Multiple sign-changing solutions for semilinear subelliptic Dirichlet problem

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 ${△}_X=-{\sum }_{i=1}^m{X}_i^{\ast }{X}_i$ is the self-adjoint sub-elliptic operator associated with Hörmander vector fields $X=(X_1,X_2,\dots ,X_m)$, $\hat{I}\text{c◯}$ is an open bounded subset in ${\mathbb{R}}^n$, and $H_{X,0}^1(\hat{I}\text{c◯})$ 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 $-{△}_X$, 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.

47 pages