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

Abstract

In this article, we study the following non local weighted problem
$$g\big(\int_{B}(w(x)|\Delta u|^{\frac{N}{2}})dx\big)\Delta(w(x)|\Delta u|^{\frac{N}{2}-2} \Delta u) =|u|^{q-2}u +\ f(x,u) \quad\mbox{ in }\quad B, \quad u=\frac{\partial u}{\partial n}=0 \quad\mbox{ on } \quad\partial B,$$
where $B$ is the unit ball in $\mathbb{R}^{N}$ 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.