Polyharmonic Kirchhoff type equations with singular exponential nonlinearities

Pawan Kumar  Mishra; Sarika  Goyal; K.  Sreenadh

doi:10.3934/cpaa.2016009

Article Contents

2016, Volume 15, Issue 5: 1689-1717. Doi: 10.3934/cpaa.2016009

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Polyharmonic Kirchhoff type equations with singular exponential nonlinearities

1.
Indian Institute of Technology Delhi, Hauz Khas, New Delhi, 110016

Received: August 31, 2015

Revised: March 31, 2016

Published: August 2016

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Abstract

In this article, we study the existence of non-negative solutions of the following polyharmonic Kirchhoff type problem with critical singular exponential nolinearity \begin{eqnarray} -M\left(\int_\Omega |\nabla^m u|^{\frac{n}{m}}dx\right)\Delta_{\frac{n}{m}}^{m} u = \frac{f(x,u)}{|x|^\alpha} \; \text{in}\; \Omega{,} \\ \quad u = \nabla u=\cdots= {\nabla}^{m-1} u=0 \quad \text{on} \quad \partial \Omega{,} \end{eqnarray} where $\Omega\subset R^n$ is a bounded domain with smooth boundary, $0 < \alpha < n$, $n\geq 2m\geq 2$ and $f(x,u)$ behaves like $e^{|u|^{\frac{n}{n-m}}}$ as $|u|\to\infty$. Using mountain pass structure and {the} concentration compactness principle, we show the existence of a nontrivial solution.
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.

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Mathematics Subject Classification: Primary: 35J35, 35J60; Secondary: 35J92.

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