Boundary Concentrated Solutions for an Elliptic Equation with Subcritical Nonlinearity
In this paper, we consider the nonlinear Neumann problem <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mrow><mo>(</mo><msub><mi>Q</mi><mi>ε</mi></msub&...
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2025-04-01
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| author | Sadeem Al-Harbi Mohamed Ben Ayed |
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| description | In this paper, we consider the nonlinear Neumann problem <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mrow><mo>(</mo><msub><mi>Q</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>, with <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><mo>Ω</mo></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>∂</mo><mi>u</mi><mo>/</mo><mo>∂</mo><mi>ν</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>∂</mo><mo>Ω</mo></mrow></semantics></math></inline-formula>, where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mo>Ω</mo></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>, with <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 smooth positive 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>. Assuming the flatness of the boundary near the critical points of the restriction of the function <i>V</i> on the boundary, we construct boundary peak solutions with isolated bubbles, leading to a multiplicity result for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><msub><mi>Q</mi><mi>ε</mi></msub><mo>)</mo></mrow></semantics></math></inline-formula>. The proof of our results relies on expanding the gradient of the associated functional and testing the equation with the appropriate vector fields, which yields constraints for the concentration points and blow-up rates. A thorough analysis of these constraints leads to our results. |
| format | Article |
| id | doaj-art-5ed29bfdbc0f4da2b8648ad13da28cc0 |
| institution | Kabale University |
| issn | 2075-1680 |
| language | English |
| publishDate | 2025-04-01 |
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| series | Axioms |
| spelling | doaj-art-5ed29bfdbc0f4da2b8648ad13da28cc02025-08-20T03:47:48ZengMDPI AGAxioms2075-16802025-04-0114534610.3390/axioms14050346Boundary Concentrated Solutions for an Elliptic Equation with Subcritical NonlinearitySadeem Al-Harbi0Mohamed Ben Ayed1Department of Mathematics, College of Science, Qassim University, Buraydah 51542, Saudi ArabiaDepartment of Mathematics, College of Science, Qassim University, Buraydah 51542, Saudi ArabiaIn this paper, we consider the nonlinear Neumann problem <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mrow><mo>(</mo><msub><mi>Q</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>, with <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><mo>Ω</mo></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>∂</mo><mi>u</mi><mo>/</mo><mo>∂</mo><mi>ν</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>∂</mo><mo>Ω</mo></mrow></semantics></math></inline-formula>, where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mo>Ω</mo></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>, with <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 smooth positive 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>. Assuming the flatness of the boundary near the critical points of the restriction of the function <i>V</i> on the boundary, we construct boundary peak solutions with isolated bubbles, leading to a multiplicity result for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><msub><mi>Q</mi><mi>ε</mi></msub><mo>)</mo></mrow></semantics></math></inline-formula>. The proof of our results relies on expanding the gradient of the associated functional and testing the equation with the appropriate vector fields, which yields constraints for the concentration points and blow-up rates. A thorough analysis of these constraints leads to our results.https://www.mdpi.com/2075-1680/14/5/346partial differential equationsneumann elliptic problemscritical sobolev exponent |
| spellingShingle | Sadeem Al-Harbi Mohamed Ben Ayed Boundary Concentrated Solutions for an Elliptic Equation with Subcritical Nonlinearity Axioms partial differential equations neumann elliptic problems critical sobolev exponent |
| title | Boundary Concentrated Solutions for an Elliptic Equation with Subcritical Nonlinearity |
| title_full | Boundary Concentrated Solutions for an Elliptic Equation with Subcritical Nonlinearity |
| title_fullStr | Boundary Concentrated Solutions for an Elliptic Equation with Subcritical Nonlinearity |
| title_full_unstemmed | Boundary Concentrated Solutions for an Elliptic Equation with Subcritical Nonlinearity |
| title_short | Boundary Concentrated Solutions for an Elliptic Equation with Subcritical Nonlinearity |
| title_sort | boundary concentrated solutions for an elliptic equation with subcritical nonlinearity |
| topic | partial differential equations neumann elliptic problems critical sobolev exponent |
| url | https://www.mdpi.com/2075-1680/14/5/346 |
| work_keys_str_mv | AT sadeemalharbi boundaryconcentratedsolutionsforanellipticequationwithsubcriticalnonlinearity AT mohamedbenayed boundaryconcentratedsolutionsforanellipticequationwithsubcriticalnonlinearity |