Nonlinear degenerate Navier problem involving the weighted biharmonic operator with measure data in weighted Sobolev spaces

In this paper, we prove the existence and uniqueness of weak solution for a nonlinear degenerate Navier problem involving the weighted biharmonic operator of the following form: Δ [ ϕ ( z ) a ( z , Δ w ) ] - div [ ϑ 1 ( z ) K ( z , ∇ w ) + ϑ 2 ( z ) L ( z , w , ∇ w ) ] + ϑ 2 ( z ) L 0 ( z , w , ∇ w ) = h 0 - ∑ j = 1 n D j h j , where ϕ , ϑ 1 and ϑ 2 are weight functions, a : D ¯ × R n ⟶ R n , K : D × R n ⟶ R n , L : D × R × R n ⟶ R n , and L 0 : D × R × R n ⟶ R are Carathéodory applications that verified some conditions, and h 0 ∈ L 1 ( D ) and h j ∈ L p ′ ( D , ϑ 1 1 - p ′ ) ( j = 1 , … , n ) .