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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| Main Authors: | , |
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| Format: | Article |
| Language: | English |
| Published: |
MDPI AG
2025-04-01
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| Series: | Axioms |
| Subjects: | |
| Online Access: | https://www.mdpi.com/2075-1680/14/5/346 |
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| Summary: | 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. |
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| ISSN: | 2075-1680 |