# On the Kirchhoff type equations in $\mathbb{R}^{N}$

## TL;DR

This paper investigates the existence, multiplicity, and nonexistence of positive solutions for a nonlinear Kirchhoff type equation in \\mathbb{R}^N, emphasizing the geometric properties of the associated energy functional and how solutions depend on parameters and dimension.

## Contribution

It provides new insights into the geometric analysis of the energy functional and establishes conditions for positive solutions based on the dimension and parameters, extending previous results.

## Key findings

- Unique positive solution for 1 ≤ N ≤ 4.
- At least two positive solutions for N ≥ 5.
- Dependence of solution existence on parameters a and dimension N.

## Abstract

Consider a nonlinear Kirchhoff type equation as follows \begin{equation*} \left\{ \begin{array}{ll} -\left( a\int_{\mathbb{R}^{N}}|\nabla u|^{2}dx+b\right) \Delta u+u=f(x)\left\vert u\right\vert ^{p-2}u & \text{ in }\mathbb{R}^{N}, \\ u\in H^{1}(\mathbb{R}^{N}), & \end{array}% \right. \end{equation*}% where $N\geq 1,a,b>0,2<p<\min \left\{ 4,2^{\ast }\right\}$($2^{\ast }=\infty $ for $N=1,2$ and $2^{\ast }=2N/(N-2)$ for $N\geq 3)$ and the function $f\in C(\mathbb{R}^{N})\cap L^{\infty }(\mathbb{R}^{N})$. Distinguishing from the existing results in the literature, we are more interested in the geometric properties of the energy functional related to the above problem. Furthermore, the nonexistence, existence, unique and multiplicity of positive solutions are proved dependent on the parameter $a$ and the dimension $N.$ In particular, we conclude that a unique positive solution exists for $1\leq N\leq4$ while at least two positive solutions are permitted for $N\geq5$.