A nonlocal elliptic system with nonlinear singular terms
<p>In this work we deal with the nonlocal elliptic system: <math><mrow><mo>(</mo><mi>S</mi><mo>)</mo></mrow><mrow><mo>{</mo><mo>(</mo><mo>−</mo><mi>Δ</mi><mo>)</mo><mrow&g...
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2025
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| Summary: | <p>In this work we deal with the nonlocal elliptic system: <math><mrow><mo>(</mo><mi>S</mi><mo>)</mo></mrow><mrow><mo>{</mo><mo>(</mo><mo>−</mo><mi>Δ</mi><mo>)</mo><mrow><msub><mi>s</mi><mn>1</mn></msub></mrow><mi>u</mi><mo>=</mo><mi>θ</mi><mi>v</mi><mi>q</mi><mi>u</mi><mrow><mn>1</mn><mo>−</mo><mi>θ</mi></mrow><mrow><mi>in</mi></mrow> <mi>Ω</mi><mo>,</mo><mo>(</mo><mo>−</mo><mi>Δ</mi><mo>)</mo><mrow><msub><mi>s</mi><mn>2</mn></msub></mrow><mi>v</mi><mo>=</mo><mi>q</mi><mi>v</mi><mrow><mi>q</mi><mo>−</mo><mn>1</mn></mrow><mi>u</mi><mi>θ</mi><mrow><mi>in</mi></mrow> <mi>Ω</mi><mo>,</mo><mi>u</mi><mo>=</mo><mi>v</mi><mo>=</mo><mn>0</mn>in <mrow><mi>R</mi></mrow><mi>N</mi><mo>∖</mo><mi>Ω</mi><mo>,</mo><mi>u</mi><mo>,</mo><mi>v</mi><mo>></mo><mn>0</mn>in <mi>Ω</mi><mo>,</mo><mo></mo></mrow></math> where <math><mi>Ω</mi><mo>⊂</mo><mrow><mi>R</mi></mrow><mi>N</mi></math> is a bounded regular domain (<math><mrow><mi>C</mi></mrow><mn>2</mn></math> is sufficient), <math><mi>N</mi><mo>></mo><mn>2</mn><msub><mi>s</mi><mn>2</mn></msub></math>, <math><msub><mi>s</mi><mn>1</mn></msub><mo>,</mo><msub><mi>s</mi><mn>2</mn></msub><mo>∈</mo><mo>(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>)</mo></math> with <math><msub><mi>s</mi><mn>2</mn></msub><mo>≠</mo><msub><mi>s</mi><mn>1</mn></msub></math>, <math><mn>0</mn><mo><</mo><mi>θ</mi><mo><</mo><mn>1</mn></math> and <i>q</i>>0. This work extends previous results obtained in the local case (<math><msub><mi>s</mi><mn>1</mn></msub><mo>=</mo><msub><mi>s</mi><mn>2</mn></msub><mo>=</mo><mn>1</mn></math>) see Boccardo L, Orsina L. [A variational semilinear singular system. Nonlinear Anal Theory Methods Appl. 2011;74(12):3849–3860. doi: <a href="http://doi.org/10.1016/j.na.2011.01.017" target="_blank">10.1016/j.na.2011.01.017</a>]. Under some suitable conditions on the parameters on <math><msub><mi>s</mi><mn>1</mn></msub><mo>,</mo><msub><mi>s</mi><mn>2</mn></msub><mo>,</mo><mi>θ</mi></math> and <i>q</i>, we obtain the existence results by using approximation methods, variational techniques and the classical minimization, a nonexistence result has also been treated.</p> |
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