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In this paper, we consider the following Kirchhoff type problem$$\left\{\aligned&-\biggl(a + b\int_{\mathbb{R}^N} |\nabla u|^2 dx \biggr)\Delta u + V(x) u = |u|^{p-2}u &\text{ in } \mathbb{R}^N,\cr &u\inH^1(\mathbb{R}^N), \endaligned\right. \eqno{(\mathcal{P}_{a,b})} $$ where$N\geq3$, $2<p<2^*=\frac{2N}{N-2}$, $a,b>0$ are parameters and $V(x)$ is apotential function. Under some mild conditions on $V(x)$, we prove that$(\mathcal{P}_{a,b})$ has a positive solution for $b$ small enough by thevariational 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 thebest of our knowledge. By combining the scaling technique, we also give aglobal description on the structure of the positive solutions to the autonomousform of $(\mathcal{P}_{a,b})$, that is $V(x)\equiv\lambda>0$. This result canbe seen as a partial complement of the studies in \cite{A12,A13}.

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