Critical groups at infinity for p-Laplacian equations with indefinite nonlinearities

In this paper, by a kind of decomposition lemma and Künneth formula we study the critical groups at infinity for the associated functional of the following p-Laplacian equation with indefinite nonlinearities { − Δ p u = λ | u | p − 2 u + a + ( x ) f ( u ) − a − ( x ) | u | q − 2 u + h ( x , u ) in Ω , u = 0 on ∂ Ω . We also obtain the existence of multiple nontrivial solutions of this equation via Morse theory.