On a p -Kirchhoff equation via Fountain Theorem and Dual Fountain Theorem

In this paper, we show the existence of infinite solutions to the Kirchhoff type quasilinear elliptic equation [ M ( ∫ Ω ( ∣ ∇ u ∣ p + λ ( x ) ∣ u ∣ p ) d x ) ] p − 1 ( − Δ p u + λ ( x ) ∣ u ∣ p − 2 u ) = f ( x , u ) in a smooth bounded domain Ω ⊂ R N with nonlinear boundary condition ∣ ∇ u ∣ p − 2 ∂ u ∂ ν = η ∣ u ∣ p − 2 on ∂ Ω . The method we used here is based on “Fountain Theorem” and “Dual Fountain Theorem”.