<abstract xmlns="http://www.w3.org/1999/xhtml">In this paper, we show the existence of solutions for the nonlinear elliptic equations of the form<disp-formula><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_mjpaa-2020-0002_eq_001.png"></graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mrow><mrow><mo>{</mo><mrow><mtable><mtr><mtd><mrow><mo>-</mo><mtext>div</mtext><mi> </mi><mi>a</mi><mrow><mo>(</mo><mrow><mi>x</mi><mo>,</mo><mi>u</mi><mo>,</mo><mo>∇</mo><mi>u</mi></mrow><mo>)</mo></mrow><mo>=</mo><mi>f</mi><mo>,</mo></mrow></mtd></mtr><mtr><mtd><mrow><mi>u</mi><mo>∈</mo><msubsup><mrow><mi>W</mi></mrow><mn>0</mn><mn>1</mn></msubsup><mi>L</mi><mi>ϕ</mi><mrow><mo>(</mo><mi>Ω</mi><mo>)</mo></mrow><mo>∩</mo><msup><mrow><mi>L</mi></mrow><mo>∞</mo></msup><mrow><mo>(</mo><mi>Ω</mi><mo>)</mo></mrow><mo>,</mo></mrow></mtd></mtr></mtable></mrow></mrow></mrow></math><tex-math>\left\{ {\matrix{ { - {\rm{div}}\,a\left( {x,u,\nabla u} \right) = f,} \hfill \cr {u \in W_0^1L\varphi \left( \Omega \right) \cap {L^\infty }\left( \Omega \right),} \hfill \cr } } \right.</tex-math></alternatives></disp-formula>where <inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_mjpaa-2020-0002_eq_002.png"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mrow><mi>a</mi><mrow><mo>(</mo><mrow><mi>x</mi><mo>,</mo><mi>s</mi><mo>,</mo><mi>ξ</mi></mrow><mo>)</mo></mrow><mo>⋅</mo><mi>ξ</mi><mo>≥</mo><msubsup><mrow><mover><mi>ϕ</mi><mo>¯</mo></mover></mrow><mi>x</mi><mrow><mo>-</mo><mn>1</mn></mrow></msubsup><mrow><mo>(</mo><mrow><mi>ϕ</mi><mrow><mo>(</mo><mrow><mi>x</mi><mo>,</mo><mi>h</mi><mrow><mo>(</mo><mrow><mrow><mo>|</mo><mi>s</mi><mo>|</mo></mrow></mrow><mo>)</mo></mrow></mrow><mo>)</mo></mrow></mrow><mo>)</mo></mrow><mi>ϕ</mi><mrow><mo>(</mo><mrow><mi>x</mi><mo>,</mo><mrow><mo>|</mo><mi>ξ</mi><mo>|</mo></mrow></mrow><mo>)</mo></mrow></mrow></math><tex-math>a\left( {x,s,\xi } \right) \cdot \xi \ge \bar \varphi _x^{ - 1}\left( {\varphi \left( {x,h\left( {\left| s \right|} \right)} \right)} \right)\varphi \left( {x,\left| \xi \right|} \right)</tex-math></alternatives></inline-formula> and <italic>h</italic> : ℝ+→]0, 1] is a continuous decreasing function with unbounded primitive. The second term <italic>f</italic> belongs to <italic>LN</italic>(Ω) or to <italic>Lm</italic>(Ω), with <inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_mjpaa-2020-0002_eq_003.png"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mrow><mi>m</mi><mo>=</mo><mfrac><mrow><mi>r</mi><mi>N</mi></mrow><mrow><mi>r</mi><mo>+</mo><mn>1</mn></mrow></mfrac></mrow></math><tex-math>m = {{rN} \over {r + 1}}</tex-math></alternatives></inline-formula> for some <italic>r</italic> > 0 and <italic>φ</italic> is a Musielak function satisfying the Δ2-condition.</abstract>

In this paper, we show the existence of solutions for the nonlinear elliptic equations of the form{-div a(x,u,∇u)=f,u∈W01Lϕ(Ω)∩L∞(Ω),\left\{ {\matrix{   { - {\rm{div}}\,a\left( {x,u,\nabla u} \right) = f,} \hfill  \cr    {u \in W_0^1L\varphi \left( \Omega  \right) \cap {L^\infty }\left( \Omega  \right),} \hfill  \cr  } } \right.where a(x,s,ξ)⋅ξ≥ϕ¯x-1(ϕ(x,h(|s|)))ϕ(x,|ξ|)a\left( {x,s,\xi } \right) \cdot \xi  \ge \bar \varphi _x^{ - 1}\left( {\varphi \left( {x,h\left( {\left| s \right|} \right)} \right)} \right)\varphi \left( {x,\left| \xi  \right|} \right) and h : ℝ+→]0, 1] is a continuous decreasing function with unbounded primitive. The second term f belongs to LN(Ω) or to Lm(Ω), with m=rNr+1m = {{rN} \over {r + 1}} for some r > 0 and φ is a Musielak function satisfying the Δ2-condition.

Existence of a weak bounded solution for nonlinear degenerate elliptic equations in Musielak-Orlicz spaces

Department of Mathematics, Faculty of Sciences Dhar El Mahraz

Department of Mathematics, Faculty of Sciences El Jadida

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 show the existence of solutions for the nonlinear elliptic equations of the form{-div a(x,u,∇u)=f,u∈W01Lϕ(Ω)∩L∞(Ω),\left\{ {\matrix{   { -...

Existence of a weak bounded solution for nonlinear degenerate elliptic equations in Musielak-Orlicz spaces

Published Online: May 29, 2020

Page range: 16 - 33

Received: Nov 20, 2019

Accepted: Feb 26, 2020

DOI: https://doi.org/10.2478/mjpaa-2020-0002

Keywords
Elliptic equations, Musielak-Orlicz-Sobolev spaces, -estimates, Bounded solutions

© 2020 M. Bourahma et al., published by Sciendo

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

Existence of a weak bounded solution for nonlinear degenerate elliptic equations in Musielak-Orlicz spaces

Published Online: May 29, 2020

Page range: 16 - 33

Received: Nov 20, 2019

Accepted: Feb 26, 2020

DOI: https://doi.org/10.2478/mjpaa-2020-0002

KeywordsElliptic equations, Musielak-Orlicz-Sobolev spaces, -estimates, Bounded solutions

© 2020 M. Bourahma et al., published by Sciendo

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

Keywords
Elliptic equations, Musielak-Orlicz-Sobolev spaces, -estimates, Bounded solutions