Multiple Solutions to p-Biharmonic Equations of Kirchhoff Type with Vanishing Potential. (17th February 2023) - Chung, N. T.; Ghanmi, A.; Kenzizi, T.
Abstract: In this paper, we study the p -biharmonic equation of Kirchhoff type M0001 Δ p 2 u − ( a + b ∫ R N | ∇ u | p d x ) Δ p u + V ( x ) | u | p − 2 u = K ( x ) f ( u ) + λ g ( x ) | u | q − 2 u, x in R N ; u in W 2, p ( R N ) ∩ W 0 1, p ( R N ) . where N ≥ 5, 1 < q < p < N 2, a > 0, b ≥ 0, λ is a positive parameter, Δ p u = div ( | ∇ u | p − 2 ∇ u ) is the p -Laplacian operator and Δ p 2 u = Δ ( | Δ u | p − 2 Δ u ) is the p -biharmonic operator, V, K, g are nonnegative functions, V is vanishing at infinity in the sense that lim | x | → + ∞ V ( x ) = 0 . When the nonlinear term f ( u ) satisfies some suitable conditions, we prove that the above problem has at least two nontrivial solutions using the mountain pass theorem combined with the Ekeland variational principle.,