Modified quasilinear equations with strongly singular and critical exponential nonlinearity

Abstract

In this article, we study global multiplicity result for a class of modified quasilinear singular equations involving the critical exponential growth: −Δu−Δ(u2)u=λ(α(x)u−q+f(x,u))inΩ,u>0inΩ,u=0on∂Ω,\left\{\begin{array}{rcl}-\Delta u-\Delta \left({u}^{2})u& =& \lambda (\alpha \left(x){u}^{-q}+f\left(x,u))\hspace{0.33em}\hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}\Omega ,\\ u& \gt & 0\hspace{1em}\hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}\Omega ,\\ u& =& 0\hspace{1em}\hspace{0.1em}\text{on}\hspace{0.1em}\hspace{0.33em}\partial \Omega ,\end{array}\right. where Ω\Omega is a smooth bounded domain in R2{{\mathbb{R}}}^{2}, 00{\Lambda }^{* }\gt 0 such that for all λ∈(0,Λ*)\lambda \in \left(0,{\Lambda }^{* }), the problem has at least two positive solutions, for λ=Λ*\lambda ={\Lambda }^{* }, the problem achieves at least one positive solution for λ>Λ*,\lambda \gt {\Lambda }^{* }, and the problem has no solutions.

Keywords

• modified quasilinear operator
• singular nonlinearity
• sub-super solution
• critical exponential nonlinearity
• trudinger-moser inequality
• 35j20
• 35j60