Polyharmonic Kirchhoff type equations with singular exponential nonlinearities

Abstract

In this article, we study the existence of non-negative solutions of the following polyharmonic Kirchhoff type problem with critical singular exponential nolinearity −M Ω |∇ m u| n m dx ∆ m n m u = f (x,u) |x| α in Ω, u = ∇u = • • • = ∇ m−1 u = 0 on ∂Ω, where Ω ⊂ R n is a bounded domain with smooth boundary, 0 < α < n, n ≥ 2m ≥ 2 and f (x, u) behaves like e |u| n n−m as |u| → ∞. 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 nonnegative solutions.