Ground state solutions for a Kirchhoff type equation involving p-biharmonic operator with exponential growth non-linearity

In this article, we study the following non local weighted problem $$g(∫B(w(x)|Δ u|N/2)dx)Δ(w(x)|Δ u|N/2-2 Δ u) =|u|q-2u +\ f(x,u) in B, u=∂ u/∂ n=0 on ∂ B,$$ where $B$ is the unit ball in $RN$ and $ w(x)$ is a singular weight of logarithm type. The non-linearity is a combination of a reaction source $f(x,u)$ which is critical in view of exponential inequality of Adams' type and a polynomial function. The Kirchhoff function $g$ is positive and continuous. The energy function loses compactness in the critical case. To remedy this, we introduce a new asymptotic condition for non-linearity and go through an intermediate problem. Using the Nehari manifold method, the quantitative deformation lemma and results from degree theory, we establish the existence of a ground-state solution.