Weighted Second Order Adams Inequality in the Whole Space \(\mathbb {R}^{4}\)

The main result of this paper is to establish a weighted second-order Adams-type inequality on the whole set of \(\mathbb {R}^{4}\) ${\mathbb{R}}^4$ . As an application of this result, we prove the existence of a solution for a Kirchhoff-type equation involving non-linearity with subcritical or critical exponential growth. In the critical case, the associated energy loses its compactness. To avoid this problem, we add an asymptotic condition to the nonlinearity