# ON A CLASS OF QUASILINEAR ELLIPTIC EQUATION WITH INDEFINITE WEIGHTS ON GRAPHS

Man, Shoudong;Zhang, Guoqing

• Accepted : 2019.04.01
• Published : 2019.07.01
• 29 5

#### Abstract

Suppose that G = (V, E) is a connected locally finite graph with the vertex set V and the edge set E. Let ${\Omega}{\subset}V$ be a bounded domain. Consider the following quasilinear elliptic equation on graph G $$\{-{\Delta}_{pu}={\lambda}K(x){\mid}u{\mid}^{p-2}u+f(x,u),\;x{\in}{\Omega}^{\circ},\\u=0,\;x{\in}{\partial}{\Omega},$$ where ${\Omega}^{\circ}$ and ${\partial}{\Omega}$ denote the interior and the boundary of ${\Omega}$, respectively, ${\Delta}_p$ is the discrete p-Laplacian, K(x) is a given function which may change sign, ${\lambda}$ is the eigenvalue parameter and f(x, u) has exponential growth. We prove the existence and monotonicity of the principal eigenvalue of the corresponding eigenvalue problem. Furthermore, we also obtain the existence of a positive solution by using variational methods.

#### Keywords

indefinite weights;quasilinear elliptic equation on graphs;eigenvalue problem on graphs

#### Acknowledgement

Supported by : National Natural Science Foundation of China