Existence of ground state solutions for weighted biharmonic problem involving non linear exponential growth

In this article, we study the following problem $$\Delta(w(x)\Delta u) = \ f(x,u) \quad\mbox{ in }\quad B, \quad u=\frac{\partial u}{\partial n}=0 on } \quad\partial B,$$ where $B$ is unit ball of $\mathbb{R}^{4}$ and $ w(x)$ a singular weight logarithm type. The reaction source $f(x,u)$ radial function with respect to $x$ it critical view exponential inequality Adams' existence result proved by using constrained minimization Nehari set coupled quantitative deformation lemma degree theory results.