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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| Main Authors: | , , , |
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| Format: | Article |
| Language: | English |
| Published: |
MDPI AG
2025-01-01
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| Series: | Axioms |
| Subjects: | |
| Online Access: | https://www.mdpi.com/2075-1680/14/1/58 |
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| Summary: | 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. |
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| ISSN: | 2075-1680 |