New Insights on Keller–Osserman Conditions for Semilinear Systems
In this article, we consider a semilinear elliptic system involving gradient terms of the form <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mfenced separators="" open="{" close=""&...
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| Format: | Article |
| Language: | English |
| Published: |
MDPI AG
2024-12-01
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| Series: | Mathematics |
| Subjects: | |
| Online Access: | https://www.mdpi.com/2227-7390/13/1/83 |
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| Summary: | In this article, we consider a semilinear elliptic system involving gradient terms of the form <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mfenced separators="" open="{" close=""><mtable><mtr><mtd columnalign="left"><mrow><mo>Δ</mo><mi>y</mi><mfenced open="(" close=")"><mi>x</mi></mfenced><mo>+</mo><msub><mi>λ</mi><mn>1</mn></msub><mfenced separators="" open="|" close="|"><mo>∇</mo><mi>y</mi><mfenced open="(" close=")"><mi>x</mi></mfenced></mfenced><mo>=</mo><mi>p</mi><mfenced open="(" close=")"><mfenced open="|" close="|"><mi>x</mi></mfenced></mfenced><mi>f</mi><mfenced separators="" open="(" close=")"><mi>y</mi><mfenced open="(" close=")"><mi>x</mi></mfenced><mo>,</mo><mi>z</mi><mfenced open="(" close=")"><mi>x</mi></mfenced></mfenced></mrow></mtd><mtd columnalign="left"><mrow><mi>i</mi><mi>f</mi></mrow></mtd><mtd columnalign="left"><mrow><mi>x</mi><mo>∈</mo><mo>Ω</mo><mo>,</mo></mrow></mtd></mtr><mtr><mtd columnalign="left"><mrow><mo>Δ</mo><mi>z</mi><mfenced open="(" close=")"><mi>x</mi></mfenced><mo>+</mo><msub><mi>λ</mi><mn>2</mn></msub><mfenced separators="" open="|" close="|"><mo>∇</mo><mi>z</mi><mfenced open="(" close=")"><mi>x</mi></mfenced></mfenced><mo>=</mo><mi>q</mi><mfenced open="(" close=")"><mfenced open="|" close="|"><mi>x</mi></mfenced></mfenced><mi>g</mi><mfenced separators="" open="(" close=")"><mi>y</mi><mfenced open="(" close=")"><mi>x</mi></mfenced></mfenced></mrow></mtd><mtd columnalign="left"><mrow><mi>i</mi><mi>f</mi></mrow></mtd><mtd columnalign="left"><mrow><mi>x</mi><mo>∈</mo><mo>Ω</mo><mo>,</mo></mrow></mtd></mtr></mtable></mfenced></semantics></math></inline-formula> where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>λ</mi><mn>1</mn></msub></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>λ</mi><mn>2</mn></msub><mo>∈</mo><mfenced separators="" open="[" close=")"><mn>0</mn><mo>,</mo><mo>∞</mo></mfenced></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mo>Ω</mo></semantics></math></inline-formula> is either a ball of radius <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>R</mi><mo>></mo><mn>0</mn></mrow></semantics></math></inline-formula> or the entire space <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>. Based on certain standard assumptions regarding the potential functions <i>p</i> and <i>q</i>, we introduce new conditions on the nonlinearities <i>f</i> and <i>g</i> to investigate the existence of entire large solutions for the given system. The method employed is successive approximation. Additionally, for specific cases of <i>p</i>, <i>q</i>, <i>f</i> and <i>g</i>, we employ Python code to plot the graph of both the numerical solution and the exact solution. |
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| ISSN: | 2227-7390 |