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

Abstract

This paper is devoted to study the existence and multiplicity of nontrivial solutions for the following boundary value problem

$ \left\{\begin{array}{ll}-\operatorname{div}\left(\omega(x)|\nabla u(x)|^{N-2} \nabla u(x)\right)=f(x, u)+\epsilon h(x), & \text { in } B \\u=0 & \text { on } \partial B\end{array}\right.$

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 \times \mathbb{R}$ and has double exponential crit-ical growth, which behaves like exp$ \left\{e^{\alpha|u|^{\frac{N}{N-1}}}\right\}$as |u| → ∞ for some α >0. Moreover, ϵ> 0, and the radial function h belongs to the dual space of $ W_{0, \text { rad }}^{1, N}(B), h \neq 0$.

Keywords

N-Laplacian / Trudinger-Moser type inequality / double exponential critical growth / variational methods