Polyharmonic Kirchhoff-type problem without Ambrosetti–Rabinowitz condition

Abstract

In this paper, we will study the polyharmonic Kirchhoff-type problem with singular exponential nonlinearity without the Ambrosetti–Rabinowitz condition: { − M ( ∫ Ω | ∇ m u | n m ) Δ n m m u = f ( x , u ) | x | σ in Ω , u = ∇ u = ⋯ = ∇ m − 1 u = 0 on ∂ Ω , where Ω ⊂ R n is a bounded domain with smooth boundary, 0 < σ < n , n ≥ 2 m ≥ 2 , M is a Kirchhoff function and f ( x , u ) has critical exponential growth. We use a suitable version of the Mountain Pass Theorem to prove the existence of a positive ground state solution for this problem.