Implicit Equations Involving the p-Laplace Operator

In this work we study the existence of solutions u ∈ W 0 1 , p ( Ω ) to the implicit elliptic problem f ( x , u , ∇ u , Δ p u ) = 0 in Ω , where Ω is a bounded domain in R N , N ≥ 2 , with smooth boundary ∂ Ω , 1 < p < ∞ , and f : Ω × R × R N × R → R . We choose the particular case when the function f can be expressed in the form f ( x , z , w , y ) = φ ( x , z , w ) - ψ ( y ) , where the function ψ depends only on the p -Laplacian Δ p u . We also present some applications of our results.