In this paper, we consider the following nonlinear \(L_{\varphi }(\Omega )\)-elliplic problem:$$\begin{aligned} -\,\text{ div } a(x,u,\nabla u) =f -\text{ div } \phi (u)\quad \text{ in } \ \Omega , \end{aligned}$$in Musielak–Orlicz–Sobolev spaces, when only large monotonicity is satisfied, with \(f\in L^{1}(\Omega )\) and \(\phi (\cdot )\in C^{0}({\mathbb {R}},{\mathbb {R}}^{N}).\) By assuming that \(\varphi (x,t)\) is a generalized Orlicz function, such that the conjugate function \(\psi (x,t)\) satisfies the \(\Delta _2\)-condition. We establish the existence of T-\(L_{\varphi }(\Omega )\)-solutions for our nonlinear problem via Minty’s lemma.

中文翻译：

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关于Minty引理的某些非线性椭圆问题解的存在性。

在本文中，我们考虑以下非线性\（L _ {\ varphi}（\ Omega）\）椭圆问题：$$ \ begin {aligned}-\，\ text {div} a（x，u，\ nabla u ）= f-\ text {div} \ phi（u）\ quad \ text {in} \ \ Omega，\ end {aligned} $$在Musielak–Orlicz–Sobolev空间中，当仅满足大单调性时，\（ f \ in L ^ {1}（\ Omega）\）和\（\ phi（\ cdot）\ in C ^ {0}（{\ mathbb {R}}，{\ mathbb {R}} ^ {N} ）。）通过假设\（\ varphi（x，t）\）是广义Orlicz函数，使得共轭函数\（\ psi（x，t）\）满足\（\ Delta _2 \） -健康）状况。我们确定T- \（L _ {\ varphi}（\ Omega）\）的存在-通过Minty引理解非线性问题。