Ground states of biharmonic equations with logarithmic weights and critical exponential growth

Abstract

In this paper, we investigate a fourth-order weighted equation of the form Δ ( w ( x ) | Δu | N 2 − 2 Δu ) = f ( x , u ) within the unit ball B of R N . Our focus lies on establishing the existence of solutions under the critical exponential growth condition, as per Adams' inequalities. Specifically, we address the logarithmic weighted biharmonic equation under Dirichlet boundary conditions in the unit ball B of R N . Our proof relies on a combination of minimax techniques and the Nehari method to establish the existence of a ground state solution. Notably, our analysis reveals a loss of compactness in the associated energy functional at a certain level. To surmount this non-compactness issue, we introduce an appropriate asymptotic condition that enables us to bypass the non-compact levels of the functional.