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

Abstract

Consider a nonlinear Kirchhoff type equation as follows \begin{equation*} \begin{cases} \displaystyle - \Big ( a\int_{\mathbb{R}^{N}}|\nabla u|^{2}dx+b \Big ) \Delta u+u=f(x)\left\vert u\right\vert ^{p-2}u & \text{ in }\mathbb{R}^{N}, \\[2mm] u\in H^{1}(\mathbb{R}^{N}), & \end{cases} \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 $f \in C(\mathbb{R}^{N})\cap L^{\infty }(\mathbb{R}^{N})$. Unlike the existing results in the literature, we are more interested in the geometric properties of the energy functional related to the above problem on $H^{1}( \mathbb{R}^{N})$. As a consequence, we prove the nonexistence, existence and multiplicity of positive solutions, respectively, depending on the parameter $a$ and the dimension $N$. In particular, for the autonomous case, we conclude that a unique positive solution exists for $1\leq N\leq4$ while at least two positive solutions are permitted for $N\geq5$.