Existence of Classical Solutions for Nonlinear Elliptic Equations with Gradient Terms
This paper deals with the existence of solutions of the elliptic equation with nonlinear gradient term <inline-formula<<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"<<semantics<<mrow<<mo<−</mo<<mo<Δ</mo<<mi<...
Ausführliche Beschreibung
Autor*in: |
Yongxiang Li [verfasserIn] Weifeng Ma [verfasserIn] |
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Format: |
E-Artikel |
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Sprache: |
Englisch |
Erschienen: |
2022 |
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Schlagwörter: |
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Übergeordnetes Werk: |
In: Entropy - MDPI AG, 2003, 24(2022), 12, p 1829 |
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Übergeordnetes Werk: |
volume:24 ; year:2022 ; number:12, p 1829 |
Links: |
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DOI / URN: |
10.3390/e24121829 |
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Katalog-ID: |
DOAJ083179798 |
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520 | |a This paper deals with the existence of solutions of the elliptic equation with nonlinear gradient term <inline-formula<<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"<<semantics<<mrow<<mo<−</mo<<mo<Δ</mo<<mi<u</mi<<mo<=</mo<<mi<f</mi<<mo<(</mo<<mi<x</mi<<mo<,</mo<<mspace width="0.166667em"<</mspace<<mi<u</mi<<mo<,</mo<<mspace width="0.166667em"<</mspace<<mo<∇</mo<<mi<u</mi<<mo<)</mo<</mrow<</semantics<</math<</inline-formula< on <inline-formula<<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"<<semantics<<mo<Ω</mo<</semantics<</math<</inline-formula< restricted by the boundary condition <inline-formula<<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"<<semantics<<mrow<<msub<<mrow<<mi<u</mi<<mo<|</mo<</mrow<<mrow<<mo<∂</mo<<mo<Ω</mo<</mrow<</msub<<mo<=</mo<<mn<0</mn<</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 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 sufficiently smooth boundary <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<, <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<, and <inline-formula<<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"<<semantics<<mrow<<mi<f</mi<<mo<:</mo<<mover<<mo<Ω</mo<<mo<¯</mo<</mover<<mo<×</mo<<mi mathvariant="double-struck"<R</mi<<mo<×</mo<<msup<<mi mathvariant="double-struck"<R</mi<<mi<N</mi<</msup<<mo<→</mo<<mi mathvariant="double-struck"<R</mi<</mrow<</semantics<</math<</inline-formula< is continuous. The existence results of classical solutions and positive solutions are obtained under some inequality conditions on the nonlinearity <inline-formula<<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"<<semantics<<mrow<<mi<f</mi<<mo<(</mo<<mi<x</mi<<mo<,</mo<<mspace width="0.166667em"<</mspace<<mi<ξ</mi<<mo<,</mo<<mspace width="0.166667em"<</mspace<<mi<η</mi<<mo<)</mo<</mrow<</semantics<</math<</inline-formula< when <inline-formula<<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"<<semantics<<mrow<<mo<|</mo<<mo<(</mo<<mi<ξ</mi<<mo<,</mo<<mspace width="0.166667em"<</mspace<<mi<η</mi<<mo<)</mo<<mo<|</mo<</mrow<</semantics<</math<</inline-formula< is small or large enough. | ||
650 | 4 | |a elliptic equation | |
650 | 4 | |a gradient term | |
650 | 4 | |a classical solution | |
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10.3390/e24121829 doi (DE-627)DOAJ083179798 (DE-599)DOAJ0497c60252094f9bad7c7d91b3ca56e3 DE-627 ger DE-627 rakwb eng QB460-466 QC1-999 Yongxiang Li verfasserin aut Existence of Classical Solutions for Nonlinear Elliptic Equations with Gradient Terms 2022 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier This paper deals with the existence of solutions of the elliptic equation with nonlinear gradient term <inline-formula<<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"<<semantics<<mrow<<mo<−</mo<<mo<Δ</mo<<mi<u</mi<<mo<=</mo<<mi<f</mi<<mo<(</mo<<mi<x</mi<<mo<,</mo<<mspace width="0.166667em"<</mspace<<mi<u</mi<<mo<,</mo<<mspace width="0.166667em"<</mspace<<mo<∇</mo<<mi<u</mi<<mo<)</mo<</mrow<</semantics<</math<</inline-formula< on <inline-formula<<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"<<semantics<<mo<Ω</mo<</semantics<</math<</inline-formula< restricted by the boundary condition <inline-formula<<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"<<semantics<<mrow<<msub<<mrow<<mi<u</mi<<mo<|</mo<</mrow<<mrow<<mo<∂</mo<<mo<Ω</mo<</mrow<</msub<<mo<=</mo<<mn<0</mn<</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 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 sufficiently smooth boundary <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<, <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<, and <inline-formula<<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"<<semantics<<mrow<<mi<f</mi<<mo<:</mo<<mover<<mo<Ω</mo<<mo<¯</mo<</mover<<mo<×</mo<<mi mathvariant="double-struck"<R</mi<<mo<×</mo<<msup<<mi mathvariant="double-struck"<R</mi<<mi<N</mi<</msup<<mo<→</mo<<mi mathvariant="double-struck"<R</mi<</mrow<</semantics<</math<</inline-formula< is continuous. The existence results of classical solutions and positive solutions are obtained under some inequality conditions on the nonlinearity <inline-formula<<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"<<semantics<<mrow<<mi<f</mi<<mo<(</mo<<mi<x</mi<<mo<,</mo<<mspace width="0.166667em"<</mspace<<mi<ξ</mi<<mo<,</mo<<mspace width="0.166667em"<</mspace<<mi<η</mi<<mo<)</mo<</mrow<</semantics<</math<</inline-formula< when <inline-formula<<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"<<semantics<<mrow<<mo<|</mo<<mo<(</mo<<mi<ξ</mi<<mo<,</mo<<mspace width="0.166667em"<</mspace<<mi<η</mi<<mo<)</mo<<mo<|</mo<</mrow<</semantics<</math<</inline-formula< is small or large enough. elliptic equation gradient term classical solution positive solution Science Q Astrophysics Physics Weifeng Ma verfasserin aut In Entropy MDPI AG, 2003 24(2022), 12, p 1829 (DE-627)316340359 (DE-600)2014734-X 10994300 nnns volume:24 year:2022 number:12, p 1829 https://doi.org/10.3390/e24121829 kostenfrei https://doaj.org/article/0497c60252094f9bad7c7d91b3ca56e3 kostenfrei https://www.mdpi.com/1099-4300/24/12/1829 kostenfrei https://doaj.org/toc/1099-4300 Journal toc kostenfrei GBV_USEFLAG_A SYSFLAG_A GBV_DOAJ GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_65 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_95 GBV_ILN_105 GBV_ILN_110 GBV_ILN_151 GBV_ILN_161 GBV_ILN_170 GBV_ILN_206 GBV_ILN_213 GBV_ILN_230 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_2005 GBV_ILN_2009 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2055 GBV_ILN_2111 GBV_ILN_4012 GBV_ILN_4037 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4249 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4335 GBV_ILN_4338 GBV_ILN_4367 GBV_ILN_4700 AR 24 2022 12, p 1829 |
spelling |
10.3390/e24121829 doi (DE-627)DOAJ083179798 (DE-599)DOAJ0497c60252094f9bad7c7d91b3ca56e3 DE-627 ger DE-627 rakwb eng QB460-466 QC1-999 Yongxiang Li verfasserin aut Existence of Classical Solutions for Nonlinear Elliptic Equations with Gradient Terms 2022 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier This paper deals with the existence of solutions of the elliptic equation with nonlinear gradient term <inline-formula<<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"<<semantics<<mrow<<mo<−</mo<<mo<Δ</mo<<mi<u</mi<<mo<=</mo<<mi<f</mi<<mo<(</mo<<mi<x</mi<<mo<,</mo<<mspace width="0.166667em"<</mspace<<mi<u</mi<<mo<,</mo<<mspace width="0.166667em"<</mspace<<mo<∇</mo<<mi<u</mi<<mo<)</mo<</mrow<</semantics<</math<</inline-formula< on <inline-formula<<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"<<semantics<<mo<Ω</mo<</semantics<</math<</inline-formula< restricted by the boundary condition <inline-formula<<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"<<semantics<<mrow<<msub<<mrow<<mi<u</mi<<mo<|</mo<</mrow<<mrow<<mo<∂</mo<<mo<Ω</mo<</mrow<</msub<<mo<=</mo<<mn<0</mn<</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 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 sufficiently smooth boundary <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<, <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<, and <inline-formula<<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"<<semantics<<mrow<<mi<f</mi<<mo<:</mo<<mover<<mo<Ω</mo<<mo<¯</mo<</mover<<mo<×</mo<<mi mathvariant="double-struck"<R</mi<<mo<×</mo<<msup<<mi mathvariant="double-struck"<R</mi<<mi<N</mi<</msup<<mo<→</mo<<mi mathvariant="double-struck"<R</mi<</mrow<</semantics<</math<</inline-formula< is continuous. The existence results of classical solutions and positive solutions are obtained under some inequality conditions on the nonlinearity <inline-formula<<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"<<semantics<<mrow<<mi<f</mi<<mo<(</mo<<mi<x</mi<<mo<,</mo<<mspace width="0.166667em"<</mspace<<mi<ξ</mi<<mo<,</mo<<mspace width="0.166667em"<</mspace<<mi<η</mi<<mo<)</mo<</mrow<</semantics<</math<</inline-formula< when <inline-formula<<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"<<semantics<<mrow<<mo<|</mo<<mo<(</mo<<mi<ξ</mi<<mo<,</mo<<mspace width="0.166667em"<</mspace<<mi<η</mi<<mo<)</mo<<mo<|</mo<</mrow<</semantics<</math<</inline-formula< is small or large enough. elliptic equation gradient term classical solution positive solution Science Q Astrophysics Physics Weifeng Ma verfasserin aut In Entropy MDPI AG, 2003 24(2022), 12, p 1829 (DE-627)316340359 (DE-600)2014734-X 10994300 nnns volume:24 year:2022 number:12, p 1829 https://doi.org/10.3390/e24121829 kostenfrei https://doaj.org/article/0497c60252094f9bad7c7d91b3ca56e3 kostenfrei https://www.mdpi.com/1099-4300/24/12/1829 kostenfrei https://doaj.org/toc/1099-4300 Journal toc kostenfrei GBV_USEFLAG_A SYSFLAG_A GBV_DOAJ GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_65 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_95 GBV_ILN_105 GBV_ILN_110 GBV_ILN_151 GBV_ILN_161 GBV_ILN_170 GBV_ILN_206 GBV_ILN_213 GBV_ILN_230 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_2005 GBV_ILN_2009 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2055 GBV_ILN_2111 GBV_ILN_4012 GBV_ILN_4037 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4249 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4335 GBV_ILN_4338 GBV_ILN_4367 GBV_ILN_4700 AR 24 2022 12, p 1829 |
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10.3390/e24121829 doi (DE-627)DOAJ083179798 (DE-599)DOAJ0497c60252094f9bad7c7d91b3ca56e3 DE-627 ger DE-627 rakwb eng QB460-466 QC1-999 Yongxiang Li verfasserin aut Existence of Classical Solutions for Nonlinear Elliptic Equations with Gradient Terms 2022 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier This paper deals with the existence of solutions of the elliptic equation with nonlinear gradient term <inline-formula<<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"<<semantics<<mrow<<mo<−</mo<<mo<Δ</mo<<mi<u</mi<<mo<=</mo<<mi<f</mi<<mo<(</mo<<mi<x</mi<<mo<,</mo<<mspace width="0.166667em"<</mspace<<mi<u</mi<<mo<,</mo<<mspace width="0.166667em"<</mspace<<mo<∇</mo<<mi<u</mi<<mo<)</mo<</mrow<</semantics<</math<</inline-formula< on <inline-formula<<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"<<semantics<<mo<Ω</mo<</semantics<</math<</inline-formula< restricted by the boundary condition <inline-formula<<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"<<semantics<<mrow<<msub<<mrow<<mi<u</mi<<mo<|</mo<</mrow<<mrow<<mo<∂</mo<<mo<Ω</mo<</mrow<</msub<<mo<=</mo<<mn<0</mn<</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 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 sufficiently smooth boundary <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<, <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<, and <inline-formula<<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"<<semantics<<mrow<<mi<f</mi<<mo<:</mo<<mover<<mo<Ω</mo<<mo<¯</mo<</mover<<mo<×</mo<<mi mathvariant="double-struck"<R</mi<<mo<×</mo<<msup<<mi mathvariant="double-struck"<R</mi<<mi<N</mi<</msup<<mo<→</mo<<mi mathvariant="double-struck"<R</mi<</mrow<</semantics<</math<</inline-formula< is continuous. The existence results of classical solutions and positive solutions are obtained under some inequality conditions on the nonlinearity <inline-formula<<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"<<semantics<<mrow<<mi<f</mi<<mo<(</mo<<mi<x</mi<<mo<,</mo<<mspace width="0.166667em"<</mspace<<mi<ξ</mi<<mo<,</mo<<mspace width="0.166667em"<</mspace<<mi<η</mi<<mo<)</mo<</mrow<</semantics<</math<</inline-formula< when <inline-formula<<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"<<semantics<<mrow<<mo<|</mo<<mo<(</mo<<mi<ξ</mi<<mo<,</mo<<mspace width="0.166667em"<</mspace<<mi<η</mi<<mo<)</mo<<mo<|</mo<</mrow<</semantics<</math<</inline-formula< is small or large enough. elliptic equation gradient term classical solution positive solution Science Q Astrophysics Physics Weifeng Ma verfasserin aut In Entropy MDPI AG, 2003 24(2022), 12, p 1829 (DE-627)316340359 (DE-600)2014734-X 10994300 nnns volume:24 year:2022 number:12, p 1829 https://doi.org/10.3390/e24121829 kostenfrei https://doaj.org/article/0497c60252094f9bad7c7d91b3ca56e3 kostenfrei https://www.mdpi.com/1099-4300/24/12/1829 kostenfrei https://doaj.org/toc/1099-4300 Journal toc kostenfrei GBV_USEFLAG_A SYSFLAG_A GBV_DOAJ GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_65 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_95 GBV_ILN_105 GBV_ILN_110 GBV_ILN_151 GBV_ILN_161 GBV_ILN_170 GBV_ILN_206 GBV_ILN_213 GBV_ILN_230 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_2005 GBV_ILN_2009 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2055 GBV_ILN_2111 GBV_ILN_4012 GBV_ILN_4037 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4249 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4335 GBV_ILN_4338 GBV_ILN_4367 GBV_ILN_4700 AR 24 2022 12, p 1829 |
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10.3390/e24121829 doi (DE-627)DOAJ083179798 (DE-599)DOAJ0497c60252094f9bad7c7d91b3ca56e3 DE-627 ger DE-627 rakwb eng QB460-466 QC1-999 Yongxiang Li verfasserin aut Existence of Classical Solutions for Nonlinear Elliptic Equations with Gradient Terms 2022 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier This paper deals with the existence of solutions of the elliptic equation with nonlinear gradient term <inline-formula<<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"<<semantics<<mrow<<mo<−</mo<<mo<Δ</mo<<mi<u</mi<<mo<=</mo<<mi<f</mi<<mo<(</mo<<mi<x</mi<<mo<,</mo<<mspace width="0.166667em"<</mspace<<mi<u</mi<<mo<,</mo<<mspace width="0.166667em"<</mspace<<mo<∇</mo<<mi<u</mi<<mo<)</mo<</mrow<</semantics<</math<</inline-formula< on <inline-formula<<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"<<semantics<<mo<Ω</mo<</semantics<</math<</inline-formula< restricted by the boundary condition <inline-formula<<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"<<semantics<<mrow<<msub<<mrow<<mi<u</mi<<mo<|</mo<</mrow<<mrow<<mo<∂</mo<<mo<Ω</mo<</mrow<</msub<<mo<=</mo<<mn<0</mn<</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 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 sufficiently smooth boundary <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<, <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<, and <inline-formula<<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"<<semantics<<mrow<<mi<f</mi<<mo<:</mo<<mover<<mo<Ω</mo<<mo<¯</mo<</mover<<mo<×</mo<<mi mathvariant="double-struck"<R</mi<<mo<×</mo<<msup<<mi mathvariant="double-struck"<R</mi<<mi<N</mi<</msup<<mo<→</mo<<mi mathvariant="double-struck"<R</mi<</mrow<</semantics<</math<</inline-formula< is continuous. The existence results of classical solutions and positive solutions are obtained under some inequality conditions on the nonlinearity <inline-formula<<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"<<semantics<<mrow<<mi<f</mi<<mo<(</mo<<mi<x</mi<<mo<,</mo<<mspace width="0.166667em"<</mspace<<mi<ξ</mi<<mo<,</mo<<mspace width="0.166667em"<</mspace<<mi<η</mi<<mo<)</mo<</mrow<</semantics<</math<</inline-formula< when <inline-formula<<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"<<semantics<<mrow<<mo<|</mo<<mo<(</mo<<mi<ξ</mi<<mo<,</mo<<mspace width="0.166667em"<</mspace<<mi<η</mi<<mo<)</mo<<mo<|</mo<</mrow<</semantics<</math<</inline-formula< is small or large enough. elliptic equation gradient term classical solution positive solution Science Q Astrophysics Physics Weifeng Ma verfasserin aut In Entropy MDPI AG, 2003 24(2022), 12, p 1829 (DE-627)316340359 (DE-600)2014734-X 10994300 nnns volume:24 year:2022 number:12, p 1829 https://doi.org/10.3390/e24121829 kostenfrei https://doaj.org/article/0497c60252094f9bad7c7d91b3ca56e3 kostenfrei https://www.mdpi.com/1099-4300/24/12/1829 kostenfrei https://doaj.org/toc/1099-4300 Journal toc kostenfrei GBV_USEFLAG_A SYSFLAG_A GBV_DOAJ GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_65 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_95 GBV_ILN_105 GBV_ILN_110 GBV_ILN_151 GBV_ILN_161 GBV_ILN_170 GBV_ILN_206 GBV_ILN_213 GBV_ILN_230 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_2005 GBV_ILN_2009 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2055 GBV_ILN_2111 GBV_ILN_4012 GBV_ILN_4037 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4249 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4335 GBV_ILN_4338 GBV_ILN_4367 GBV_ILN_4700 AR 24 2022 12, p 1829 |
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10.3390/e24121829 doi (DE-627)DOAJ083179798 (DE-599)DOAJ0497c60252094f9bad7c7d91b3ca56e3 DE-627 ger DE-627 rakwb eng QB460-466 QC1-999 Yongxiang Li verfasserin aut Existence of Classical Solutions for Nonlinear Elliptic Equations with Gradient Terms 2022 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier This paper deals with the existence of solutions of the elliptic equation with nonlinear gradient term <inline-formula<<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"<<semantics<<mrow<<mo<−</mo<<mo<Δ</mo<<mi<u</mi<<mo<=</mo<<mi<f</mi<<mo<(</mo<<mi<x</mi<<mo<,</mo<<mspace width="0.166667em"<</mspace<<mi<u</mi<<mo<,</mo<<mspace width="0.166667em"<</mspace<<mo<∇</mo<<mi<u</mi<<mo<)</mo<</mrow<</semantics<</math<</inline-formula< on <inline-formula<<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"<<semantics<<mo<Ω</mo<</semantics<</math<</inline-formula< restricted by the boundary condition <inline-formula<<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"<<semantics<<mrow<<msub<<mrow<<mi<u</mi<<mo<|</mo<</mrow<<mrow<<mo<∂</mo<<mo<Ω</mo<</mrow<</msub<<mo<=</mo<<mn<0</mn<</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 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 sufficiently smooth boundary <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<, <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<, and <inline-formula<<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"<<semantics<<mrow<<mi<f</mi<<mo<:</mo<<mover<<mo<Ω</mo<<mo<¯</mo<</mover<<mo<×</mo<<mi mathvariant="double-struck"<R</mi<<mo<×</mo<<msup<<mi mathvariant="double-struck"<R</mi<<mi<N</mi<</msup<<mo<→</mo<<mi mathvariant="double-struck"<R</mi<</mrow<</semantics<</math<</inline-formula< is continuous. The existence results of classical solutions and positive solutions are obtained under some inequality conditions on the nonlinearity <inline-formula<<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"<<semantics<<mrow<<mi<f</mi<<mo<(</mo<<mi<x</mi<<mo<,</mo<<mspace width="0.166667em"<</mspace<<mi<ξ</mi<<mo<,</mo<<mspace width="0.166667em"<</mspace<<mi<η</mi<<mo<)</mo<</mrow<</semantics<</math<</inline-formula< when <inline-formula<<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"<<semantics<<mrow<<mo<|</mo<<mo<(</mo<<mi<ξ</mi<<mo<,</mo<<mspace width="0.166667em"<</mspace<<mi<η</mi<<mo<)</mo<<mo<|</mo<</mrow<</semantics<</math<</inline-formula< is small or large enough. elliptic equation gradient term classical solution positive solution Science Q Astrophysics Physics Weifeng Ma verfasserin aut In Entropy MDPI AG, 2003 24(2022), 12, p 1829 (DE-627)316340359 (DE-600)2014734-X 10994300 nnns volume:24 year:2022 number:12, p 1829 https://doi.org/10.3390/e24121829 kostenfrei https://doaj.org/article/0497c60252094f9bad7c7d91b3ca56e3 kostenfrei https://www.mdpi.com/1099-4300/24/12/1829 kostenfrei https://doaj.org/toc/1099-4300 Journal toc kostenfrei GBV_USEFLAG_A SYSFLAG_A GBV_DOAJ GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_65 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_95 GBV_ILN_105 GBV_ILN_110 GBV_ILN_151 GBV_ILN_161 GBV_ILN_170 GBV_ILN_206 GBV_ILN_213 GBV_ILN_230 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_2005 GBV_ILN_2009 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2055 GBV_ILN_2111 GBV_ILN_4012 GBV_ILN_4037 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4249 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4335 GBV_ILN_4338 GBV_ILN_4367 GBV_ILN_4700 AR 24 2022 12, p 1829 |
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In Entropy 24(2022), 12, p 1829 volume:24 year:2022 number:12, p 1829 |
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Yongxiang Li @@aut@@ Weifeng Ma @@aut@@ |
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Yongxiang Li |
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Yongxiang Li misc QB460-466 misc QC1-999 misc elliptic equation misc gradient term misc classical solution misc positive solution misc Science misc Q misc Astrophysics misc Physics Existence of Classical Solutions for Nonlinear Elliptic Equations with Gradient Terms |
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QB460-466 QC1-999 Existence of Classical Solutions for Nonlinear Elliptic Equations with Gradient Terms elliptic equation gradient term classical solution positive solution |
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Existence of Classical Solutions for Nonlinear Elliptic Equations with Gradient Terms |
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Existence of Classical Solutions for Nonlinear Elliptic Equations with Gradient Terms |
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existence of classical solutions for nonlinear elliptic equations with gradient terms |
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Existence of Classical Solutions for Nonlinear Elliptic Equations with Gradient Terms |
abstract |
This paper deals with the existence of solutions of the elliptic equation with nonlinear gradient term <inline-formula<<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"<<semantics<<mrow<<mo<−</mo<<mo<Δ</mo<<mi<u</mi<<mo<=</mo<<mi<f</mi<<mo<(</mo<<mi<x</mi<<mo<,</mo<<mspace width="0.166667em"<</mspace<<mi<u</mi<<mo<,</mo<<mspace width="0.166667em"<</mspace<<mo<∇</mo<<mi<u</mi<<mo<)</mo<</mrow<</semantics<</math<</inline-formula< on <inline-formula<<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"<<semantics<<mo<Ω</mo<</semantics<</math<</inline-formula< restricted by the boundary condition <inline-formula<<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"<<semantics<<mrow<<msub<<mrow<<mi<u</mi<<mo<|</mo<</mrow<<mrow<<mo<∂</mo<<mo<Ω</mo<</mrow<</msub<<mo<=</mo<<mn<0</mn<</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 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 sufficiently smooth boundary <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<, <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<, and <inline-formula<<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"<<semantics<<mrow<<mi<f</mi<<mo<:</mo<<mover<<mo<Ω</mo<<mo<¯</mo<</mover<<mo<×</mo<<mi mathvariant="double-struck"<R</mi<<mo<×</mo<<msup<<mi mathvariant="double-struck"<R</mi<<mi<N</mi<</msup<<mo<→</mo<<mi mathvariant="double-struck"<R</mi<</mrow<</semantics<</math<</inline-formula< is continuous. The existence results of classical solutions and positive solutions are obtained under some inequality conditions on the nonlinearity <inline-formula<<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"<<semantics<<mrow<<mi<f</mi<<mo<(</mo<<mi<x</mi<<mo<,</mo<<mspace width="0.166667em"<</mspace<<mi<ξ</mi<<mo<,</mo<<mspace width="0.166667em"<</mspace<<mi<η</mi<<mo<)</mo<</mrow<</semantics<</math<</inline-formula< when <inline-formula<<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"<<semantics<<mrow<<mo<|</mo<<mo<(</mo<<mi<ξ</mi<<mo<,</mo<<mspace width="0.166667em"<</mspace<<mi<η</mi<<mo<)</mo<<mo<|</mo<</mrow<</semantics<</math<</inline-formula< is small or large enough. |
abstractGer |
This paper deals with the existence of solutions of the elliptic equation with nonlinear gradient term <inline-formula<<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"<<semantics<<mrow<<mo<−</mo<<mo<Δ</mo<<mi<u</mi<<mo<=</mo<<mi<f</mi<<mo<(</mo<<mi<x</mi<<mo<,</mo<<mspace width="0.166667em"<</mspace<<mi<u</mi<<mo<,</mo<<mspace width="0.166667em"<</mspace<<mo<∇</mo<<mi<u</mi<<mo<)</mo<</mrow<</semantics<</math<</inline-formula< on <inline-formula<<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"<<semantics<<mo<Ω</mo<</semantics<</math<</inline-formula< restricted by the boundary condition <inline-formula<<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"<<semantics<<mrow<<msub<<mrow<<mi<u</mi<<mo<|</mo<</mrow<<mrow<<mo<∂</mo<<mo<Ω</mo<</mrow<</msub<<mo<=</mo<<mn<0</mn<</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 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 sufficiently smooth boundary <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<, <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<, and <inline-formula<<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"<<semantics<<mrow<<mi<f</mi<<mo<:</mo<<mover<<mo<Ω</mo<<mo<¯</mo<</mover<<mo<×</mo<<mi mathvariant="double-struck"<R</mi<<mo<×</mo<<msup<<mi mathvariant="double-struck"<R</mi<<mi<N</mi<</msup<<mo<→</mo<<mi mathvariant="double-struck"<R</mi<</mrow<</semantics<</math<</inline-formula< is continuous. The existence results of classical solutions and positive solutions are obtained under some inequality conditions on the nonlinearity <inline-formula<<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"<<semantics<<mrow<<mi<f</mi<<mo<(</mo<<mi<x</mi<<mo<,</mo<<mspace width="0.166667em"<</mspace<<mi<ξ</mi<<mo<,</mo<<mspace width="0.166667em"<</mspace<<mi<η</mi<<mo<)</mo<</mrow<</semantics<</math<</inline-formula< when <inline-formula<<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"<<semantics<<mrow<<mo<|</mo<<mo<(</mo<<mi<ξ</mi<<mo<,</mo<<mspace width="0.166667em"<</mspace<<mi<η</mi<<mo<)</mo<<mo<|</mo<</mrow<</semantics<</math<</inline-formula< is small or large enough. |
abstract_unstemmed |
This paper deals with the existence of solutions of the elliptic equation with nonlinear gradient term <inline-formula<<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"<<semantics<<mrow<<mo<−</mo<<mo<Δ</mo<<mi<u</mi<<mo<=</mo<<mi<f</mi<<mo<(</mo<<mi<x</mi<<mo<,</mo<<mspace width="0.166667em"<</mspace<<mi<u</mi<<mo<,</mo<<mspace width="0.166667em"<</mspace<<mo<∇</mo<<mi<u</mi<<mo<)</mo<</mrow<</semantics<</math<</inline-formula< on <inline-formula<<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"<<semantics<<mo<Ω</mo<</semantics<</math<</inline-formula< restricted by the boundary condition <inline-formula<<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"<<semantics<<mrow<<msub<<mrow<<mi<u</mi<<mo<|</mo<</mrow<<mrow<<mo<∂</mo<<mo<Ω</mo<</mrow<</msub<<mo<=</mo<<mn<0</mn<</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 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 sufficiently smooth boundary <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<, <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<, and <inline-formula<<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"<<semantics<<mrow<<mi<f</mi<<mo<:</mo<<mover<<mo<Ω</mo<<mo<¯</mo<</mover<<mo<×</mo<<mi mathvariant="double-struck"<R</mi<<mo<×</mo<<msup<<mi mathvariant="double-struck"<R</mi<<mi<N</mi<</msup<<mo<→</mo<<mi mathvariant="double-struck"<R</mi<</mrow<</semantics<</math<</inline-formula< is continuous. The existence results of classical solutions and positive solutions are obtained under some inequality conditions on the nonlinearity <inline-formula<<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"<<semantics<<mrow<<mi<f</mi<<mo<(</mo<<mi<x</mi<<mo<,</mo<<mspace width="0.166667em"<</mspace<<mi<ξ</mi<<mo<,</mo<<mspace width="0.166667em"<</mspace<<mi<η</mi<<mo<)</mo<</mrow<</semantics<</math<</inline-formula< when <inline-formula<<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"<<semantics<<mrow<<mo<|</mo<<mo<(</mo<<mi<ξ</mi<<mo<,</mo<<mspace width="0.166667em"<</mspace<<mi<η</mi<<mo<)</mo<<mo<|</mo<</mrow<</semantics<</math<</inline-formula< is small or large enough. |
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Existence of Classical Solutions for Nonlinear Elliptic Equations with Gradient Terms |
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