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

Abstract

The higher order Kirchhoff type equation $${\int }_{{\mathbb{R}}^{2m}}(|{\nabla }^{m}u{|}^2+\sum _{\gamma =0}^{m-1}{a}_{\gamma }(x)|{\nabla }^{\gamma }u{|}^2)dx\left((-\Delta {)}^{m}u+\sum _{\gamma =0}^{m-1}(-1{)}^{\gamma }{\nabla }^{\gamma }\cdot (a_{\gamma }(x){\nabla }^{\gamma }u)\right)=\frac{f(x,u)}{|x{|}^{\beta }}+\epsilon h(x)\text{in}{\mathbb{R}}^{2m}$$ is considered in this paper. We assume that the nonlinearity of the equation has exponential critical growth and prove that, for a positive $\epsilon$ which is small enough, there are two distinct nontrivial solutions to the equation. When $\epsilon =0$, we also prove that the equation has a nontrivial mountain-pass type solution.

14 pages