Solutions for a quasilinear elliptic problem with indefinite nonlinearity with critical growth

Abstract

We are interested in nonhomogeneous problems with a nonlinearity that changes sign and may possess a critical growth as follows { − div ⁡ ( a ( | ∇ u | p ) | ∇ u | p − 2 ∇ u ) = λ | u | q − 2 u + W ( x ) | u | r − 2 u in Ω , u = 0 on ∂ Ω , where Ω ⊂ R N is a bounded domain with smooth boundary ∂ Ω , N ≥ 2 , 1 < p ≤ q < N , q < r ≤ q ∗ , λ ∈ R and function W is a weight function which changes sign in Ω. Using variational methods, we prove the existence of four solutions: two solutions which do not change sign and two solutions which change sign exactly once in Ω.