Singular solutions for divergence-form elliptic equations involving regular variation theory: Existence and classification
We generalise and sharpen several recent results in the literature regarding the existence and complete classification of the isolated singularities for a broad class of nonlinear elliptic equations of the form \begin{equation} -{\rm div}\,(\mathcal A(|x|) \,|\nabla u|^{p-2} \nabla u)+b(x)\,h(u)=0\q...
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| creator | Chang, Ting-Ying Cîrstea, Florica |
| description | We generalise and sharpen several recent results in the literature regarding
the existence and complete classification of the isolated singularities for a
broad class of nonlinear elliptic equations of the form \begin{equation} -{\rm
div}\,(\mathcal A(|x|) \,|\nabla u|^{p-2} \nabla u)+b(x)\,h(u)=0\quad \text{in
} B_1\setminus\{0\}, \end{equation} where $B_r$ denotes the open ball with
radius $r>0$ centred at zero in $\mathbb{R}^N$ $(N\geq 2)$. We assume that
$\mathcal{A} \in C^1(0,1]$, $b\in C(\bar{B_1}\setminus\{0\})$ and $h\in
C[0,\infty)$ are positive functions associated with regularly varying functions
of index $\vartheta$, $\sigma$ and $q$ at $0$, $0$ and $\infty$ respectively,
satisfying $q>p-1>0$ and $\vartheta-\sigma |
| doi_str_mv | 10.48550/arxiv.1602.03612 |
| format | Article |
| fullrecord | <record><control><sourceid>arxiv_GOX</sourceid><recordid>TN_cdi_arxiv_primary_1602_03612</recordid><sourceformat>XML</sourceformat><sourcesystem>PC</sourcesystem><sourcerecordid>1602_03612</sourcerecordid><originalsourceid>FETCH-LOGICAL-a672-4bf3a3b04faabf644b2eb4b89e6a3e75ed6658719e54e2ee4c55c613b49f351c3</originalsourceid><addsrcrecordid>eNotkM1OwzAQhH3hgAoPwAm_QEIS_yThhqryI1XiQO_R2lmXldyk2EnU8vS0CafVaGc-aYaxhzxLZaVU9gThRFOa66xIM6Hz4pb9flG3Hz0EHns_DtR3kbs-8JYmDHvsLCYXeeDoPR0Hshx_Rlhs1E29ny5xHnBBTBBofvLhG_twfuabE8XhSuHQtdx6iJEc2dl0x24c-Ij3_3fFdq-b3fo92X6-faxftgnoskikcQKEyaQDME5LaQo00lQ1ahBYKmy1VlWZ16gkFojSKmV1LoysnVC5FSv2uGDn8s0x0AHCubmO0MwjiD_CNVvA</addsrcrecordid><sourcetype>Open Access Repository</sourcetype><iscdi>true</iscdi><recordtype>article</recordtype></control><display><type>article</type><title>Singular solutions for divergence-form elliptic equations involving regular variation theory: Existence and classification</title><source>arXiv.org</source><creator>Chang, Ting-Ying ; Cîrstea, Florica</creator><creatorcontrib>Chang, Ting-Ying ; Cîrstea, Florica</creatorcontrib><description>We generalise and sharpen several recent results in the literature regarding
the existence and complete classification of the isolated singularities for a
broad class of nonlinear elliptic equations of the form \begin{equation} -{\rm
div}\,(\mathcal A(|x|) \,|\nabla u|^{p-2} \nabla u)+b(x)\,h(u)=0\quad \text{in
} B_1\setminus\{0\}, \end{equation} where $B_r$ denotes the open ball with
radius $r>0$ centred at zero in $\mathbb{R}^N$ $(N\geq 2)$. We assume that
$\mathcal{A} \in C^1(0,1]$, $b\in C(\bar{B_1}\setminus\{0\})$ and $h\in
C[0,\infty)$ are positive functions associated with regularly varying functions
of index $\vartheta$, $\sigma$ and $q$ at $0$, $0$ and $\infty$ respectively,
satisfying $q>p-1>0$ and $\vartheta-\sigma<p<N+\vartheta$. We prove that the
condition $b(x) \,h(\Phi)\not \in L^1(B_{1/2})$ is sharp for the removability
of all singularities at zero for the positive solutions of our problem, where
$\Phi$ denotes the "fundamental solution" of $-{\rm div}\,(\mathcal A(|x|)\,
|\nabla u|^{p-2} \nabla u)=\delta_0$ (the Dirac mass at zero) in $B_1$, subject
to $\Phi|_{\partial B_1}=0$. If $b(x) \,h(\Phi)\in L^1(B_{1/2})$, we show that
any non-removable singularity at zero for a positive solution to our equation
is either weak (i.e., $\lim_{|x|\to 0} u(x)/\Phi(|x|)\in (0,\infty)$) or strong
($ \lim_{|x|\to 0} u(x)/\Phi(|x|)=\infty$). The main difficulty and novelty of
this paper, for which we develop new techniques, come from the explicit
asymptotic behaviour of the strong singularity solutions in the critical case,
which had previously remained open even for $\mathcal{A}=1$. We also study the
existence and uniqueness of the positive solution to our problem with a
prescribed admissible behaviour at zero and a Dirichlet condition on $\partial
B_1$.</description><identifier>DOI: 10.48550/arxiv.1602.03612</identifier><language>eng</language><subject>Mathematics - Analysis of PDEs</subject><creationdate>2016-02</creationdate><rights>http://arxiv.org/licenses/nonexclusive-distrib/1.0</rights><oa>free_for_read</oa><woscitedreferencessubscribed>false</woscitedreferencessubscribed></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><link.rule.ids>228,230,776,881</link.rule.ids><linktorsrc>$$Uhttps://arxiv.org/abs/1602.03612$$EView_record_in_Cornell_University$$FView_record_in_$$GCornell_University$$Hfree_for_read</linktorsrc><backlink>$$Uhttps://doi.org/10.48550/arXiv.1602.03612$$DView paper in arXiv$$Hfree_for_read</backlink></links><search><creatorcontrib>Chang, Ting-Ying</creatorcontrib><creatorcontrib>Cîrstea, Florica</creatorcontrib><title>Singular solutions for divergence-form elliptic equations involving regular variation theory: Existence and classification</title><description>We generalise and sharpen several recent results in the literature regarding
the existence and complete classification of the isolated singularities for a
broad class of nonlinear elliptic equations of the form \begin{equation} -{\rm
div}\,(\mathcal A(|x|) \,|\nabla u|^{p-2} \nabla u)+b(x)\,h(u)=0\quad \text{in
} B_1\setminus\{0\}, \end{equation} where $B_r$ denotes the open ball with
radius $r>0$ centred at zero in $\mathbb{R}^N$ $(N\geq 2)$. We assume that
$\mathcal{A} \in C^1(0,1]$, $b\in C(\bar{B_1}\setminus\{0\})$ and $h\in
C[0,\infty)$ are positive functions associated with regularly varying functions
of index $\vartheta$, $\sigma$ and $q$ at $0$, $0$ and $\infty$ respectively,
satisfying $q>p-1>0$ and $\vartheta-\sigma<p<N+\vartheta$. We prove that the
condition $b(x) \,h(\Phi)\not \in L^1(B_{1/2})$ is sharp for the removability
of all singularities at zero for the positive solutions of our problem, where
$\Phi$ denotes the "fundamental solution" of $-{\rm div}\,(\mathcal A(|x|)\,
|\nabla u|^{p-2} \nabla u)=\delta_0$ (the Dirac mass at zero) in $B_1$, subject
to $\Phi|_{\partial B_1}=0$. If $b(x) \,h(\Phi)\in L^1(B_{1/2})$, we show that
any non-removable singularity at zero for a positive solution to our equation
is either weak (i.e., $\lim_{|x|\to 0} u(x)/\Phi(|x|)\in (0,\infty)$) or strong
($ \lim_{|x|\to 0} u(x)/\Phi(|x|)=\infty$). The main difficulty and novelty of
this paper, for which we develop new techniques, come from the explicit
asymptotic behaviour of the strong singularity solutions in the critical case,
which had previously remained open even for $\mathcal{A}=1$. We also study the
existence and uniqueness of the positive solution to our problem with a
prescribed admissible behaviour at zero and a Dirichlet condition on $\partial
B_1$.</description><subject>Mathematics - Analysis of PDEs</subject><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2016</creationdate><recordtype>article</recordtype><sourceid>GOX</sourceid><recordid>eNotkM1OwzAQhH3hgAoPwAm_QEIS_yThhqryI1XiQO_R2lmXldyk2EnU8vS0CafVaGc-aYaxhzxLZaVU9gThRFOa66xIM6Hz4pb9flG3Hz0EHns_DtR3kbs-8JYmDHvsLCYXeeDoPR0Hshx_Rlhs1E29ny5xHnBBTBBofvLhG_twfuabE8XhSuHQtdx6iJEc2dl0x24c-Ij3_3fFdq-b3fo92X6-faxftgnoskikcQKEyaQDME5LaQo00lQ1ahBYKmy1VlWZ16gkFojSKmV1LoysnVC5FSv2uGDn8s0x0AHCubmO0MwjiD_CNVvA</recordid><startdate>20160210</startdate><enddate>20160210</enddate><creator>Chang, Ting-Ying</creator><creator>Cîrstea, Florica</creator><scope>AKZ</scope><scope>GOX</scope></search><sort><creationdate>20160210</creationdate><title>Singular solutions for divergence-form elliptic equations involving regular variation theory: Existence and classification</title><author>Chang, Ting-Ying ; Cîrstea, Florica</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-a672-4bf3a3b04faabf644b2eb4b89e6a3e75ed6658719e54e2ee4c55c613b49f351c3</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2016</creationdate><topic>Mathematics - Analysis of PDEs</topic><toplevel>online_resources</toplevel><creatorcontrib>Chang, Ting-Ying</creatorcontrib><creatorcontrib>Cîrstea, Florica</creatorcontrib><collection>arXiv Mathematics</collection><collection>arXiv.org</collection></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext_linktorsrc</fulltext></delivery><addata><au>Chang, Ting-Ying</au><au>Cîrstea, Florica</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>Singular solutions for divergence-form elliptic equations involving regular variation theory: Existence and classification</atitle><date>2016-02-10</date><risdate>2016</risdate><abstract>We generalise and sharpen several recent results in the literature regarding
the existence and complete classification of the isolated singularities for a
broad class of nonlinear elliptic equations of the form \begin{equation} -{\rm
div}\,(\mathcal A(|x|) \,|\nabla u|^{p-2} \nabla u)+b(x)\,h(u)=0\quad \text{in
} B_1\setminus\{0\}, \end{equation} where $B_r$ denotes the open ball with
radius $r>0$ centred at zero in $\mathbb{R}^N$ $(N\geq 2)$. We assume that
$\mathcal{A} \in C^1(0,1]$, $b\in C(\bar{B_1}\setminus\{0\})$ and $h\in
C[0,\infty)$ are positive functions associated with regularly varying functions
of index $\vartheta$, $\sigma$ and $q$ at $0$, $0$ and $\infty$ respectively,
satisfying $q>p-1>0$ and $\vartheta-\sigma<p<N+\vartheta$. We prove that the
condition $b(x) \,h(\Phi)\not \in L^1(B_{1/2})$ is sharp for the removability
of all singularities at zero for the positive solutions of our problem, where
$\Phi$ denotes the "fundamental solution" of $-{\rm div}\,(\mathcal A(|x|)\,
|\nabla u|^{p-2} \nabla u)=\delta_0$ (the Dirac mass at zero) in $B_1$, subject
to $\Phi|_{\partial B_1}=0$. If $b(x) \,h(\Phi)\in L^1(B_{1/2})$, we show that
any non-removable singularity at zero for a positive solution to our equation
is either weak (i.e., $\lim_{|x|\to 0} u(x)/\Phi(|x|)\in (0,\infty)$) or strong
($ \lim_{|x|\to 0} u(x)/\Phi(|x|)=\infty$). The main difficulty and novelty of
this paper, for which we develop new techniques, come from the explicit
asymptotic behaviour of the strong singularity solutions in the critical case,
which had previously remained open even for $\mathcal{A}=1$. We also study the
existence and uniqueness of the positive solution to our problem with a
prescribed admissible behaviour at zero and a Dirichlet condition on $\partial
B_1$.</abstract><doi>10.48550/arxiv.1602.03612</doi><oa>free_for_read</oa></addata></record> |
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| source | arXiv.org |
| subjects | Mathematics - Analysis of PDEs |
| title | Singular solutions for divergence-form elliptic equations involving regular variation theory: Existence and classification |
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