Implicit Equations Involving the p-Laplace Operator

Abstract

AbstractIn this work we study the existence of solutions $$u \in W^{1,p}_0(\Omega )$$ u ∈ W 0 1 , p ( Ω ) to the implicit elliptic problem $$ f(x, u, \nabla u, \Delta _p u)= 0$$ f ( x , u , ∇ u , Δ p u ) = 0 in $$ \Omega $$ Ω , where $$ \Omega $$ Ω is a bounded domain in $$ {\mathbb {R}}^N $$ R N , $$ N \ge 2 $$ N ≥ 2 , with smooth boundary $$ \partial \Omega $$ ∂ Ω , $$ 1< p< \infty $$ 1 < p < ∞ , and $$ f:\Omega \times {\mathbb {R}}\times {\mathbb {R}}^N \times {\mathbb {R}}\rightarrow {\mathbb {R}}$$ f : Ω × R × R N × R → R . We choose the particular case when the function f can be expressed in the form $$ f(x, z, w, y)= \varphi (x, z, w)- \psi (y) $$ f ( x , z , w , y ) = φ ( x , z , w ) - ψ ( y ) , where the function $$ \psi $$ ψ depends only on the p-Laplacian $$ \Delta _p u $$ Δ p u . We also present some applications of our results.