In this note, we show multiplicity of positive weak solutions for nonlinear equations with a first stratum Hardy potential on the Heisenberg and stratified groups. The mountain pass theorem, Sobolev-Folland-Stein and first stratum Hardy inequalities play key roles in our proofs.

中文翻译：

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Heisenberg群上具有Hardy势的非线性方程的正解的多重性

在此注释中，我们显示了具有第一层Hardy势的非线性方程组在Heisenberg群和分层群上的正弱解的多重性。山口定理，Sobolev-Folland-Stein和第一层Hardy不等式在我们的证明中起关键作用。