Positive solutions to an elliptic equation in $\mathbb{R}^N$ of the Kirchhoff type

In this paper, we consider the following Kirchhoff type problem \left\{\aligned&-\biggl(a + b\int_{\mathbb{R}^N} |\nabla u|^2 dx \biggr) \Delta u + V(x) u = |u|^{p-2}u &\text{ in } \mathbb{R}^N,\cr &u\in H^1(\mathbb{R}^N), \endaligned\right. \eqno{(\mathcal{P}_{a,b})} where $N\geq3$, $2<p<2^*=\frac{2N}{N-2}$, $a,b>0$ are parameters and $V(x)$ is a potential function. Under some mild conditions on $V(x)$, we prove that $(\mathcal{P}_{a,b})$ has a positive solution for $b$ small enough by the variational method, a non-existence result is also established in the cases $N\geq4$. Our results in the case $N=3$ partial improve the results in \cite{G15,LY14} and our results in the cases $N\geq4$ are totally new to the best of our knowledge. By combining the scaling technique, we also give a global description on the structure of the positive solutions to the autonomous form of $(\mathcal{P}_{a,b})$, that is $V(x)\equiv\lambda>0$. This result can be seen as a partial complement of the studies in \cite{A12,A13}.