Regularizing Effects for a Singular Elliptic Problem

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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Bibliographic Details
Main Authors: Ida de Bonis, Maria Michaela Porzio
Format: Article
Language:English
Published: MDPI AG 2025-01-01
Series:Axioms
Subjects:
nonlinear elliptic equations
singular lower order terms
irregular data
Online Access:https://www.mdpi.com/2075-1680/14/1/47
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Summary: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.
ISSN:2075-1680

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