Infinitely Many Solutions for $N-$Kirchhoff Equation with Critical Exponential Growth in ${\mathbb {R}}^N$

Abstract

In this paper, we study the existence of infinitely many solutions to $N-$Kirchhoff type equation

$$\begin{aligned}&\left( a+\mu \left( \int _{\mathbb {R}^N}(|\nabla u|^N+V(x)|u|^N)\mathrm{d}x\right) ^{\tau }\right) \nonumber \\&\quad \times \;(-\Delta _Nu+V(x)|u|^{N-2}u)=f(x,u), \quad x\in \mathbb {R}^N, \end{aligned}$$

(0.1)

where $N\ge 2, f(x,u)=\lambda h(x)|u|^{m-2}u+g(u)(1<m<N)$ and g(u) behaves like $\exp (\alpha _0|u|^{\frac{N}{N-1}})$ when $|u|\rightarrow \infty$. The potential function $V(x)>0$ is continuous and bounded in $\mathbb {R}^N$, $\lambda$ is a nonnegative real parameter, $h(x)\in L^{\sigma }(\mathbb {R}^N)$ with $\sigma =\frac{N}{N-m}.$ Using variational methods and some special techniques, we prove that there exists $\lambda _0>0$ such that problem (0.1) admits infinitely many nonnegative high-energy solutions provided that $\lambda \in [0,\lambda _0)$. Also, we prove that problem (0.1) has infinitely many nonnegative solutions for $f(x,u)=h|u|^{m-2}u,(1<m<N)$.