On the solvability of asymptotically linear systems at resonance

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• 作者：Maya Chhetri^a ; ^{maya@uncg.edu" class="auth_mail" title="E-mail the corresponding author} ; Petr Girg^b ; ^c ; ¹ ; ^{pgirg@kma.zcu.cz" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Elliptic systems ; Asymptotically linear ; Resonance ; Lyapunov&ndash ; Schmidt method
• 刊名：Journal of Mathematical Analysis and Applications
• 出版年：2016
• 出版时间：15 October 2016
• 年：2016
• 卷：442
• 期：2
• 页码：583-599
• 全文大小：415 K

文摘

This paper is concerned with the solvability of the system at resonance at the simple eigenvalue 1202&_mathId=si2.gif&_user=111111111&_pii=S0022247X16301202&_rdoc=1&_issn=0022247X&md5=10fa02837e815007b215b24816a47b8d" title="Click to view the MathML source">ν₁${\nu }_1$ of the corresponding linear eigenvalue problem. Here 1202&_mathId=si3.gif&_user=111111111&_pii=S0022247X16301202&_rdoc=1&_issn=0022247X&md5=8ae0dd8763c41fad942a26ed749fbe96" title="Click to view the MathML source">Ω⊂R^N$\mathrm{\Omega }\subset {\mathbb{R}}^N$ (1202&_mathId=si4.gif&_user=111111111&_pii=S0022247X16301202&_rdoc=1&_issn=0022247X&md5=cbc1a2b1cd650303dd6cf482e0093cf3" title="Click to view the MathML source">N≥1$N\ge 1$) is a bounded domain with 1202&_mathId=si5.gif&_user=111111111&_pii=S0022247X16301202&_rdoc=1&_issn=0022247X&md5=401a9c6cf87cd48e29734c1cc77d1d66" title="Click to view the MathML source">C^2,η$C^{2,\eta }$-boundary ∂Ω, 1202&_mathId=si6.gif&_user=111111111&_pii=S0022247X16301202&_rdoc=1&_issn=0022247X&md5=d0661ed5af46b236584ad630a056311a" title="Click to view the MathML source">η∈(0,1)$\eta \in (0,1)$ (a bounded interval if 1202&_mathId=si7.gif&_user=111111111&_pii=S0022247X16301202&_rdoc=1&_issn=0022247X&md5=c6c719b23b79262d9cb88861eeea47fe" title="Click to view the MathML source">N=1$N=1$) and 1202&_mathId=si8.gif&_user=111111111&_pii=S0022247X16301202&_rdoc=1&_issn=0022247X&md5=be810004b9b99eff999d913fc14fa40b" title="Click to view the MathML source">θ₁${\theta }_1$, 1202&_mathId=si9.gif&_user=111111111&_pii=S0022247X16301202&_rdoc=1&_issn=0022247X&md5=7c7b990bcb1828d6c955e5e6472e1acf" title="Click to view the MathML source">θ₂${\theta }_2$ are positive constants. The nonlinear perturbations 1202&_mathId=si10.gif&_user=111111111&_pii=S0022247X16301202&_rdoc=1&_issn=0022247X&md5=19fa4733c5090cea36ddba3a88b09ea0" title="Click to view the MathML source">f(x,u,v),g(x,u,v):Ω×R²→R$f(x,u,v),g(x,u,v):\mathrm{\Omega }\times {\mathbb{R}}^2\to \mathbb{R}$ are Carathéodory functions that are sublinear at infinity. We employ the Lyapunov–Schmidt method to provide sufficient conditions on 1202&_mathId=si11.gif&_user=111111111&_pii=S0022247X16301202&_rdoc=1&_issn=0022247X&md5=c6da766c509044bcf4e2444e2a7de8d7" title="Click to view the MathML source">h₁,h₂∈L^r(Ω)$h_1,h_2\in L^r(\mathrm{\Omega })$; 1202&_mathId=si12.gif&_user=111111111&_pii=S0022247X16301202&_rdoc=1&_issn=0022247X&md5=abe5a93d21b505ee2bdfca8d9c50786b" title="Click to view the MathML source">r>N$r>N$, to guarantee the solvability of the system.