<abstract xmlns="http://www.w3.org/1999/xhtml">In this paper, we prove the existence of entropy solutions for anisotropic elliptic unilateral problem associated to the equations of the form<disp-formula><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_mjpaa-2018-0014_eq_001.png"></graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mrow><mo>-</mo><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>N</mi></munderover><mrow><msub><mrow><mo>∂</mo></mrow><mi>i</mi></msub><msub><mrow><mi>a</mi></mrow><mi>i</mi></msub><mo>(</mo><mi>x</mi><mo>,</mo><mi>u</mi><mo>,</mo><mo>∇</mo><mi>u</mi><mo>)</mo><mo>-</mo></mrow><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>N</mi></munderover><mrow><msub><mrow><mo>∂</mo></mrow><mi>i</mi></msub><msub><mrow><mi>φ</mi></mrow><mi>i</mi></msub><mo>(</mo><mi>u</mi><mo>)</mo><mo>=</mo><mi>f</mi><mo>,</mo></mrow></mrow></math><tex-math>$$ - \sum\limits_{i = 1}^N {{\partial _i}{a_i}(x,u,\nabla u) - } \sum\limits_{i = 1}^N {{\partial _i}{\phi _i}(u) = f,} $$</tex-math></alternatives></disp-formula>where the right hand side <italic>f</italic> belongs to <italic>L</italic>1(Ω). The operator <inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_mjpaa-2018-0014_eq_002.png"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mrow><mo>-</mo><msubsup><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>N</mi></msubsup><mrow><msub><mrow><mo>∂</mo></mrow><mi>i</mi></msub><msub><mrow><mi>a</mi></mrow><mi>i</mi></msub><mrow><mo>(</mo><mrow><mi>x</mi><mo>,</mo><mi>u</mi><mo>,</mo><mo>∇</mo><mi>u</mi></mrow><mo>)</mo></mrow></mrow></mrow></math><tex-math>$- \sum\nolimits_{i = 1}^N {{\partial _i}{a_i}\left( {x,u,\nabla u} \right)} $</tex-math></alternatives></inline-formula> is a Leray-Lions anisotropic operator and <italic>ϕi</italic> ∈ <italic>C</italic>0(ℝ,ℝ).</abstract>

In this paper, we prove the existence of entropy solutions for anisotropic elliptic unilateral problem associated to the equations of the form-∑i=1N∂iai(x,u,∇u)-∑i=1N∂iφi(u)=f,$$ - \sum\limits_{i = 1}^N {{\partial _i}{a_i}(x,u,\nabla u) - } \sum\limits_{i = 1}^N {{\partial _i}{\phi _i}(u) = f,} $$where the right hand side f belongs to L1(Ω). The operator -∑i=1N∂iai(x,u,∇u)$- \sum\nolimits_{i = 1}^N {{\partial _i}{a_i}\left( {x,u,\nabla u} \right)} $ is a Leray-Lions anisotropic operator and ϕi ∈ C0(ℝ,ℝ).

Existence of Solutions for Some Nonlinear Elliptic Anisotropic Unilateral Problems with Lower Order Terms

Moroccan Journal of Pure and Applied Analysis

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

In this paper, we prove the existence of entropy solutions for anisotropic elliptic unilateral problem associated to the equations of the...

Existence of Solutions for Some Nonlinear Elliptic Anisotropic Unilateral Problems with Lower Order Terms

Online veröffentlicht: 16. Mai 2019

Seitenbereich: 171 - 188

Eingereicht: 25. Feb. 2019

Akzeptiert: 18. Apr. 2019

DOI: https://doi.org/10.1515/mjpaa-2018-0014

Schlüsselwörter
Entropy solutions, Anisotropic elliptic equations, Anisotropic Sobolev space

© 2018 Youssef Akdim et al., published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

Existence of Solutions for Some Nonlinear Elliptic Anisotropic Unilateral Problems with Lower Order Terms

Online veröffentlicht: 16. Mai 2019

Seitenbereich: 171 - 188

Eingereicht: 25. Feb. 2019

Akzeptiert: 18. Apr. 2019

DOI: https://doi.org/10.1515/mjpaa-2018-0014

SchlüsselwörterEntropy solutions, Anisotropic elliptic equations, Anisotropic Sobolev space

© 2018 Youssef Akdim et al., published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

Schlüsselwörter
Entropy solutions, Anisotropic elliptic equations, Anisotropic Sobolev space