Interior Peak Solutions for a Semilinear Dirichlet Problem
In this paper, we consider the semilinear Dirichlet problem <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mrow><mo>(</mo><msub><mi mathvariant="script">P</mi&g...
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2025-01-01
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| author | Hissah Alharbi Hibah Alkhuzayyim Mohamed Ben Ayed Khalil El Mehdi |
| author_facet | Hissah Alharbi Hibah Alkhuzayyim Mohamed Ben Ayed Khalil El Mehdi |
| author_sort | Hissah Alharbi |
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| description | In this paper, we consider the semilinear Dirichlet problem <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mrow><mo>(</mo><msub><mi mathvariant="script">P</mi><mi>ε</mi></msub><mo>)</mo></mrow><mo>:</mo><mo>−</mo><mo>Δ</mo><mi>u</mi><mo>+</mo><mi>V</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mi>u</mi><mo>=</mo><msup><mi>u</mi><mrow><mstyle scriptlevel="0" displaystyle="true"><mfrac><mrow><mi>n</mi><mo>+</mo><mn>2</mn></mrow><mrow><mi>n</mi><mo>−</mo><mn>2</mn></mrow></mfrac></mstyle><mo>−</mo><mi>ε</mi></mrow></msup></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>u</mi><mo>></mo><mn>0</mn></mrow></semantics></math></inline-formula> in <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>Ω</mo></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>u</mi><mo>=</mo><mn>0</mn></mrow></semantics></math></inline-formula> on ∂<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>Ω</mo></mrow></semantics></math></inline-formula>, where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>Ω</mo></mrow></semantics></math></inline-formula> is a bounded regular domain in <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi mathvariant="double-struck">R</mi><mi>n</mi></msup></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>n</mi><mo>≥</mo><mn>4</mn></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ε</mi></semantics></math></inline-formula> is a small positive parameter, and <i>V</i> is a non-constant positive <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>C</mi><mn>2</mn></msup></semantics></math></inline-formula>-function on <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mover><mo>Ω</mo><mo>¯</mo></mover></semantics></math></inline-formula>. We construct interior peak solutions with isolated bubbles. This leads to a multiplicity result for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><msub><mi mathvariant="script">P</mi><mi>ε</mi></msub><mo>)</mo></mrow></semantics></math></inline-formula>. The proof of our results relies on precise expansions of the gradient of the Euler–Lagrange functional associated with <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><msub><mi mathvariant="script">P</mi><mi>ε</mi></msub><mo>)</mo></mrow></semantics></math></inline-formula>, along with a suitable projection of the bubbles. This projection and its associated estimates are new and play a crucial role in tackling such types of problems. |
| format | Article |
| id | doaj-art-064f80fa4b974b1e9213cf68aeafe868 |
| institution | Kabale University |
| issn | 2075-1680 |
| language | English |
| publishDate | 2025-01-01 |
| publisher | MDPI AG |
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| series | Axioms |
| spelling | doaj-art-064f80fa4b974b1e9213cf68aeafe8682025-01-24T13:22:17ZengMDPI AGAxioms2075-16802025-01-011415810.3390/axioms14010058Interior Peak Solutions for a Semilinear Dirichlet ProblemHissah Alharbi0Hibah Alkhuzayyim1Mohamed Ben Ayed2Khalil El Mehdi3Department of Mathematics, College of Science, Qassim University, Buraydah 51452, Saudi ArabiaDepartment of Mathematics, College of Science, Qassim University, Buraydah 51452, Saudi ArabiaDepartment of Mathematics, College of Science, Qassim University, Buraydah 51452, Saudi ArabiaDepartment of Mathematics, College of Science, Qassim University, Buraydah 51452, Saudi ArabiaIn this paper, we consider the semilinear Dirichlet problem <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mrow><mo>(</mo><msub><mi mathvariant="script">P</mi><mi>ε</mi></msub><mo>)</mo></mrow><mo>:</mo><mo>−</mo><mo>Δ</mo><mi>u</mi><mo>+</mo><mi>V</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mi>u</mi><mo>=</mo><msup><mi>u</mi><mrow><mstyle scriptlevel="0" displaystyle="true"><mfrac><mrow><mi>n</mi><mo>+</mo><mn>2</mn></mrow><mrow><mi>n</mi><mo>−</mo><mn>2</mn></mrow></mfrac></mstyle><mo>−</mo><mi>ε</mi></mrow></msup></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>u</mi><mo>></mo><mn>0</mn></mrow></semantics></math></inline-formula> in <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>Ω</mo></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>u</mi><mo>=</mo><mn>0</mn></mrow></semantics></math></inline-formula> on ∂<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>Ω</mo></mrow></semantics></math></inline-formula>, where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>Ω</mo></mrow></semantics></math></inline-formula> is a bounded regular domain in <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi mathvariant="double-struck">R</mi><mi>n</mi></msup></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>n</mi><mo>≥</mo><mn>4</mn></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ε</mi></semantics></math></inline-formula> is a small positive parameter, and <i>V</i> is a non-constant positive <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>C</mi><mn>2</mn></msup></semantics></math></inline-formula>-function on <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mover><mo>Ω</mo><mo>¯</mo></mover></semantics></math></inline-formula>. We construct interior peak solutions with isolated bubbles. This leads to a multiplicity result for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><msub><mi mathvariant="script">P</mi><mi>ε</mi></msub><mo>)</mo></mrow></semantics></math></inline-formula>. The proof of our results relies on precise expansions of the gradient of the Euler–Lagrange functional associated with <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><msub><mi mathvariant="script">P</mi><mi>ε</mi></msub><mo>)</mo></mrow></semantics></math></inline-formula>, along with a suitable projection of the bubbles. This projection and its associated estimates are new and play a crucial role in tackling such types of problems.https://www.mdpi.com/2075-1680/14/1/58partial differential equationsvariational analysisnonlinear analysiscritical Sobolev exponent |
| spellingShingle | Hissah Alharbi Hibah Alkhuzayyim Mohamed Ben Ayed Khalil El Mehdi Interior Peak Solutions for a Semilinear Dirichlet Problem Axioms partial differential equations variational analysis nonlinear analysis critical Sobolev exponent |
| title | Interior Peak Solutions for a Semilinear Dirichlet Problem |
| title_full | Interior Peak Solutions for a Semilinear Dirichlet Problem |
| title_fullStr | Interior Peak Solutions for a Semilinear Dirichlet Problem |
| title_full_unstemmed | Interior Peak Solutions for a Semilinear Dirichlet Problem |
| title_short | Interior Peak Solutions for a Semilinear Dirichlet Problem |
| title_sort | interior peak solutions for a semilinear dirichlet problem |
| topic | partial differential equations variational analysis nonlinear analysis critical Sobolev exponent |
| url | https://www.mdpi.com/2075-1680/14/1/58 |
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