeddings
  16. ...and 4 more sections

Key Result

Theorem 1.2

Let $\mathcal{A}$ be a selfadjoint operator in $L^2(M)$ satisfying the assumption (A). Suppose that $p>2$ and $q \in L^\infty(M \times \mathbb{S}^1)$, $q \ge 0$ are chosen with the properties that $q>0$ on some open subset of $M \times \mathbb{S}^1$ and that conditions $(CE)_p$ and $(CC)_q$ hold. Th

Theorems & Definitions (65)

  • Definition 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Remark 1.6
  • Theorem 1.7
  • Theorem 1.8
  • Remark 1.9
  • Theorem 2.1
  • ...and 55 more