Multiple solutions of a perturbed Yamabe-type equation on graph
3
조회
0
다운로드
0
인용
0
참고문헌
키워드
초록
Let $u$ be a function on a locally finite graph $G=(V, E)$ and $\Omega$ be a bounded subset of $V$. Let $\varepsilon>0$, $p>2$ and $0\leq\lambda<\lambda_1(\Omega)$ be constants, where $\lambda_1(\Omega)$ is the first eigenvalue of the discrete Laplacian, and $h: V\ra\mathbb{R}$ be a function satisfying $h\geq 0$ and $h\not\equiv 0$. We consider a perturbed Yamabe equation, say \begin{equation*}\left\{\begin{array}{lll} -\Delta u-\lambda u=|u|^{p-2}u+\varepsilon h, &{\rm in}& \Omega,\\ u=0,&{\rm on}&\p\Omega,\end{array}\ri. \end{equation*} where $\Omega$ and $\p\Omega$ denote the interior and the boundary of $\Omega$, respectively. Using variational methods, we prove that there exists some positive constant $\varepsilon_0>0$ such that for all $\varepsilon\in(0,\varepsilon_0)$, the above equation has two distinct solutions. Moreover, we consider a more general nonlinear equation \begin{equation*}\left\{\begin{array}{lll} -\Delta u=f(u)+\varepsilon h, &{\rm in}& \Omega,\\ u=0, &{\rm on}&\p\Omega,\end{array}\ri. \end{equation*} and prove similar result for certain nonlinear term $f(u)$.
Abstract
Let $u$ be a function on a locally finite graph $G=(V, E)$ and $\Omega$ be a bounded subset of $V$. Let $\varepsilon>0$, $p>2$ and $0\leq\lambda<\lambda_1(\Omega)$ be constants, where $\lambda_1(\Omega)$ is the first eigenvalue of the discrete Laplacian, and $h: V\ra\mathbb{R}$ be a function satisfying $h\geq 0$ and $h\not\equiv 0$. We consider a perturbed Yamabe equation, say \begin{equation*}\left\{\begin{array}{lll} -\Delta u-\lambda u=|u|^{p-2}u+\varepsilon h, &{\rm in}& \Omega,\\ u=0,&{\rm on}&\p\Omega,\end{array}\ri. \end{equation*} where $\Omega$ and $\p\Omega$ denote the interior and the boundary of $\Omega$, respectively. Using variational methods, we prove that there exists some positive constant $\varepsilon_0>0$ such that for all $\varepsilon\in(0,\varepsilon_0)$, the above equation has two distinct solutions. Moreover, we consider a more general nonlinear equation \begin{equation*}\left\{\begin{array}{lll} -\Delta u=f(u)+\varepsilon h, &{\rm in}& \Omega,\\ u=0, &{\rm on}&\p\Omega,\end{array}\ri. \end{equation*} and prove similar result for certain nonlinear term $f(u)$.
참고문헌 (22)
- Adimurthi, "An interpolation of Hardy inequality and Trundinger-Moser inequality in RN and its applications", Int. Math. Res. Not. IMRN, 2010(13), pp.2394~2426 (2010)[DOI]
- S. Akduman, "Nonlinear Schrödinger equation with growing potential on infinite metric graphs", Nonlinear Anal, 184, pp.258~272 (2019)[DOI]
- A. Ambrosetti, "Dual variational methods in critical point theory and applications", J. Functional Analysis, 14, pp.349~381 (1973)[DOI]
- J. M. B. do O, "N-Laplacian equations in RN with critical growth", Abstr. Appl. Anal, 2(3-4), pp.301~315 (1997)[DOI]
- J. M. do O, "On a quasilinear nonhomogeneous elliptic equation with critical growth in RN", J. Differential Equations, 246(4), pp.1363~1386 (2009)[DOI]
- Huabin Ge, "Kazdan-Warner equation on infinite graphs", 대한수학회지, 55(5), pp.1091~1101 (2018), 대한수학회
- H. Ge, "The 1-Yamabe equation on graphs", Commun. Contemp. Math, 21(8), pp.1850040~ (2019)[DOI]
- A. Grigor’yan, "Yamabe type equations on graphs", J. Differential Equations, 261(9), pp.4924~4943 (2016)[DOI]
- A. Grigor’yan, "Kazdan-Warner equation on graph", Calc. Var. Partial Differential Equations, 55(4), pp.13~ (2016)[DOI]
- A. Grigor’yan, "Existence of positive solutions to some nonlinear equations on locally finite graphs", Sci. China Math, 60(7), pp.1311~1324 (2017)[DOI]
- X. Han, "Existence and convergence of solutions for nonlinear biharmonic equations on graphs", J. Differential Equations, 268(7), pp.3936~3961 (2020)[DOI]
- P. Horn, "Volume doubling, Poincaré inequality and Gauss-ian heat kernel estimate for non-negatively curved graphs", J. Reine Angew. Math, 757, pp.89~130 (2019)[DOI]
- S. Hou, "Multiple solutions of a nonlinear biharmonic equation on graphs", ~ (2021)
- A. Huang, "Existence of solutions to mean field equations on graphs", Comm. Math. Phys, 377(1), pp.613~621 (2020)[DOI]
- M. Keller, "The Kazdan-Warner equation on canonically compactifiable graphs", Calc. Var. Partial Differential Equations, 57(2), pp.18~ (2018)[DOI]
- S. Liu, "Multiple solutions of Kazdan-Warner equation on graphs in the negative case", Calc. Var. Partial Differential Equations, 59(5), pp.15~ (2020)[DOI]
- C. Liu, "Positive solutions of Yamabe-type equations with function coefficients on graphs", J. Math. Anal. Appl, 473(2), pp.1343~1357 (2019)[DOI]
- S. Man, "On a class of nonlinear Schrödinger equations on finite graphs", Bull. Aust. Math. Soc, 101(3), pp.477~487 (2020)[DOI]
- C. Tian, "Global stability in a networked SIR epidemic model", Appl. Math. Lett, 107, pp.106444~ (2020)[DOI]
- Y. Yang, "Existence of positive solutions to quasi-linear elliptic equations with exponen-tial growth in the whole Euclidean space", J. Funct. Anal, 262(4), pp.1679~1704 (2012)[DOI]
- X. Zhang, "Positive solutions of p-th Yamabe type equations on infinite graphs", Proc. Amer. Math. Soc, 147(4), pp.1421~1427 (2019)[DOI]
- N. Zhang, "Convergence of ground state solutions for nonlinear Schrödinger equations on graphs", Sci. China Math, 61(8), pp.1481~1494 (2018)[DOI]