The Existence of Global Solution for a Class of Semilinear Equations on Heisenberg Group

Abstract

Based on the concepts of a generalized critical point and the corresponding generalized P.S. condition introduced by Duong Minh Duc[1], we have proved a new $Z_2$ index theorem and get a result on multiplicity of generalized critical points. Using the result and a quite standard variational method, it is found that the equation $$-{\Delta }_{H^n}u=|u{|}^{p-1}ux\in H^n$$ has infinite positive solutions. Our approach can also be applied to study more general nonlinear problems.

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