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

Abstract

<jats:title>Abstract</jats:title>
<jats:p>In this paper, we show the existence of solutions for the nonlinear elliptic equations of the form</jats:p>
<jats:p>
<jats:disp-formula>
<jats:alternatives>
<jats:graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_mjpaa-2020-0002_eq_001.png"/>
<m:math xmlns:m="http://www.w3.org/1998/Math/MathML">
<m:mrow>
<m:mrow>
<m:mo>{</m:mo>
<m:mrow>
<m:mtable>
<m:mtr>
<m:mtd>
<m:mrow>
<m:mo>-</m:mo>
<m:mtext>div</m:mtext>
<m:mi> </m:mi>
<m:mi>a</m:mi>
<m:mrow>
<m:mo>(</m:mo>
<m:mrow>
<m:mi>x</m:mi>
<m:mo>,</m:mo>
<m:mi>u</m:mi>
<m:mo>,</m:mo>
<m:mo>∇</m:mo>
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<m:mo>=</m:mo>
<m:mi>f</m:mi>
<m:mo>,</m:mo>
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<m:mi>u</m:mi>
<m:mo>∈</m:mo>
<m:msubsup>
<m:mrow>
<m:mi>W</m:mi>
</m:mrow>
<m:mn>0</m:mn>
<m:mn>1</m:mn>
</m:msubsup>
<m:mi>L</m:mi>
<m:mi>ϕ</m:mi>
<m:mrow>
<m:mo>(</m:mo>
<m:mi>Ω</m:mi>
<m:mo>)</m:mo>
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<m:mo>∩</m:mo>
<m:msup>
<m:mrow>
<m:mi>L</m:mi>
</m:mrow>
<m:mo>∞</m:mo>
</m:msup>
<m:mrow>
<m:mo>(</m:mo>
<m:mi>Ω</m:mi>
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<m:mo>,</m:mo>
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</m:mtd>
</m:mtr>
</m:mtable>
</m:mrow>
</m:mrow>
</m:mrow>
</m:math>
<jats: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.</jats:tex-math>
</jats:alternatives>
</jats:disp-formula>
</jats:p>
<jats:p>where <jats:inline-formula>
<jats:alternatives>
<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_mjpaa-2020-0002_eq_002.png"/>
<m:math xmlns:m="http://www.w3.org/1998/Math/MathML">
<m:mrow>
<m:mi>a</m:mi>
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<m:mi>x</m:mi>
<m:mo>,</m:mo>
<m:mi>s</m:mi>
<m:mo>,</m:mo>
<m:mi>ξ</m:mi>
</m:mrow>
<m:mo>)</m:mo>
</m:mrow>
<m:mo>⋅</m:mo>
<m:mi>ξ</m:mi>
<m:mo>≥</m:mo>
<m:msubsup>
<m:mrow>
<m:mover>
<m:mi>ϕ</m:mi>
<m:mo>¯</m:mo>
</m:mover>
</m:mrow>
<m:mi>x</m:mi>
<m:mrow>
<m:mo>-</m:mo>
<m:mn>1</m:mn>
</m:mrow>
</m:msubsup>
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<m:mo>(</m:mo>
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<m:mi>ϕ</m:mi>
<m:mrow>
<m:mo>(</m:mo>
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<m:mi>x</m:mi>
<m:mo>,</m:mo>
<m:mi>h</m:mi>
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<m:mo>|</m:mo>
<m:mi>s</m:mi>
<m:mo>|</m:mo>
</m:mrow>
</m:mrow>
<m:mo>)</m:mo>
</m:mrow>
</m:mrow>
<m:mo>)</m:mo>
</m:mrow>
</m:mrow>
<m:mo>)</m:mo>
</m:mrow>
<m:mi>ϕ</m:mi>
<m:mrow>
<m:mo>(</m:mo>
<m:mrow>
<m:mi>x</m:mi>
<m:mo>,</m:mo>
<m:mrow>
<m:mo>|</m:mo>
<m:mi>ξ</m:mi>
<m:mo>|</m:mo>
</m:mrow>
</m:mrow>
<m:mo>)</m:mo>
</m:mrow>
</m:mrow>
</m:math>
<jats: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)</jats:tex-math>
</jats:alternatives>
</jats:inline-formula> and <jats:italic>h</jats:italic> : ℝ+→]0, 1] is a continuous decreasing function with unbounded primitive. The second term <jats:italic>f</jats:italic> belongs to <jats:italic>L<jats:sup>N</jats:sup>
</jats:italic>(Ω) or to <jats:italic>L<jats:sup>m</jats:sup>
</jats:italic>(Ω), with <jats:inline-formula>
<jats:alternatives>
<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_mjpaa-2020-0002_eq_003.png"/>
<m:math xmlns:m="http://www.w3.org/1998/Math/MathML">
<m:mrow>
<m:mi>m</m:mi>
<m:mo>=</m:mo>
<m:mfrac>
<m:mrow>
<m:mi>r</m:mi>
<m:mi>N</m:mi>
</m:mrow>
<m:mrow>
<m:mi>r</m:mi>
<m:mo>+</m:mo>
<m:mn>1</m:mn>
</m:mrow>
</m:mfrac>
</m:mrow>
</m:math>
<jats:tex-math>m = {{rN} \over {r + 1}}</jats:tex-math>
</jats:alternatives>
</jats:inline-formula> for some <jats:italic>r</jats:italic> &gt; 0 and <jats:italic>φ</jats:italic> is a Musielak function satisfying the Δ<jats:sub>2</jats:sub>-condition.</jats:p>