<abstract xmlns="http://www.w3.org/1999/xhtml">

<p>We investigate the existence of non-trivial weak solutions for the following <italic>p</italic>(<italic>x</italic>)-Kirchhoff bi-nonlocal elliptic problem driven by both <italic>p</italic>(<italic>x</italic>)-Laplacian and <italic>p</italic>(<italic>x</italic>)-Biharmonic operators
<disp-formula>
<alternatives>
<graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_mjpaa-2023-0028_eq_001.png"></graphic>
<math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block"><mrow><mrow><mo>{</mo> <mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><mi>M</mi><mrow><mo>(</mo><mi>σ</mi><mo>)</mo></mrow><mrow><mo>(</mo><mrow><msubsup><mrow><mi mathvariant="normal">Δ</mi></mrow><mrow><mi>p</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mrow><mn>2</mn></msubsup><mi>u</mi><mo>-</mo><msub><mrow><mi mathvariant="normal">Δ</mi></mrow><mrow><mi>p</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mrow></msub><mi>u</mi></mrow><mo>)</mo></mrow><mo>=</mo><mi>λ</mi><mi>ϑ</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><msup><mrow><mrow><mrow><mo>|</mo> <mi>u</mi> <mo>|</mo></mrow></mrow></mrow><mrow><mi>q</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>-</mo><mn>2</mn></mrow></msup><mi>u</mi><msup><mrow><mrow><mrow><mo>(</mo><mrow><mrow><msub><mo>∫</mo><mi mathvariant="normal">Ω</mi></msub><mrow><mfrac><mrow><mi>ϑ</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mrow><mrow><mi>q</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mrow></mfrac><msup><mrow><mrow><mrow><mo>|</mo> <mi>u</mi> <mo>|</mo></mrow></mrow></mrow><mrow><mi>q</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mrow></msup><mi>d</mi><mi>x</mi></mrow></mrow></mrow><mo>)</mo></mrow></mrow></mrow><mi>r</mi></msup><mi> </mi><mtext>in</mtext><mi> </mi><mi mathvariant="normal">Ω</mi><mo>,</mo></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow><mi>u</mi><mo>∈</mo><msup><mrow><mi>W</mi></mrow><mrow><mn>2</mn><mo>,</mo><mi>p</mi><mrow><mo>(</mo><mo>.</mo><mo>)</mo></mrow></mrow></msup><mrow><mo>(</mo><mi mathvariant="normal">Ω</mi><mo>)</mo></mrow><mo>∩</mo><msubsup><mrow><mi>W</mi></mrow><mn>0</mn><mrow><mn>1</mn><mo>,</mo><mi>p</mi><mrow><mo>(</mo><mo>.</mo><mo>)</mo></mrow></mrow></msubsup><mrow><mo>(</mo><mi mathvariant="normal">Ω</mi><mo>)</mo></mrow><mo>,</mo></mrow></mtd></mtr></mtable></mrow> </mrow></mrow></math>
<tex-math>\left\{ {\matrix{   {M\left( \sigma  \right)\left( {\Delta _{p\left( x \right)}^2u - {\Delta _{p\left( x \right)}}u} \right) = \lambda \vartheta \left( x \right){{\left| u \right|}^{q\left( x \right) - 2}}u{{\left( {\int_\Omega  {{{\vartheta \left( x \right)} \over {q\left( x \right)}}{{\left| u \right|}^{q\left( x \right)}}dx} } \right)}^r}\,{\rm{in}}\,\Omega ,} \hfill  \cr    {u \in {W^{2,p\left( . \right)}}\left( \Omega  \right) \cap W_0^{1,p\left( . \right)}\left( \Omega  \right),} \hfill  \cr  } } \right.</tex-math>
</alternatives>
</disp-formula>
under some suitable conditions on the continuous functions <italic>p</italic>, <italic>q</italic>, the non-negative function <italic>ϑ</italic> and <italic>M</italic>(<italic>σ</italic>), where
<disp-formula>
<alternatives>
<graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_mjpaa-2023-0028_eq_002.png"></graphic>
<math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block"><mrow><mi>σ</mi><mo>:</mo><mo>=</mo><mrow><msub><mo>∫</mo><mi mathvariant="normal">Ω</mi></msub><mrow><mfrac><mrow><msup><mrow><mrow><mrow><mo>|</mo> <mrow><mi mathvariant="normal">Δ</mi><mi>u</mi></mrow> <mo>|</mo></mrow></mrow></mrow><mrow><mi>p</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mrow></msup></mrow><mrow><mi>p</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mrow></mfrac><mo>+</mo><mfrac><mrow><msup><mrow><mrow><mrow><mo>|</mo> <mrow><mo>∇</mo><mi>u</mi></mrow> <mo>|</mo></mrow></mrow></mrow><mrow><mi>p</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mrow></msup></mrow><mrow><mi>p</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mrow></mfrac><mi>d</mi><mi>x</mi><mo>.</mo></mrow></mrow></mrow></math>
<tex-math>\sigma : = \int_\Omega  {{{{{\left| {\Delta u} \right|}^{p\left( x \right)}}} \over {p\left( x \right)}} + {{{{\left| {\nabla u} \right|}^{p\left( x \right)}}} \over {p\left( x \right)}}dx.}</tex-math>
</alternatives>
</disp-formula>
Our main results is obtained by employing variational techniques and the well-known symmetric mountain pass lemma.</p>
</abstract>

We investigate the existence of non-trivial weak solutions for the following p(x)-Kirchhoff bi-nonlocal elliptic problem driven by both p(x)-Laplacian and p(x)-Biharmonic operators



{ M(σ)(Δp(x)2u-Δp(x)u)=λϑ(x)| u |q(x)-2u(∫Ωϑ(x)q(x)| u |q(x)dx)r in Ω,u∈W2,p(.)(Ω)∩W01,p(.)(Ω), 
\left\{ {\matrix{   {M\left( \sigma  \right)\left( {\Delta _{p\left( x \right)}^2u - {\Delta _{p\left( x \right)}}u} \right) = \lambda \vartheta \left( x \right){{\left| u \right|}^{q\left( x \right) - 2}}u{{\left( {\int_\Omega  {{{\vartheta \left( x \right)} \over {q\left( x \right)}}{{\left| u \right|}^{q\left( x \right)}}dx} } \right)}^r}\,{\rm{in}}\,\Omega ,} \hfill  \cr    {u \in {W^{2,p\left( . \right)}}\left( \Omega  \right) \cap W_0^{1,p\left( . \right)}\left( \Omega  \right),} \hfill  \cr  } } \right.


under some suitable conditions on the continuous functions p, q, the non-negative function ϑ and M(σ), where



σ:=∫Ω| Δu |p(x)p(x)+| ∇u |p(x)p(x)dx.
\sigma : = \int_\Omega  {{{{{\left| {\Delta u} \right|}^{p\left( x \right)}}} \over {p\left( x \right)}} + {{{{\left| {\nabla u} \right|}^{p\left( x \right)}}} \over {p\left( x \right)}}dx.}


Our main results is obtained by employing variational techniques and the well-known symmetric mountain pass lemma.

<italic>p</italic>(<italic>x</italic>)-Kirchhoff bi-nonlocal elliptic problem driven by both <italic>p</italic>(<italic>x</italic>)-Laplacian and <italic>p</italic>(<italic>x</italic>)-Biharmonic operators

AMNEA Group, Laboratory MAIS, Department of Mathematics, FSTE, BP 509, Boutalamine Errachidia

Moroccan Journal of Pure and Applied Analysis

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

()-Kirchhoff bi-nonlocal elliptic problem driven by both ()-Laplacian and ()-Biharmonic operators

We investigate the existence of non-trivial weak solutions for the following p(x)-Kirchhoff bi-nonlocal elliptic problem driven by both p(x)-Laplacian and...

p(x)-Kirchhoff bi-nonlocal elliptic problem driven by both p(x)-Laplacian and p(x)-Biharmonic operators

Publié en ligne: 29 sept. 2023

Pages: 407 - 419

Reçu: 06 févr. 2023

Accepté: 29 sept. 2023

DOI: https://doi.org/10.2478/mjpaa-2023-0028

Mots clés
Variable exponents, Variational methods, ()-Kirchhoff, Symmetric mountain pass lemma

© 2023 Mohsine Jennane et al., published by Sciendo

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

p(x)-Kirchhoff bi-nonlocal elliptic problem driven by both p(x)-Laplacian and p(x)-Biharmonic operators

Publié en ligne: 29 sept. 2023

Pages: 407 - 419

Reçu: 06 févr. 2023

Accepté: 29 sept. 2023

DOI: https://doi.org/10.2478/mjpaa-2023-0028

Mots clésVariable exponents, Variational methods, ()-Kirchhoff, Symmetric mountain pass lemma

© 2023 Mohsine Jennane et al., published by Sciendo

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

Mots clés
Variable exponents, Variational methods, ()-Kirchhoff, Symmetric mountain pass lemma