Morse theory for indefinite nonlinear elliptic problems

Using the heat flow as a deformation, a Morse theory for the solutions of the nonlinear elliptic equation:−Δu−λu=a+(x)|u|q−1u−a−(x)|u|p−1u+h(x,u)−Δu−λu=a+(x)|u|q−1u−a−(x)|u|p−1u+h(x,u) in a bounded domain Ω⊂RNΩ⊂RN with the Dirichlet boundary condition is established, where a±⩾0a±⩾0, supp(a−)∩supp(a+)=∅supp(a−)∩supp(a+)=∅, supp(a+)≠∅supp(a+)≠∅, 11p>1. Various existence and multiplicity results of solutions are presented.