Classification of isolated singularities of positive solutions for Choquard equations

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• 作者：Huyuan Chen^a ; ^b ; ^{chenhuyuan@yeah.net" class="auth_mail" title="E-mail the corresponding author} ; Feng Zhou^c ; ^{fzhou@math.ecnu.edu.cn" class="auth_mail" title="E-mail the corresponding author}
• 关键词：35J75 ; 35B40
• 刊名：Journal of Differential Equations
• 出版年：2016
• 出版时间：15 December 2016
• 年：2016
• 卷：261
• 期：12
• 页码：6668-6698
• 全文大小：383 K

文摘

In this paper we classify the isolated singularities of positive solutions to Choquard equation where 62&_mathId=si2.gif&_user=111111111&_pii=S0022039616302662&_rdoc=1&_issn=00220396&md5=ced54b00bf0ba4dcb47e352c9f76ee48" title="Click to view the MathML source">p>0,q≥1,N≥3,α∈(0,N)$p>0,q\ge 1,N\ge 3,\alpha \in (0,N)$ and 62&_mathId=si3.gif&_user=111111111&_pii=S0022039616302662&_rdoc=1&_issn=00220396&md5=6c4508edd628990e46416a75d40eeddf">62-si3.gif">$I_{\alpha }[u^p](x)={\int }_{{\mathbb{R}}^N}\frac{u{(y)}^p}{|x-y{|}^{N-\alpha }}dy$. We show that any positive solution u is a solution of in the distributional sense for some 62&_mathId=si41.gif&_user=111111111&_pii=S0022039616302662&_rdoc=1&_issn=00220396&md5=894843b84bd2c879dc74cd36d9e1ad48" title="Click to view the MathML source">k≥0$k\ge 0$, where 62&_mathId=si6.gif&_user=111111111&_pii=S0022039616302662&_rdoc=1&_issn=00220396&md5=ecce76a6e1a415ef2e631248b422a822" title="Click to view the MathML source">δ₀${\delta }_0$ is the Dirac mass at the origin. We prove the existence of singular solutions in the subcritical case: 62&_mathId=si7.gif&_user=111111111&_pii=S0022039616302662&_rdoc=1&_issn=00220396&md5=2e22c12f11b5abbe7e5d5c614633b2c0">62-si7.gif">$p+q<\frac{N+\alpha }{N-2}\text{and}p<\frac{N}{N-2},q<\frac{N}{N-2}$ and prove that either the solution u   has removable singularity at the origin or satisfies 62&_mathId=si8.gif&_user=111111111&_pii=S0022039616302662&_rdoc=1&_issn=00220396&md5=d27636cb478504b030eb89ff4d0e5db8" title="Click to view the MathML source">lim_|x|→0⁺⁡u(x)|x|^N−2=C_N${\lim }_{|x|\to 0^{+}}u(x)|x{|}^{N-2}=C_N$ which is a positive constant. In the supercritical case: 62&_mathId=si9.gif&_user=111111111&_pii=S0022039616302662&_rdoc=1&_issn=00220396&md5=70146721e42f9535f8941e9ba640062a">62-si9.gif">$p+q\ge \frac{N+\alpha }{N-2}\text{or}p\ge \frac{N}{N-2},\text{or}q\ge \frac{N}{N-2}$ we prove that 62&_mathId=si10.gif&_user=111111111&_pii=S0022039616302662&_rdoc=1&_issn=00220396&md5=510479bb49b7722168cf1be59081acdc" title="Click to view the MathML source">k=0$k=0$.