Three nontrivial solutions of boundary value problems for semilinear $\Delta_{\gamma}-$Laplace equation

In this paper, we study the existence of multiple solutions for the boundary value problem\begin{equation}\Delta_{\gamma} u+f(x,u)=0 \quad \mbox{ in } \Omega, \quad \quad u=0 \quad \mbox{ on } \partial \Omega, \notag\end{equation}where $\Omega$ is a bounded domain with smooth boundary in $\mathbb{R}^N \ (N \ge 2)$ and $\Delta_{\gamma}$ is the subelliptic operator of the type $$\Delta_\gamma: =\sum\limits_{j=1}^{N}\partial_{x_j} \left(\gamma_j^2 \partial_{x_j} \right), \ \partial_{x_j}=\frac{\partial }{\partial x_{j}}, \gamma = (\gamma_1, \gamma_2, ..., \gamma_N), $$the nonlinearity $f(x , \xi)$ is subcritical growth and may be not satisfy the Ambrosetti-Rabinowitz (AR) condition. We establish the existence of three nontrivial solutions by using Morse theory.

Singular perturbations for a subelliptic operator

We study some classes of singular perturbation problems where the dynamics of the fast variables evolve in the whole space obeying to an infinitesimal operator which is subelliptic and ergodic. We prove that the corresponding ergodic problem admits a solution which is globally Lipschitz continuous and it has at most a logarithmic growth at infinity. The main result of this paper establishes that, as ϵ → 0, the value functions of the singular perturbation problems converge locally uniformly to the solution of an effective problem whose operator and terminal data are explicitly given in terms of the invariant measure for the ergodic operator.

Local solvability and hypoellipticity for pseudodifferential operators of Egorov type with infinite degeneracy

Let P be a pseudodifferential operator of the formwhere s, b ≥ 0 are even integers and is odd function with f′(t) > 0 (t ≠ 0). Here . We shall call P an operator of Egorov type because P with f(t) = tk, (k odd) is an important model of subelliptic operators studied by Egorov [1] and Hörmander [3], [4, Chapter 27]. Roughly speaking, any subelliptic operator can be reduced to this operator or Mizohata one after several steps of microlocalization arguments. In this paper we shall study the hypoellipticity of P and the local solvability of adjoint operator P* in the case where f(t) vanishes infinitely at the origin and moreover consider the case where ts and are replaced by functions with zero of infinite order.