Infinitely many solutions for quasilinear Schrödinger equation with concave-convex nonlinearities

In this work, we study the existence of infinitely many solutions to the following quasilinear Schrödinger equations with a parameter α and a concave-convex nonlinearity: 0.1 − Δ p u + V ( x ) | u | p − 2 u − Δ p ( | u | 2 α ) | u | 2 α − 2 u = λ h 1 ( x ) | u | m − 2 u + h 2 ( x ) | u | q − 2 u , x ∈ R N , where Δ p u = div ( | ∇ u | p − 2 ∇ u ) , 1 < p < N , λ ≥ 0 , and 1 < m < p < 2 α p < q < 2 α p ∗ = 2 α p N N − p . The functions V ( x ) , h 1 ( x ) , and h 2 ( x ) satisfy some suitable conditions. Using variational methods and some special techniques, we prove that there exists λ 0 > 0 such that Eq. ( 0.1 ) admits infinitely many high energy solutions in W 1 , p ( R N ) provided that λ ∈ [ 0 , λ 0 ] .