Polyharmonic Kirchhoff type equations with singular exponential nonlinearities

Abstract

\noi In this article, we study the existence of non-negative solutions of the following polyharmonic Kirchhoff type problem with critical singular exponential nolinearity $$ \quad \left\{ \begin{array}{lr} \quad -M\left(\displaystyle\int_Ω|\nabla^m u|^{\frac{n}{m}}dx\right)Δ_{\frac{n}{m}}^{m} u = \frac{f(x,u)}{|x|^α} \; \text{in}\; \Om{,} \quad \quad u = \nabla u=\cdot\cdot\cdot= {\nabla}^{m-1} u=0 \quad \text{on} \quad \partial \Om{,} \end{array} \right. $$ where $\Om\subset \mb R^n$ is a bounded domain with smooth boundary, $n\ge 2m\ge 2$ and $f(x,u)$ behaves like $e^{|u{|}^{\frac{n}{n-m}}}$ as $|u|\ra\infty$. Using mountain pass structure and {the} concentration compactness principle, we show the existence of a nontrivial solution. %{OR}\\ In the later part of the paper, we also discuss the above problem with convex-concave type sign changing nonlinearity. Using {the} Nehari manifold technique, we show the existence and multiplicity of non-negative solutions. \medskip

Communications in pure and applied analysis (2016)