On sharpened singular Adams' type inequalities and applications to $(p,\frac{n}{2})$-biharmonic equations

Abstract: The purpose of this article is to establish a sharp version of Adams' type inequality in a suitable higher order function space with singular weight in $\mathbb{R}^n$. In addition, we also provide the proof of a sharp singular concentration-compactness principle due to Lions' as an improvement of this singular Adams' inequality. It is well known that such types of inequalities can be classified in two different ways, for instance, critical sharp singular Adams' type inequality (involving Sobolev full norm as constraint) and subcritical sharp singular Adams' type inequality (involving Sobolev partial norm as constraint). Further, we shall establish that critical and subcritical sharp singular Adams' type inequalities are surprisingly equivalent. To be more precisely, we also discuss about the asymptotic behavior of the lower and upper bounds of subcritical sharp singular Adams' type inequality and establish a relation between the suprema of such types of critical and subcritical sharp inequalities. Despite this, we shall demonstrate a new compact embedding, which plays a crucial role in our arguments. Moreover, as an application of these results, by employing the mountain pass theorem, we study the existence of nontrivial solutions to a class of nonhomogeneous quasilinear elliptic equations involving $(p,\frac{n}{2})$-biharmonic operator with singular exponential growth as follows $$ \Delta^2_p u+\Delta^2_{\frac{n}{2}} u=\frac{g(x,u)}{|x|^\gamma}\quad\text{in}\quad \mathbb{R}^n, $$ with $n\geq 4$, $1<p<\frac{n}{2}$, $\gamma\in(0,n)$ and the nonlinear term $g:\mathbb{R}^n\times \mathbb{R}\to \mathbb{R}$ is a Carath\'eodory function, which behaves like $\exp{(\alpha|s|^{\frac{n}{n-2}})}$ as $|s|\to~+\infty$ for some $\alpha\>0$.