In this article, we study the following non local problem $$g\big(\int_{ B}w(x) |\Delta u|^{ 2}\big)\Delta(w(x)\Delta u) =|u|^{ q-2}u +\ f(x,u) \quad\mbox{ in }\quad B, \quad u=\frac{ \partial u}{ \partial n}=0 \quad\mbox{ on } \quad\partial B,$$ where $B$ is the unit ball in $\mathbb{ R}^{ 4}$ 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. By using the Nehari manifold method , the quantitative deformation lemma and degree theory results, we establish the existence of a ground state solution.
Categories: math.AP