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

TL;DR

This paper proves the existence of multiple solutions for a higher order Kirchhoff type equation with exponential critical growth in \\mathbb{R}^{2m}, demonstrating solutions' existence under small perturbations and at the unperturbed case.

Contribution

It establishes the existence of two solutions for small \\epsilon > 0 and a mountain-pass solution when \\epsilon=0, for a Kirchhoff problem with exponential critical growth.

Findings

Existence of two solutions for small \\epsilon > 0.

Existence of a mountain-pass solution at \\epsilon=0.

Solutions are nontrivial and distinct.

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.