Ground state solutions for weighted biharmonic Kirchhoff problem via the Nehari manifold approach

Abstract

In this article, we delve into the investigation of the following non-local problem within the unit ball B 1 of R 4 : g ( ∫ B 1 w β ( x ) | Δu | 2 ) Δ w β 2 u = | u | q − 2 u + f ( x , u ) in B 1 , u = ∂ n u = 0 on ∂ B 1 , where Δ w β 2 . = Δ ( w β ( x ) Δ . ) is the weighted Biharmonic operator, where w β ( x ) represents a logarithmically singular weight, and the non-linearity involves a combination of a reaction source f ( x , u ) , critical in the context of Adams' type exponential inequality and a polynomial function. The Kirchhoff function g is characterized by being positive and continuous. Employing the Nehari manifold method, the quantitative deformation lemma and degree theory results, we establish the existence of a ground state solution.