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

在这项工作中，我们研究隐式椭圆问题\（f（x，u，\ nabla u，\ Delta _p u）的解\（u \ in W ^ {1，p} _0（\ Omega）\ ）的存在性= 0 \）在\（\ Omega \）中，其中\（\ Omega \）是\（{\ mathbb {R}} ^ N \），\（N \ ge 2 \）中的边界域，且边界光滑\（\ partial \ Omega \），\（1 <p <\ infty \）和\（f：\ Omega \ times {\ mathbb {R}} \ times {\ mathbb {R}} ^ N \ times { \ mathbb {R}} \ rightarrow {\ mathbb {R}} \）。当函数f可以表示为\（f（x，z，w，y）= \ varphi（x，z，w）-\ psi（y）\）时，我们选择特殊情况，其中函数\（\ psi \）仅取决于p -Laplacian \（\ Delta _p u \）。我们还介绍了我们的结果的一些应用。