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Existence of solutions for some nonlinear elliptic problems involving Minty’s lemma
Ricerche di Matematica ( IF 1.1 ) Pub Date : 2018-09-21 , DOI: 10.1007/s11587-018-0423-7
Mohammed Al-Hawmi , Abdelmoujib Benkirane , Hassane Hjiaj , Abdelfattah Touzani

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:

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,\( f \ in L ^ {1}(\ Omega)\)\(\ 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。
更新日期:2018-09-21
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