Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
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ON A CLASS OF QUASILINEAR ELLIPTIC EQUATION WITH INDEFINITE WEIGHTS ON GRAPHS

  • Man, Shoudong (Department of Mathematics Tianjin University of Finance and Economics) ;
  • Zhang, Guoqing (College of Sciences University of Shanghai for Science and Technology)
  • Received : 2018.07.04
  • Accepted : 2019.04.01
  • Published : 2019.07.01
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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.

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References

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