Regularizing Effects for a Singular Elliptic Problem
In this paper, we prove existence and regularity results for a nonlinear elliptic problem of p-Laplacian type with a singular potential like <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mstyle scriptlevel="...
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2025-01-01
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| author | Ida de Bonis Maria Michaela Porzio |
| author_facet | Ida de Bonis Maria Michaela Porzio |
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| description | In this paper, we prove existence and regularity results for a nonlinear elliptic problem of p-Laplacian type with a singular potential like <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mstyle scriptlevel="0" displaystyle="true"><mfrac><mi>f</mi><msup><mi>u</mi><mi>γ</mi></msup></mfrac></mstyle></semantics></math></inline-formula> and a lower order term <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>b</mi><mi>u</mi></mrow></semantics></math></inline-formula>, where <i>u</i> is the solution and <i>b</i> and <i>f</i> are only assumed to be summable functions. We show that, despite the lack of regularity of the data, for suitable choices of the function <i>b</i> in the lower order term, a strong regularizing effect appears. In particular we exhibit the existence of bounded solutions. Worth notice is that this result fails if <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>b</mi><mo>≡</mo><mn>0</mn></mrow></semantics></math></inline-formula>, i.e., in absence of the lower order term. Moreover, we show that, if the singularity is “not too large” (i.e., <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>γ</mi><mo>≤</mo><mn>1</mn></mrow></semantics></math></inline-formula>), such a regular solution is also unique. |
| format | Article |
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| institution | Kabale University |
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| language | English |
| publishDate | 2025-01-01 |
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| spelling | doaj-art-6e033575ed6942f4946e431ea66f31a32025-01-24T13:22:15ZengMDPI AGAxioms2075-16802025-01-011414710.3390/axioms14010047Regularizing Effects for a Singular Elliptic ProblemIda de Bonis0Maria Michaela Porzio1Dipartimento di Pianificazione, Design, Tecnologia dell’Architettura, Sapienza Università di Roma, Via Flaminia 70, 00196 Roma, ItalyDipartimento di Pianificazione, Design, Tecnologia dell’Architettura, Sapienza Università di Roma, Via Flaminia 70, 00196 Roma, ItalyIn this paper, we prove existence and regularity results for a nonlinear elliptic problem of p-Laplacian type with a singular potential like <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mstyle scriptlevel="0" displaystyle="true"><mfrac><mi>f</mi><msup><mi>u</mi><mi>γ</mi></msup></mfrac></mstyle></semantics></math></inline-formula> and a lower order term <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>b</mi><mi>u</mi></mrow></semantics></math></inline-formula>, where <i>u</i> is the solution and <i>b</i> and <i>f</i> are only assumed to be summable functions. We show that, despite the lack of regularity of the data, for suitable choices of the function <i>b</i> in the lower order term, a strong regularizing effect appears. In particular we exhibit the existence of bounded solutions. Worth notice is that this result fails if <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>b</mi><mo>≡</mo><mn>0</mn></mrow></semantics></math></inline-formula>, i.e., in absence of the lower order term. Moreover, we show that, if the singularity is “not too large” (i.e., <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>γ</mi><mo>≤</mo><mn>1</mn></mrow></semantics></math></inline-formula>), such a regular solution is also unique.https://www.mdpi.com/2075-1680/14/1/47nonlinear elliptic equationssingular lower order termsirregular data |
| spellingShingle | Ida de Bonis Maria Michaela Porzio Regularizing Effects for a Singular Elliptic Problem Axioms nonlinear elliptic equations singular lower order terms irregular data |
| title | Regularizing Effects for a Singular Elliptic Problem |
| title_full | Regularizing Effects for a Singular Elliptic Problem |
| title_fullStr | Regularizing Effects for a Singular Elliptic Problem |
| title_full_unstemmed | Regularizing Effects for a Singular Elliptic Problem |
| title_short | Regularizing Effects for a Singular Elliptic Problem |
| title_sort | regularizing effects for a singular elliptic problem |
| topic | nonlinear elliptic equations singular lower order terms irregular data |
| url | https://www.mdpi.com/2075-1680/14/1/47 |
| work_keys_str_mv | AT idadebonis regularizingeffectsforasingularellipticproblem AT mariamichaelaporzio regularizingeffectsforasingularellipticproblem |