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.

Keywords

• singular potential
• variational methods
• infinitely many solutions
• heisenberg group
• 35r03
• 35j20
• 35j15
• 58e30
• 35j61