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

The higher order Kirchhoff type equation $\int_{\mathbb{R}^{2m}}(|\nabla^m u|^2 +\sum_{γ=0}^{m-1}a_γ(x)|\nabla^γu|^2)dx \left((-Δ)^m u+\sum_{γ=0}^{m-1}(-1)^γ\nabla^γ\cdot(a_γ(x)\nabla^γu)\right) =\frac{f(x,u)}{|x|^β}+εh(x)\ \ \text{in}\ \ \mathbb{R}^{2m}$ is considered in this paper. We assume that the nonlinearity of the equation has exponential critical growth and prove that, for a positive $ε$ 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.

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