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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 $G$ $$ \left \{ \begin{array}{lcr} -\Delta_{p}u= \lambda K(x)|u|^{p-2}u+ f(x,u), \ x\in\Omega^{\circ},\\ u=0, \ x\in\partial \Omega, \end{array} \right. $$ 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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참고문헌 (16)
참고문헌 신청
A. Ambrosetti / 1973 / Dual variational methods in critical point theory and applications / J. Functional Analysis 14:349~381
M. Degiovanni / 2007 / Linking over cones and nontrivial solutions for p-Laplace equations with p-superlinear nonlinearity / Ann. Inst. H. Poincar´e Anal. Non Lin´eaire 24(6):907~919
X. Fan / 2000 / Linking and existence results for perturbations of the p-Laplacian / Nonlinear Anal 42(8):1413~1420
H. Ge / 2018 / A p-th Yamabe equation on graph / Proc. Amer. Math. Soc 146(5):2219~2224
H. Ge / 2018 / The 1-Yamabe equation on graph / Commun. in Contemporary Math 20(5):32~38
M. Degiovanni / 2007 / Linking over cones and nontrivial solutions for p-Laplace equations with p-superlinear nonlinearity / Ann. Inst. H. Poincar´e Anal. Non Lin´eaire 24(6):907~919
X. Fan / 2000 / Linking and existence results for perturbations of the p-Laplacian / Nonlinear Anal 42(8):1413~1420
H. Ge / 2018 / A p-th Yamabe equation on graph / Proc. Amer. Math. Soc 146(5):2219~2224
H. Ge / 2018 / The 1-Yamabe equation on graph / Commun. in Contemporary Math 20(5):32~38
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