Weighted logarithmic Adam’s inequalities defined on the whole Euclidean space \(\mathbb {R}^{4}\) and its applications to weighted biharmonic equations of Kirchhoff type

Abstract

In this article, we establish a logarithmic weighted Adams’ inequality in some weighted Sobolev space in the whole of \(\mathbb {R}^{4}\). As an application, we study a weighted fourth-order equation of Kirchhoff type, in \(\mathbb {R}^{4}\). The nonlinearity is assumed to have a critical or subcritical exponential growth according to the Adams-type inequalities already established. It is proved that there is a ground-state solution to this problem by Nehari method and the mountain pass theorem. The major difficulty is the lack of compactness of the energy due to the critical exponential growth of the nonlinear term f.