The Dirichlet problem with sublinear indefinite nonlinearities

We consider the Dirichlet problem with sublinear indefinite nonlinearities −Δu−λu=h−(x)g1(u)+h+(x)g2(u) with h−(x)≤0 and h+(x)≥0. Using decomposition of the space H01(Ω), we construct a special deformation, and then compute the Morse critical groups at zero of the energy functional. Combining with the Morse critical groups at infinity, some existences, such as the existence of a nodal solution, are established via Morse theory.