Yamabe type equations on graphs

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• 作者：Alexander Grigor'yan^a ; ^{grigor@math.uni-bielefeld.de" class="auth_mail" title="E-mail the corresponding author} ; Yong Lin^b ; ^{linyong01@ruc.edu.cn" class="auth_mail" title="E-mail the corresponding author} ; Yunyan Yang^b ; ^{yunyanyang@ruc.edu.cn" class="auth_mail" title="E-mail the corresponding author}
• 关键词：34B45 ; 35A15 ; 58E30
• 刊名：Journal of Differential Equations
• 出版年：2016
• 出版时间：5 November 2016
• 年：2016
• 卷：261
• 期：9
• 页码：4924-4943
• 全文大小：317 K

文摘

Let class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022039616301760&_mathId=si1.gif&_user=111111111&_pii=S0022039616301760&_rdoc=1&_issn=00220396&md5=5c0550a174b7c2b603ee8dfed4fab9af" title="Click to view the MathML source">G=(V,E)class="mathContainer hidden">class="mathCode">$G=(V,E)$ be a locally finite graph, class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022039616301760&_mathId=si2.gif&_user=111111111&_pii=S0022039616301760&_rdoc=1&_issn=00220396&md5=b055c95fa158e14af49b3ea4228aad65" title="Click to view the MathML source">Ω⊂Vclass="mathContainer hidden">class="mathCode">$\mathrm{\Omega }\subset V$ be a bounded domain, Δ be the usual graph Laplacian, and class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022039616301760&_mathId=si3.gif&_user=111111111&_pii=S0022039616301760&_rdoc=1&_issn=00220396&md5=1cc12c5e63b17b058342a7810b75845e" title="Click to view the MathML source">λ₁(Ω)class="mathContainer hidden">class="mathCode">${\lambda }_1(\mathrm{\Omega })$ be the first eigenvalue of −Δ with respect to Dirichlet boundary condition. Using the mountain pass theorem due to Ambrosetti–Rabinowitz, we prove that if class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022039616301760&_mathId=si220.gif&_user=111111111&_pii=S0022039616301760&_rdoc=1&_issn=00220396&md5=9e803a4c36d0ead96660bdac88fb0b85" title="Click to view the MathML source">α<λ₁(Ω)class="mathContainer hidden">class="mathCode">$\alpha <{\lambda }_1(\mathrm{\Omega })$, then for any class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022039616301760&_mathId=si5.gif&_user=111111111&_pii=S0022039616301760&_rdoc=1&_issn=00220396&md5=65e8fdd158b842879d6c79a28527a8a9" title="Click to view the MathML source">p>2class="mathContainer hidden">class="mathCode">$p>2$, there exists a positive solution to where class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022039616301760&_mathId=si231.gif&_user=111111111&_pii=S0022039616301760&_rdoc=1&_issn=00220396&md5=dcdc6edb867fd635a3cfd08febc662d7" title="Click to view the MathML source">Ω^∘class="mathContainer hidden">class="mathCode">${\mathrm{\Omega }}^{\circ }$ and ∂Ω denote the interior and the boundary of Ω respectively. Also we consider similar problems involving the p-Laplacian and poly-Laplacian by the same method. Such problems can be viewed as discrete versions of the Yamabe type equations on Euclidean space or compact Riemannian manifolds.