Existence and Non-existence of Solutions for Semilinear bi$-\Delta _{\gamma }-$Laplace Equation

Abstract

In this paper, we study existence and non-existence of weak solutions for semilinear bi$-\Delta _{\gamma }-$Laplace equation

$$\begin{aligned} \Delta ^2_\gamma u=f(x,u) \ \text { in }\Omega , \quad u= \partial _\nu u =0 \; \text { on }\partial \Omega , \end{aligned}$$

where $\Omega$ is a bounded domain with smooth boundary in $\mathbb {R}^N \ (N \ge 2), f(x,\xi )$ is a Carathéodory function and $\Delta _{\gamma }$ is the subelliptic operator of the type

$$\begin{aligned} \Delta _\gamma : =\sum \limits _{j=1}^{N}\partial _{x_j} \left( \gamma _j^2 \partial _{x_j} \right) , \quad \partial _{x_j}: =\frac{\partial }{\partial x_{j}}, \gamma = (\gamma _1, \gamma _2, ..., \gamma _N),\quad \Delta ^2_\gamma : =\Delta _\gamma (\Delta _\gamma ). \end{aligned}$$