On a Nonhomogeneous $N$-Laplacian Problem with Double Exponential Critical Growth (Journal of Partial Differential Equations, 2025)
This paper is devoted to studying the existence and multiplicity of nontrivialsolutions for the following boundary value problem $$\begin{cases} -{\rm div}(\omega(x)| abla u(x)|^{N-2} abla u(x))=f(x,u)+\epsilon h(x), & {\rm in} \ B; \\ u=0, & {\rm on} \ \partial B, \end{cases}$$where $B$ is the unit ball in $\mathbb{R}^N,$ the radial positive weight $ω(x)$ is of logarithmic typefunction, 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 $W^{1,N}_{0,rad}(B)$ $h e 0.$
Keywords: $N$-Laplacian; Trudinger-Moser type inequality; double exponential critical growth; variational methods.