On a class of quasilinear elliptic equation with indefinite weights on graphs

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 $G$ $$\{\begin{array}{l}-{\Delta }_{p}u=\lambda K(x)|u{|}^{p-2}u+f(x,u),x\in {\Omega }^{\circ },u=0,x\in \partial \Omega ,\end{array}$$ 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.

References in corpus (1)

• Convergence of ground state solutions for nonlinear Schrödinger equations on graphs