On critical systems involving fractional Laplacian

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• 作者：Zhenyu Guo^a ; ^b ; ^{guozy@163.com" class="auth_mail" title="E-mail the corresponding author} ; Senping Luo^b ; ^{luosp1989@163.com" class="auth_mail" title="E-mail the corresponding author} ; Wenming Zou^b ; ^{wzou@math.tsinghua.edu.cn" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Ground state ; Nehari manifold ; Fractional&ndash ; Sobolev critical exponent
• 刊名：Journal of Mathematical Analysis and Applications
• 出版年：2017
• 出版时间：1 February 2017
• 年：2017
• 卷：446
• 期：1
• 页码：681-706
• 全文大小：455 K

文摘

Consider the following non-local critical system where class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16305017&_mathId=si2.gif&_user=111111111&_pii=S0022247X16305017&_rdoc=1&_issn=0022247X&md5=ea3b687c18f5eb216fbba77d2574f914" title="Click to view the MathML source">(−Δ)^sclass="mathContainer hidden">class="mathCode">${(-\mathrm{\Delta })}^s$ is fractional Laplacian, class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16305017&_mathId=si3.gif&_user=111111111&_pii=S0022247X16305017&_rdoc=1&_issn=0022247X&md5=7bc0b6d5f8bdce37d5a774f8e6130790" title="Click to view the MathML source">0<s<1class="mathContainer hidden">class="mathCode">$0<s<1$ and all class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16305017&_mathId=si4.gif&_user=111111111&_pii=S0022247X16305017&_rdoc=1&_issn=0022247X&md5=994a1a16789a2a3a408dee7c611f94e2" title="Click to view the MathML source">λ₁,λ₂,μ₁,μ₂,γ>0class="mathContainer hidden">class="mathCode">${\lambda }_1,{\lambda }_2,{\mu }_1,{\mu }_2,\gamma >0$, class="mathmlsrc">title="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16305017&_mathId=si13.gif&_user=111111111&_pii=S0022247X16305017&_rdoc=1&_issn=0022247X&md5=f8e8a4269f2f1444426c8ac06fdeac3f">class="imgLazyJSB inlineImage" height="20" width="79" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0022247X16305017-si13.gif">class="mathContainer hidden">class="mathCode">$2_{⁎}:=\frac{2N}{N-2s}$ is a fractional Sobolev critical exponent, class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16305017&_mathId=si6.gif&_user=111111111&_pii=S0022247X16305017&_rdoc=1&_issn=0022247X&md5=2da5dffee4e444839c5c151e8e05d6e2" title="Click to view the MathML source">N>2sclass="mathContainer hidden">class="mathCode">$N>2s$, class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16305017&_mathId=si7.gif&_user=111111111&_pii=S0022247X16305017&_rdoc=1&_issn=0022247X&md5=8da188a4fbb87f10989532b51f71cd60" title="Click to view the MathML source">α,β>1class="mathContainer hidden">class="mathCode">$\alpha ,\beta >1$, class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16305017&_mathId=si14.gif&_user=111111111&_pii=S0022247X16305017&_rdoc=1&_issn=0022247X&md5=98c94dc624225d21dcc252298fa714b5" title="Click to view the MathML source">α+β=2_⁎class="mathContainer hidden">class="mathCode">$\alpha +\beta =2_{⁎}$, and Ω is an open bounded domain in class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16305017&_mathId=si147.gif&_user=111111111&_pii=S0022247X16305017&_rdoc=1&_issn=0022247X&md5=ce265466aa1c875411996ff59f1b5152" title="Click to view the MathML source">R^Nclass="mathContainer hidden">class="mathCode">${\mathbb{R}}^N$ with Lipschitz boundary. Under proper conditions, we establish the existence result of the ground state solution to system (0.1).