We prove generic multiplicity of solutions for a scalar field equation on compact surfaces via Morse inequalities. In particular our result improves significantly the multiplicity estimate which can be deduced from the degree-counting formula in Chen and Lin (2003) [12]. Related results are derived for the prescribed Q-curvature equation.

10aGeneric multiplicity10aGeometric PDE's10aMorse inequalities10aScalar field equations1 aDe Marchis, Francesca uhttp://www.sciencedirect.com/science/article/pii/S0022123610002697