Existence results of infinitely many weak solutions of a singular subelliptic system on the Heisenberg group

Abstract This article shows the existence and multiplicity of weak solutions for the singular subelliptic system on the Heisenberg group { -Δℍnu+a(ξ)u(| z |4+t2)12=λFu(ξ,u,v)in   Ω,-Δℍnv+b(ξ)v(| z |4+t2)12=λFv(ξ,u,v)in   Ω,u=v=0on  ∂Ω. \left\{ {\matrix{ { - {\Delta _{{\mathbb{H}^n}}}u + a\left( \xi \right){u \over {{{\left( {{{\left| z \right|}^4} + {t^2}} \right)}^{{1 \over 2}}}}} = \lambda {F_u}\left( {\xi ,u,v} \right)} \hfill & {in\,\,\,\Omega ,} \hfill \cr { - {\Delta _{{\mathbb{H}^n}}}v + b\left( \xi \right){v \over {{{\left( {{{\left| z \right|}^4} + {t^2}} \right)}^{{1 \over 2}}}}} = \lambda {F_v}\left( {\xi ,u,v} \right)} \hfill & {in\,\,\,\Omega ,} \hfill \cr {u = v = 0} \hfill & {on\,\,\partial \Omega .} \hfill \cr } } \right. The approach is based on variational methods.