On a Nonhomogeneous $N$-Laplacian Problem with Double Exponential Critical Growth

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 $W^{1,N}_{0,rad}(B)$ $h\ne 0.$