<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

Publicado en línea: 16 may 2019

Páginas: 171 - 188

Recibido: 25 feb 2019

Aceptado: 18 abr 2019

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

Palabras clave
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

Publicado en línea: 16 may 2019

Páginas: 171 - 188

Recibido: 25 feb 2019

Aceptado: 18 abr 2019

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

Palabras claveEntropy 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.

Palabras clave
Entropy solutions, Anisotropic elliptic equations, Anisotropic Sobolev space