{"id":{"repo_id":"carleton","oai_identifier":"oai:carleton.scholaris.ca:20.500.14718/38845"},"canonical_url":"https://search.dev.ndltd.org/etd/carleton/oai:carleton.scholaris.ca:20.500.14718/38845","repository":{"repo_id":"carleton","name":"Carleton University","base_url":"https://carleton.scholaris.ca/server/oai/request"},"display":{"title":"Existence of Solutions Via a New Variational Principle for Nonlocal Semilinear Elliptic Equations","abstract":"The aim of this thesis is to prove the existence of a weak solution for semilinear fractional elliptic equations given by \\begin{eqnarray*}\\left\\{ \\begin{array}{ll} (-\\Delta)^s u=|u|^{p-2} u+ f(x),&amp; \\quad x\\in \\Omega,\\\\ u=0, &amp; \\quad x \\in \\R^n \\backslash \\Omega, %0, &amp; \\quad x \\in \\partial \\Oemga, \\end{array} \\right. \\end{eqnarray*} where $(-\\Delta)^s$ denotes the fractional Laplace operator with $s \\in (0,1],$ $n &gt; 2s,$ $\\Omega$ is an open bounded domain in $\\R^n$ with $C^2$-boundary and $f \\in L^2(\\Omega).$ We are interested in extending this result for $s \\in (0, 1]$ to $p$ greater than the critical Sobolev exponent where the compact embedding fails to hold. We shall make use of a new variational principle presented in \\cite{Mo1} that allows one to deal with problems well beyond the compact structure.","abstract_html":"The aim of this thesis is to prove the existence of a weak solution for semilinear fractional elliptic equations given by \\begin{eqnarray*}\\left\\{ \\begin{array}{ll} (-\\Delta)^s u=|u|^{p-2} u+ f(x),&amp;amp; \\quad x\\in \\Omega,\\\\ u=0, &amp;amp; \\quad x \\in \\R^n \\backslash \\Omega, %0, &amp;amp; \\quad x \\in \\partial \\Oemga, \\end{array} \\right. \\end{eqnarray*} where <span class=\"etd-inline-math\">(-\\Delta)<sup>s</sup></span> denotes the fractional Laplace operator with $s \\in (0,1],$ $n &amp;gt; 2s,$ $\\Omega$ is an open bounded domain in <span class=\"etd-inline-math\">\\R<sup>n</sup></span> with <span class=\"etd-inline-math\">C<sup>2</sup></span>-boundary and <span class=\"etd-inline-math\">f \\in L<sup>2</sup>(\\Omega).</span> We are interested in extending this result for $s \\in (0, 1]$ to $p$ greater than the critical Sobolev exponent where the compact embedding fails to hold. We shall make use of a new variational principle presented in \\cite{Mo1} that allows one to deal with problems well beyond the compact structure.","abstract_has_math":true,"creators":["Wong, Kok Lin"],"institution":"Carleton University","degree_name":"Master of Science (M.Sc.)","degree_level":"Master&apos;s","degree_discipline":"Mathematics","degree_department":null,"school":null,"contributors":[],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2018,"date_issued":"2018","date_published":"2018","updated_at":"2026-07-24T01:34:25Z","subjects":[],"languages":["en"],"rights":["Copyright © 2018 the author(s). Theses may be used for non-commercial research, educational, or related academic purposes only. Such uses include personal study, research, scholarship, and teaching. Theses may only be shared by linking to Carleton University Institutional Repository and no part may be used without proper attribution to the author. No part may be used for commercial purposes directly or indirectly via a for-profit platform; no adaptation or derivative works are permitted without consent from the copyright owner."],"rights_urls":[],"identifier_entries":[{"key":"dc:identifier.doi","label":"DOI","values":["10.22215/etd/2018-13241"],"render_values":[{"text":"10.22215/etd/2018-13241","href":"https://doi.org/10.22215/etd/2018-13241","code":true}]}]},"links":{"outbound_url":"https://hdl.handle.net/20.500.14718/38845","outbound_label":"Handle","outbound_source":"dc:identifier.uri"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:creator","label":"Author","values":["Wong, Kok Lin"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date.accessioned","label":"Dc Date Accessioned","values":["2025-04-08T19:31:04Z"]},{"key":"dc:date.available","label":"Dc Date Available","values":["2025-04-08T19:31:04Z"]},{"key":"dc:date.issued","label":"Date","values":["2018"]},{"key":"dc:publisher","label":"Institution","values":["Carleton University"]},{"key":"dc:type","label":"Dc Type","values":["thesis"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Mathematics"]},{"key":"thesis:degree_level","label":"Degree Level","values":["Master&apos;s"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Master of Science (M.Sc.)"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language.iso","label":"Language (ISO)","values":["en"]},{"key":"dc:rights","label":"Dc Rights","values":["Copyright © 2018 the author(s). Theses may be used for non-commercial research, educational, or related academic purposes only. Such uses include personal study, research, scholarship, and teaching. Theses may only be shared by linking to Carleton University Institutional Repository and no part may be used without proper attribution to the author. No part may be used for commercial purposes directly or indirectly via a for-profit platform; no adaptation or derivative works are permitted without consent from the copyright owner."]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier.doi","label":"DOI","values":["10.22215/etd/2018-13241"]},{"key":"dc:identifier.uri","label":"Identifier URI","values":["https://hdl.handle.net/20.500.14718/38845"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["The aim of this thesis is to prove the existence of a weak solution for semilinear fractional elliptic equations given by \\begin{eqnarray*}\\left\\{ \\begin{array}{ll} (-\\Delta)^s u=|u|^{p-2} u+ f(x),&amp; \\quad x\\in \\Omega,\\\\ u=0, &amp; \\quad x \\in \\R^n \\backslash \\Omega, %0, &amp; \\quad x \\in \\partial \\Oemga, \\end{array} \\right. \\end{eqnarray*} where $(-\\Delta)^s$ denotes the fractional Laplace operator with $s \\in (0,1],$ $n &gt; 2s,$ $\\Omega$ is an open bounded domain in $\\R^n$ with $C^2$-boundary and $f \\in L^2(\\Omega).$ We are interested in extending this result for $s \\in (0, 1]$ to $p$ greater than the critical Sobolev exponent where the compact embedding fails to hold. We shall make use of a new variational principle presented in \\cite{Mo1} that allows one to deal with problems well beyond the compact structure."]},{"key":"dc:title","label":"Title","values":["Existence of Solutions Via a New Variational Principle for Nonlocal Semilinear Elliptic Equations"]}]}],"canonical_facts":{"dc:creator":["Wong, Kok Lin"],"dc:date.accessioned":["2025-04-08T19:31:04Z"],"dc:date.available":["2025-04-08T19:31:04Z"],"dc:date.issued":["2018"],"dc:description.abstract":["The aim of this thesis is to prove the existence of a weak solution for semilinear fractional elliptic equations given by \\begin{eqnarray*}\\left\\{ \\begin{array}{ll} (-\\Delta)^s u=|u|^{p-2} u+ f(x),&amp; \\quad x\\in \\Omega,\\\\ u=0, &amp; \\quad x \\in \\R^n \\backslash \\Omega, %0, &amp; \\quad x \\in \\partial \\Oemga, \\end{array} \\right. \\end{eqnarray*} where $(-\\Delta)^s$ denotes the fractional Laplace operator with $s \\in (0,1],$ $n &gt; 2s,$ $\\Omega$ is an open bounded domain in $\\R^n$ with $C^2$-boundary and $f \\in L^2(\\Omega).$ We are interested in extending this result for $s \\in (0, 1]$ to $p$ greater than the critical Sobolev exponent where the compact embedding fails to hold. We shall make use of a new variational principle presented in \\cite{Mo1} that allows one to deal with problems well beyond the compact structure."],"dc:identifier.doi":["10.22215/etd/2018-13241"],"dc:identifier.uri":["https://hdl.handle.net/20.500.14718/38845"],"dc:language.iso":["en"],"dc:publisher":["Carleton University"],"dc:rights":["Copyright © 2018 the author(s). Theses may be used for non-commercial research, educational, or related academic purposes only. Such uses include personal study, research, scholarship, and teaching. Theses may only be shared by linking to Carleton University Institutional Repository and no part may be used without proper attribution to the author. No part may be used for commercial purposes directly or indirectly via a for-profit platform; no adaptation or derivative works are permitted without consent from the copyright owner."],"dc:title":["Existence of Solutions Via a New Variational Principle for Nonlocal Semilinear Elliptic Equations"],"dc:type":["thesis"],"thesis:degree_discipline":["Mathematics"],"thesis:degree_level":["Master&apos;s"],"thesis:degree_name":["Master of Science (M.Sc.)"]},"updated_at":"2026-07-24T01:34:25Z"}