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In this paper, we study the semilinear subelliptic equation \[ \left\{  \begin{array}{cc}  -\triangle_{X} u=f(x,u)+g(x,u) & \mbox{in}~\Omega, \\[2mm]  u=0\hfill & \mbox{on}~\partial\Omega,  \end{array}  \right. \] where $\triangle_{X}=-\sum_{i=1}^{m}X_{i}^{*}X_{i}$ is theself-adjoint H\"{o}rmander operator associated with vector fields$X=(X_{1},X_{2},\ldots,X_{m})$ satisfying the H\"{o}rmander condition,$f(x,u)\in C(\overline{\Omega}\times \mathbb{R})$, $g(x,u)$ is aCarath\'{e}odory function on $\Omega\times \mathbb{R}$, and $\Omega$ is an openbounded domain in $\mathbb{R}^n$ with smooth boundary. Combining theperturbation from symmetry method with the approaches involving eigenvalueestimate and Morse index in estimating the min-max values, we obtain two kindsof existence results for multiple weak solutions to the problem above.Furthermore, we discuss the difference between the eigenvalue estimate approachand the Morse index approach in degenerate situations. Compared with theclassical elliptic cases, both approaches here have their own strengths in thedegenerate cases. This new phenomenon implies the results in general degeneratecases would be quite different from the situations in classical elliptic cases.

Multiplicity of solutions for semilinear subelliptic Dirichlet problem