The Existence and Uniqueness of Nonlinear Elliptic Equations with General Growth in the Gradient
In this paper, we prove the existence and uniqueness results for a weak solution to a class of Dirichlet boundary value problems whose prototype is <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>...
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2024-12-01
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| author | Angelo Alvino Vincenzo Ferone Anna Mercaldo |
| author_facet | Angelo Alvino Vincenzo Ferone Anna Mercaldo |
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| description | In this paper, we prove the existence and uniqueness results for a weak solution to a class of Dirichlet boundary value problems whose prototype is <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>−</mo><msub><mo>Δ</mo><mi>p</mi></msub><mi>u</mi><mo>=</mo><mi>β</mi><msup><mrow><mo>|</mo><mo>∇</mo><mi>u</mi><mo>|</mo></mrow><mi>q</mi></msup><mo>+</mo><mi>f</mi><mo> </mo><mrow><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi><mo> </mo><mo>Ω</mo></mrow><mspace width="0.166667em"></mspace><mo>,</mo><mo> </mo><mi>u</mi><mo>=</mo><mn>0</mn><mo> </mo><mrow><mi mathvariant="normal">o</mi><mi mathvariant="normal">n</mi><mo> </mo><mo>∂</mo><mo>Ω</mo></mrow><mspace width="0.166667em"></mspace><mo>,</mo><mspace width="4pt"></mspace></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 open subset of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mrow><mi mathvariant="double-struck">R</mi></mrow><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>2</mn></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>1</mn><mo><</mo><mi>p</mi><mo><</mo><mi>N</mi></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mo>Δ</mo><mi>p</mi></msub><mi>u</mi><mo>=</mo><mi>div</mi><mfenced separators="" open="(" close=")"><msup><mrow><mo>|</mo><mo>∇</mo><mi>u</mi><mo>|</mo></mrow><mrow><mi>p</mi><mo>−</mo><mn>2</mn></mrow></msup><mo>∇</mo><mi>u</mi></mfenced></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>p</mi><mo>−</mo><mn>1</mn><mo><</mo><mi>q</mi><mo><</mo><mi>p</mi></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>β</mi></semantics></math></inline-formula> is a positive constant and <i>f</i> is a measurable function satisfying suitable summability conditions depending on <i>q</i> and a smallness condition. |
| format | Article |
| id | doaj-art-a142abf0eb0c40688bac056e5f885f15 |
| institution | Kabale University |
| issn | 2227-7390 |
| language | English |
| publishDate | 2024-12-01 |
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| spelling | doaj-art-a142abf0eb0c40688bac056e5f885f152025-01-10T13:18:08ZengMDPI AGMathematics2227-73902024-12-011316310.3390/math13010063The Existence and Uniqueness of Nonlinear Elliptic Equations with General Growth in the GradientAngelo Alvino0Vincenzo Ferone1Anna Mercaldo2Dipartimento di Matematica e Applicazioni “R. Caccioppoli”, Università degli Studi di Napoli Federico II, 80138 Napoli, ItalyDipartimento di Matematica e Applicazioni “R. Caccioppoli”, Università degli Studi di Napoli Federico II, 80138 Napoli, ItalyDipartimento di Matematica e Applicazioni “R. Caccioppoli”, Università degli Studi di Napoli Federico II, 80138 Napoli, ItalyIn this paper, we prove the existence and uniqueness results for a weak solution to a class of Dirichlet boundary value problems whose prototype is <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>−</mo><msub><mo>Δ</mo><mi>p</mi></msub><mi>u</mi><mo>=</mo><mi>β</mi><msup><mrow><mo>|</mo><mo>∇</mo><mi>u</mi><mo>|</mo></mrow><mi>q</mi></msup><mo>+</mo><mi>f</mi><mo> </mo><mrow><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi><mo> </mo><mo>Ω</mo></mrow><mspace width="0.166667em"></mspace><mo>,</mo><mo> </mo><mi>u</mi><mo>=</mo><mn>0</mn><mo> </mo><mrow><mi mathvariant="normal">o</mi><mi mathvariant="normal">n</mi><mo> </mo><mo>∂</mo><mo>Ω</mo></mrow><mspace width="0.166667em"></mspace><mo>,</mo><mspace width="4pt"></mspace></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 open subset of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mrow><mi mathvariant="double-struck">R</mi></mrow><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>2</mn></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>1</mn><mo><</mo><mi>p</mi><mo><</mo><mi>N</mi></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mo>Δ</mo><mi>p</mi></msub><mi>u</mi><mo>=</mo><mi>div</mi><mfenced separators="" open="(" close=")"><msup><mrow><mo>|</mo><mo>∇</mo><mi>u</mi><mo>|</mo></mrow><mrow><mi>p</mi><mo>−</mo><mn>2</mn></mrow></msup><mo>∇</mo><mi>u</mi></mfenced></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>p</mi><mo>−</mo><mn>1</mn><mo><</mo><mi>q</mi><mo><</mo><mi>p</mi></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>β</mi></semantics></math></inline-formula> is a positive constant and <i>f</i> is a measurable function satisfying suitable summability conditions depending on <i>q</i> and a smallness condition.https://www.mdpi.com/2227-7390/13/1/63uniquenessnonlinear elliptic equationsexistencea priori estimates |
| spellingShingle | Angelo Alvino Vincenzo Ferone Anna Mercaldo The Existence and Uniqueness of Nonlinear Elliptic Equations with General Growth in the Gradient Mathematics uniqueness nonlinear elliptic equations existence a priori estimates |
| title | The Existence and Uniqueness of Nonlinear Elliptic Equations with General Growth in the Gradient |
| title_full | The Existence and Uniqueness of Nonlinear Elliptic Equations with General Growth in the Gradient |
| title_fullStr | The Existence and Uniqueness of Nonlinear Elliptic Equations with General Growth in the Gradient |
| title_full_unstemmed | The Existence and Uniqueness of Nonlinear Elliptic Equations with General Growth in the Gradient |
| title_short | The Existence and Uniqueness of Nonlinear Elliptic Equations with General Growth in the Gradient |
| title_sort | existence and uniqueness of nonlinear elliptic equations with general growth in the gradient |
| topic | uniqueness nonlinear elliptic equations existence a priori estimates |
| url | https://www.mdpi.com/2227-7390/13/1/63 |
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