### The Existence of Positive Solutions to Kirchhoff Type Equations in $\mathbb{R}^{N}$ With Asymptotic Nonlinearity

#### Abstract

In this paper, we are concerned with the following Kirchhoff problem

$$\left\{%

\begin{array}{ll}\vspace{0.2cm}

\left(a+\lambda\int_{\mathbb{R}^N}(|\nabla u|^2+V(x)|u|^2)\right)[-\Delta u+V(x)u]=f(x,u), & \hbox{$x\in \mathbb{R}^N$},\\

u\in H^1(\mathbb{R}^N),~~~~u>0, & \hbox{$x\in \mathbb{R}^N$},

\end{array}%

\right.$$

where $N\geq3$, $a>0$ is a constant, $\lambda>0$ is a parameter, the potential $V(x)$ may not be radially symmetric and $f(x,s)$ is asymptotically linear with respect to $s$ at infinity. Under some assumptions on $V$ and $f$, we prove the existence of a positive solution for $\lambda$ small and the nonexistence result for $\lambda$ large.

$$\left\{%

\begin{array}{ll}\vspace{0.2cm}

\left(a+\lambda\int_{\mathbb{R}^N}(|\nabla u|^2+V(x)|u|^2)\right)[-\Delta u+V(x)u]=f(x,u), & \hbox{$x\in \mathbb{R}^N$},\\

u\in H^1(\mathbb{R}^N),~~~~u>0, & \hbox{$x\in \mathbb{R}^N$},

\end{array}%

\right.$$

where $N\geq3$, $a>0$ is a constant, $\lambda>0$ is a parameter, the potential $V(x)$ may not be radially symmetric and $f(x,s)$ is asymptotically linear with respect to $s$ at infinity. Under some assumptions on $V$ and $f$, we prove the existence of a positive solution for $\lambda$ small and the nonexistence result for $\lambda$ large.

This work is licensed under a Creative Commons Attribution 3.0 License.

Journal of Mathematics Research ISSN 1916-9795 (Print) ISSN 1916-9809 (Online)

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