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Multiple sign-changing solutions for semilinear subelliptic Dirichlet problem

Hua Chen, Hong-Ge Chen, Jin-Ning Li, Xin Liao

TL;DR

This work extends multiplicity results for sign-changing solutions to semilinear subelliptic Dirichlet problems driven by Hörmander vector fields. By combining a perturbation-from-symmetry strategy with two distinct lower-bound mechanisms for sign-changing min-max values, the authors obtain unbounded sequences of sign-changing weak solutions under two different hypotheses: (A1) tied to a Dirichlet eigenvalue lower bound (via (L)), and (A2) based on augmented Morse-indices and a degenerate Cwikel–Lieb–Rozenblum inequality linked to the generalized Métivier index. The two approaches reveal essential differences from the classical elliptic theory in non-equiregular settings and provide complementary multiplicity criteria, including an intersection lemma and a Marino–Prodi perturbation adapted to degenerate subelliptic operators. The results broaden the existence theory for sign-changing solutions in degenerate subelliptic geometries and illustrate how subellipticity generates phenomena not present in the elliptic case. An explicit non-equiregular example underscores the distinct roles of (A1) and (A2).

Abstract

We study the following perturbation from symmetry problem for the semilinear subelliptic equation \[ \left\{ \begin{array}{cc} -\triangle_{X} u=f(x,u)+g(x,u) & \mbox{in}~Ω, \\[2mm] u\in H_{X,0}^{1}(Ω),\hfill \end{array} \right. \] where $\triangle_{X}=-\sum_{i=1}^{m}X_{i}^{*}X_{i}$ is the self-adjoint sub-elliptic operator associated with Hörmander vector fields $X=(X_{1},X_{2},\ldots,X_{m})$, $Ω$ is an open bounded subset in $\mathbb{R}^n$, and $H_{X,0}^{1}(Ω)$ denotes the weighted Sobolev space. We establish multiplicity results for sign-changing solutions using a perturbation method alongside refined techniques for invariant sets. The pivotal aspect lies in the estimation of the lower bounds of min-max values associated with sign-changing critical points. In this paper, we construct two distinct lower bounds of these min-max values. The first one is derived from the lower bound of Dirichlet eigenvalues of $-\triangle_{X}$, while the second one is based on the Morse-type estimates and Cwikel-Lieb-Rozenblum type inequality in degenerate cases. These lower bounds provide different sufficient conditions for multiplicity results, each with unique advantages and are not mutually inclusive, particularly in the general non-equiregular case. This novel observation suggests that in some sense, the situation for sub-elliptic equations would have essential difference from the classical elliptic framework.

Multiple sign-changing solutions for semilinear subelliptic Dirichlet problem

TL;DR

This work extends multiplicity results for sign-changing solutions to semilinear subelliptic Dirichlet problems driven by Hörmander vector fields. By combining a perturbation-from-symmetry strategy with two distinct lower-bound mechanisms for sign-changing min-max values, the authors obtain unbounded sequences of sign-changing weak solutions under two different hypotheses: (A1) tied to a Dirichlet eigenvalue lower bound (via (L)), and (A2) based on augmented Morse-indices and a degenerate Cwikel–Lieb–Rozenblum inequality linked to the generalized Métivier index. The two approaches reveal essential differences from the classical elliptic theory in non-equiregular settings and provide complementary multiplicity criteria, including an intersection lemma and a Marino–Prodi perturbation adapted to degenerate subelliptic operators. The results broaden the existence theory for sign-changing solutions in degenerate subelliptic geometries and illustrate how subellipticity generates phenomena not present in the elliptic case. An explicit non-equiregular example underscores the distinct roles of (A1) and (A2).

Abstract

We study the following perturbation from symmetry problem for the semilinear subelliptic equation \[ \left\{ \begin{array}{cc} -\triangle_{X} u=f(x,u)+g(x,u) & \mbox{in}~Ω, \\[2mm] u\in H_{X,0}^{1}(Ω),\hfill \end{array} \right. \] where is the self-adjoint sub-elliptic operator associated with Hörmander vector fields , is an open bounded subset in , and denotes the weighted Sobolev space. We establish multiplicity results for sign-changing solutions using a perturbation method alongside refined techniques for invariant sets. The pivotal aspect lies in the estimation of the lower bounds of min-max values associated with sign-changing critical points. In this paper, we construct two distinct lower bounds of these min-max values. The first one is derived from the lower bound of Dirichlet eigenvalues of , while the second one is based on the Morse-type estimates and Cwikel-Lieb-Rozenblum type inequality in degenerate cases. These lower bounds provide different sufficient conditions for multiplicity results, each with unique advantages and are not mutually inclusive, particularly in the general non-equiregular case. This novel observation suggests that in some sense, the situation for sub-elliptic equations would have essential difference from the classical elliptic framework.

Paper Structure

This paper contains 17 sections, 42 theorems, 292 equations.

Table of Contents

  1. Introduction and main results
  2. Preliminaries
  3. The framework of Hörmander vector fields
  4. Basic variational settings
  5. Perturbation from symmetry in sign-changing settings
  6. Perturbation theory
  7. Sign-changing settings
  8. The min-max values for perturbation functional $J$
  9. Estimates on augmented Morse index
  10. Deformation lemma and relative genus
  11. The sign-changing critical values of $I_{p}$
  12. The Marino-Prodi perturbation of $I_p$
  13. Lower bound estimates of Morse index
  14. Proofs of Theorem 1.1 and Theorem 1.2
  15. Proof of Theorem \ref{['thm1']}
  16. ...and 2 more sections

Key Result

Theorem 1.1

Let $X=(X_1,X_{2},\ldots,X_m)$ satisfy condition $(H)$, and let $\Omega\subset\subset U$ be a bounded open subset. Assume that for sufficiently large $k\geq 1$, the Dirichlet eigenvalue $\lambda_k$ of $-\triangle_{X}$ on $\Omega$ satisfy the lower bound assumption $(L)$. Suppose further that the fun then the problem problem1-1 possesses an unbounded sequence of sign-changing weak solutions in $H_{

Theorems & Definitions (91)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 1.1
  • Remark 1.2
  • Definition 2.1: Generalized Métivier index
  • Remark 2.1
  • Proposition 2.1: Sharp Sobolev inequality, see Chen-Chen-Li2024
  • Proposition 2.2: Rellich-Kondrachov compact embedding theorem, see Chen-Chen-Li2024
  • Proposition 2.3: Chen-Chen-Li-Liao2022
  • Proposition 2.4: Cwikel-Lieb-Rozenblum inequality, see Chen-Chen-Li-Liao2022
  • ...and 81 more