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Nonlinear Elliptic Systems with Coupled Gradient Terms
Acta Applicandae Mathematicae ( IF 1.6 ) Pub Date : 2020-04-14 , DOI: 10.1007/s10440-020-00329-7
Ahmed Attar , Rachid Bentifour , El-Haj Laamri

In this paper, we analyze the existence and non-existence of nonnegative solutions to a class of nonlinear elliptic systems of type:

$$ \left \{ \textstyle\begin{array}{r@{\quad }c@{\quad }l@{\quad }l@{\quad }l} -\Delta u & = & |\nabla v|^{q}+\lambda f & \text{in } &\Omega , \\ -\Delta v& = &|\nabla u|^{p}+\mu g &\text{in } &\Omega , \\ u=v&=& 0 & \text{on } &\partial \Omega , \\ u,v& \geq & 0 & \text{in } &\Omega , \end{array}\displaystyle \right . $$

where \(\Omega \) is a bounded domain of \(\mathbb{R}^{N}\) and \(p, q\ge 1\). \(f,g\) are nonnegative measurable functions with additional hypotheses and \(\lambda , \mu \ge 0\).

This extends previous similar results obtained in the case where the right-hand sides are potential and gradient terms, see (Abdellaoui et al. in Appl. Anal. 98(7):1289–1306, [2019], Attar and Bentifour in Electron. J. Differ. Equ. 2017:1, [2017]).



中文翻译:

耦合梯度项的非线性椭圆系统

在本文中,我们分析了一类非线性椭圆系统的非负解的存在性和不存在性:

$$ \ left \ {\ textstyle \ begin {array} {r @ {\ quad} c @ {\ quad} l @ {\ quad} l @ {\ quad} l}-\ Delta u&=&| \ nabla v | ^ {q} + \ lambda f&\ text {in}&\ Omega,\\-\ Delta v&=&| \ nabla u | ^ {p} + \ mu g&\ text {in}&\ Omega ,\\ u = v&=&0&\ text {on}&\ partial \ Omega,\\ u,v&\ geq&0&\ text {in}&\ Omega,\ end {array} \ displaystyle \ right。$$

其中\(\ Omega \)\(\ mathbb {R} ^ {N} \)\(p,q \ ge 1 \)的有界域。\(f,g \)是具有附加假设和\(\ lambda,\ mu \ ge 0 \)的非负可测量函数。

这扩展了以前在右侧是电势和梯度项的情况下获得的类似结果,请参见(Abdellaoui等人,Appl。Anal。98(7):1289-1306,[2019],Etarn中的Attar和Bentifour。 J. Differ。Equ。2017:1,[2017])。

更新日期:2020-04-18
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