Existence and non-existence of solutions to a Hamiltonian strongly degenerate elliptic system

We study the non-existence and existence of infinitely many solutions to the semilinear degenerate elliptic system \left\{\begin{aligned} &\displaystyle{-}\Delta_{\lambda}u=\lvert v\rvert^{p-1}% v&&\displaystyle\phantom{}\text{in }\Omega,\\ &\displaystyle{-}\Delta_{\lambda}v=\lvert u\rvert^{q-1}u&&\displaystyle% \phantom{}\text{in }\Omega,\\ &\displaystyle u=v=0&&\displaystyle\phantom{}\text{on }\partial\Omega,\end{% aligned}\right. in a bounded domain {\Omega\subset\mathbb{R}^{N}} with smooth boundary {\partial\Omega} . Here {p,q>1} , and {\Delta_{\lambda}} is the strongly degenerate operator of the form \Delta_{\lambda}u=\sum^{N}_{j=1}\frac{\partial}{\partial x_{j}}\Bigl{(}\lambda% _{j}^{2}(x)\frac{\partial u}{\partial x_{j}}\Bigr{)}, where {\lambda(x)=(\lambda_{1}(x),\dots,\lambda_{N}(x))} satisfies certain conditions.