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

<p>We consider the existence of solutions of the following weighted problem:
<disp-formula>
<alternatives>
<graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_auom-2021-0033_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>L</mi><mo>:</mo><mo>=</mo><mo>-</mo><mi>d</mi><mi>i</mi><mi>v</mi><mrow><mo>(</mo><mrow><mi>ρ</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><msup><mrow><mrow><mrow><mo>|</mo><mrow><mo>∇</mo><mi>u</mi></mrow><mo>|</mo></mrow></mrow></mrow><mrow><mi>N</mi><mo>-</mo><mn>2</mn></mrow></msup><mo>∇</mo><mi>u</mi></mrow><mo>)</mo></mrow><mo>+</mo><mi>ξ</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>N</mi><mo>-</mo><mn>2</mn></mrow></msup></mrow></mtd><mtd columnalign="left"><mrow><mi>u</mi><mo>=</mo><mi>f</mi><mrow><mo>(</mo><mrow><mi>x</mi><mo>,</mo><mi>u</mi></mrow><mo>)</mo></mrow></mrow></mtd><mtd columnalign="left"><mrow><mi>i</mi><mi>n</mi></mrow></mtd><mtd columnalign="left"><mi>B</mi></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow><mi>u</mi><mo>></mo><mn>0</mn></mrow></mtd><mtd columnalign="left"><mrow><mi>i</mi><mi>n</mi></mrow></mtd><mtd columnalign="left"><mi>B</mi></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow><mi>u</mi><mo>=</mo><mn>0</mn></mrow></mtd><mtd columnalign="left"><mrow><mi>o</mi><mi>n</mi></mrow></mtd><mtd columnalign="left"><mrow><mo>∂</mo><mi>B</mi><mo>,</mo></mrow></mtd></mtr></mtable></mrow></mrow></mrow></math>
<tex-math>\left\{ {\matrix{{L: =  - div\left( {\rho \left( x \right){{\left| {\nabla u} \right|}^{N - 2}}\nabla u} \right) + \xi \left( x \right){{\left| u \right|}^{N - 2}}} \hfill &amp; {u = f\left( {x,u} \right)} \hfill &amp; {in} \hfill &amp; B \hfill  \cr {} \hfill &amp; {u > 0} \hfill &amp; {in} \hfill &amp; B \hfill  \cr {} \hfill &amp; {u = 0} \hfill &amp; {on} \hfill &amp; {\partial B,} \hfill  \cr  } } \right.</tex-math>
</alternatives>
</disp-formula>
where B is the unit ball of ℝ<italic><sup>N</sup></italic>, <italic>N #62;</italic> 2, <inline-formula>
<alternatives>
<inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_auom-2021-0033_eq_002.png"></inline-graphic>
<math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>ρ</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><msup><mrow><mrow><mrow><mo>(</mo><mrow><mo>log</mo><mfrac><mi>e</mi><mrow><mrow><mo>|</mo><mi>x</mi><mo>|</mo></mrow></mrow></mfrac></mrow><mo>)</mo></mrow></mrow></mrow><mrow><mi>N</mi><mo>-</mo><mn>1</mn></mrow></msup></mrow></math>
<tex-math>\rho \left( x \right) = {\left( {\log {e \over {\left| x \right|}}} \right)^{N - 1}}</tex-math>
</alternatives>
</inline-formula> the singular logarithm weight with the limiting exponent <italic>N −</italic> 1 in the Trudinger-Moser embedding, and ξ(<italic>x</italic>) is a positif continuous potential. The nonlinearities are critical or subcritical growth in view of Trudinger-Moser inequalities of double exponential type. We prove the existence of positive solution by using Mountain Pass theorem. In the critical case, the function of Euler Lagrange does not fulfil the requirements of Palais-Smale conditions at all levels. We dodge this problem by using adapted test functions to identify this level of compactness.</p>
</abstract>

We consider the existence of solutions of the following weighted problem:



{L:=-div(ρ(x)|∇u|N-2∇u)+ξ(x)|u|N-2u=f(x,u)inBu>0inBu=0on∂B,
\left\{ {\matrix{{L: =  - div\left( {\rho \left( x \right){{\left| {\nabla u} \right|}^{N - 2}}\nabla u} \right) + \xi \left( x \right){{\left| u \right|}^{N - 2}}} \hfill & {u = f\left( {x,u} \right)} \hfill & {in} \hfill & B \hfill  \cr {} \hfill & {u > 0} \hfill & {in} \hfill & B \hfill  \cr {} \hfill & {u = 0} \hfill & {on} \hfill & {\partial B,} \hfill  \cr  } } \right.


where B is the unit ball of ℝN, N #62; 2, 


ρ(x)=(loge|x|)N-1
\rho \left( x \right) = {\left( {\log {e \over {\left| x \right|}}} \right)^{N - 1}}

 the singular logarithm weight with the limiting exponent N − 1 in the Trudinger-Moser embedding, and ξ(x) is a positif continuous potential. The nonlinearities are critical or subcritical growth in view of Trudinger-Moser inequalities of double exponential type. We prove the existence of positive solution by using Mountain Pass theorem. In the critical case, the function of Euler Lagrange does not fulfil the requirements of Palais-Smale conditions at all levels. We dodge this problem by using adapted test functions to identify this level of compactness.

Non-autonomous weighted elliptic equations with double exponential growth

University of Tunis El Manar, Campus of University

Analele ştiinţifice ale Universităţii "Ovidius" Constanţa. Seria Matematică

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

{"article-title":"Non-autonomous weighted elliptic equations with double exponential growth"}

We consider the existence of solutions of the following weighted problem:



{L:=-div(ρ(x)|∇u|N-2∇u)+ξ(x)|u|N-2u=f(x,u)inBu>0inBu=0on∂B,
\left\{ {\matrix{{L: =...

Non-autonomous weighted elliptic equations with double exponential growth

Published Online: Nov 23, 2021

Page range: 33 - 66

Received: Apr 01, 2021

Accepted: May 15, 2021

DOI: https://doi.org/10.2478/auom-2021-0033

Keywords
Moser-Trudinger inequality, double exponential growth, mountain pass, compactness level

© 2021 Sami Baraket et al., published by Sciendo

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

Non-autonomous weighted elliptic equations with double exponential growth

Published Online: Nov 23, 2021

Page range: 33 - 66

Received: Apr 01, 2021

Accepted: May 15, 2021

DOI: https://doi.org/10.2478/auom-2021-0033

KeywordsMoser-Trudinger inequality, double exponential growth, mountain pass, compactness level

© 2021 Sami Baraket et al., published by Sciendo

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

Keywords
Moser-Trudinger inequality, double exponential growth, mountain pass, compactness level