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Let $G=(V,E)$ be a connected locally finite and weighted graph, $\Delta_p$ be the $p$-th graph Laplacian. Consider the $p$-th nonlinear equation $$-\Delta_pu+h|u|^{p-2}u=f(x,u)$$ on $G$, where $p>2$, $h,f$ satisfy certain assumptions. Grigor'yan-Lin-Yang \cite{GLY2} proved the existence of the solution to the above nonlinear equation in a bounded domain $\Omega\subset V$. In this paper, we show that there exists a strictly positive solution on the infinite set $V$ to the above nonlinear equation by modifying some conditions in \cite{GLY2}. To the $m$-order differential operator $\mathcal{L}_{m,p}$, we also prove the existence of the nontrivial solution to the analogous nonlinear equation.
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참고문헌 (37)
참고문헌 신청
Stanley Alama / 1992 / Existence of solutions for semilinear elliptic equations with indefinite linear part / Journal of Differential Equations 96(1):89~115
Claudianor O. Alves / 2009 / On multiplicity and concentration of positive solutions for a class of quasilinear problems with critical exponential growth in RN / Journal of Differential Equations 246(3):1288~1311
Frank Bauer / 2014 / The dual Cheeger constant and spectra of infinite graphs / Advances in Mathematics 251:147~194
Frank Bauer / 2017 / Sharp Davies–Gaffney–Grigor’yan Lemma on graphs / Mathematische Annalen 368(3-4):1429~1437
T. B¨uhler / 2009 / Spectral clustering based on the graph p-Laplacian / Proc. 26th Annual Int. Conf. Mach. Learning :81~88
Claudianor O. Alves / 2009 / On multiplicity and concentration of positive solutions for a class of quasilinear problems with critical exponential growth in RN / Journal of Differential Equations 246(3):1288~1311
Frank Bauer / 2014 / The dual Cheeger constant and spectra of infinite graphs / Advances in Mathematics 251:147~194
Frank Bauer / 2017 / Sharp Davies–Gaffney–Grigor’yan Lemma on graphs / Mathematische Annalen 368(3-4):1429~1437
T. B¨uhler / 2009 / Spectral clustering based on the graph p-Laplacian / Proc. 26th Annual Int. Conf. Mach. Learning :81~88
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