Existence of solutions for a higher order Kirchhoff type problem with exponential critical growth

Abstract

A higher order Kirchhoff type equation ∫R2m(∣∇mu∣2+∑γ=0m−1aγ(x)∣∇γu∣2)dx((−Δ)mu+∑γ=0m−1(−1)γ∇γ⋅(aγ(x)∇γu))=f(x,u)∣x∣β+ϵh(x)in R2m is considered in this paper. We assume that the nonlinearity of the equation has exponential critical growth and prove that, for a positive ϵ which is small enough, there are two distinct nontrivial solutions to the equation. When ϵ=0, we also prove that the equation has a nontrivial mountain-pass type solution.