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Volume 38, Issue 1
On a Nonhomogeneous $N$-Laplacian Problem with Double Exponential Critical Growth

Wenjing Chen & Zexi Wang

DOI: 10.4208/jpde.v38.n1.5

J. Part. Diff. Eq., 38 (2025), pp. 80-99.

Published online: 2025-04

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  • Abstract

This paper is devoted to studying the existence and multiplicity of nontrivial solutions for the following boundary value problem $$\begin{cases} -{\rm div}(\omega(x)|\nabla u(x)|^{N-2}\nabla u(x))=f(x,u)+\epsilon h(x), & {\rm in} \ B; \\ u=0, & {\rm on} \ \partial B, \end{cases}$$where $B$ is the unit ball in $\mathbb{R}^N,$ the radial positive weight $ω(x)$ is of logarithmic type function, the functional $f(x,u)$ is continuous in $B×\mathbb{R}$ and has double exponential critical growth, which behaves like ${\rm exp}\{e^{\alpha|u|^{\frac{N}{N-1}}} \}$ as $|u| → ∞$ for some $α > 0.$ Moreover, $ϵ>0,$ and the radial function $h$ belongs to the dual space of $W^{1,N}_{0,rad}(B)$ $h\ne 0.$

  • Keywords

$N$-Laplacian, Trudinger-Moser type inequality, double exponential critical growth, variational methods.

  • AMS Subject Headings

35J20, 35J62, 35J91

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{JPDE-38-80, author = {Chen , Wenjing and Wang , Zexi}, title = {On a Nonhomogeneous $N$-Laplacian Problem with Double Exponential Critical Growth}, journal = {Journal of Partial Differential Equations}, year = {2025}, volume = {38}, number = {1}, pages = {80--99}, abstract = {

This paper is devoted to studying the existence and multiplicity of nontrivial solutions for the following boundary value problem $$\begin{cases} -{\rm div}(\omega(x)|\nabla u(x)|^{N-2}\nabla u(x))=f(x,u)+\epsilon h(x), & {\rm in} \ B; \\ u=0, & {\rm on} \ \partial B, \end{cases}$$where $B$ is the unit ball in $\mathbb{R}^N,$ the radial positive weight $ω(x)$ is of logarithmic type function, the functional $f(x,u)$ is continuous in $B×\mathbb{R}$ and has double exponential critical growth, which behaves like ${\rm exp}\{e^{\alpha|u|^{\frac{N}{N-1}}} \}$ as $|u| → ∞$ for some $α > 0.$ Moreover, $ϵ>0,$ and the radial function $h$ belongs to the dual space of $W^{1,N}_{0,rad}(B)$ $h\ne 0.$

}, issn = {2079-732X}, doi = {https://doi.org/10.4208/jpde.v38.n1.5}, url = {http://global-sci.org/intro/article_detail/jpde/23953.html} }
TY - JOUR T1 - On a Nonhomogeneous $N$-Laplacian Problem with Double Exponential Critical Growth AU - Chen , Wenjing AU - Wang , Zexi JO - Journal of Partial Differential Equations VL - 1 SP - 80 EP - 99 PY - 2025 DA - 2025/04 SN - 38 DO - http://doi.org/10.4208/jpde.v38.n1.5 UR - https://global-sci.org/intro/article_detail/jpde/23953.html KW - $N$-Laplacian, Trudinger-Moser type inequality, double exponential critical growth, variational methods. AB -

This paper is devoted to studying the existence and multiplicity of nontrivial solutions for the following boundary value problem $$\begin{cases} -{\rm div}(\omega(x)|\nabla u(x)|^{N-2}\nabla u(x))=f(x,u)+\epsilon h(x), & {\rm in} \ B; \\ u=0, & {\rm on} \ \partial B, \end{cases}$$where $B$ is the unit ball in $\mathbb{R}^N,$ the radial positive weight $ω(x)$ is of logarithmic type function, the functional $f(x,u)$ is continuous in $B×\mathbb{R}$ and has double exponential critical growth, which behaves like ${\rm exp}\{e^{\alpha|u|^{\frac{N}{N-1}}} \}$ as $|u| → ∞$ for some $α > 0.$ Moreover, $ϵ>0,$ and the radial function $h$ belongs to the dual space of $W^{1,N}_{0,rad}(B)$ $h\ne 0.$

Chen , Wenjing and Wang , Zexi. (2025). On a Nonhomogeneous $N$-Laplacian Problem with Double Exponential Critical Growth. Journal of Partial Differential Equations. 38 (1). 80-99. doi:10.4208/jpde.v38.n1.5
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