In this paper, we establish maximal domains on the real parameters \(\lambda ,\mu >0\) to existence of \(C^{1}({\mathbb {R}}^{N})\)-entire positive solutions for the quasilinear elliptic system$$\begin{aligned} \left\{ \begin{array}{l} -\Delta _p u = \eta a(x)f_1(u) + \lambda b(x)g_1(u)h_1(v)~in ~ {\mathbb {R}}^N,\\ -\Delta _p v = \theta c(x)f_2(v) + \mu d(x)g_2(v)h_2(u)~in~ {\mathbb {R}}^N,\\ u, v > 0~in~ {\mathbb {R}}^N,~~ u(x), v(x) {\mathop {\longrightarrow }\limits ^{|x|\rightarrow \infty }} 0, \end{array} \right. \end{aligned}$$where \(\Delta _p\) is the \(p-\)Laplacian operator with \(1< p< N \) (\(3\le N\)); \(0<a, b, c, d\in C({\mathbb {R}}^{N})\); either \(\eta =\theta =1\) or \(\eta =\lambda ,\theta =\mu \) and \(f_i, g_i, h_i~(i=1,2)\) are positive continuous functions that satisfy some technical conditions, which allow \(f_{i}\) to behave in a singular way at 0 and \(g_ih_i\) as a \((p-1)\)-superlinear term at 0 and infinity. The main difficulties in approaching our problem come from its non-variational structure, building ordered sub-supersolutions and from the lack of a well-defined spectral theory. Using appropriate truncation, a generalization of the first eigenvalue in \({\mathbb {R}}^{N}\) and a priori estimates, we are able to prove our principal results.

中文翻译：

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奇异拟线性椭圆系统整体正解存在的$$（\ lambda，\ mu）$$（λ，μ）参数的最大域

在本文中，我们在实参数\（\ lambda，\ mu> 0 \）上建立最大域，以存在\（C ^ {1}（{\ mathbb {R}} ^ {N}）\）- entire拟线性椭圆系统的正解$$ \ begin {aligned} \ left \ {\ begin {array} {l}-\ Delta _p u = \ eta a（x）f_1（u）+ \ lambda b（x）g_1 （u）h_1（v）〜in〜{\ mathbb {R}} ^ N，\\-\ Delta _p v = \ theta c（x）f_2（v）+ \ mu d（x）g_2（v）h_2 （u）〜in〜{\ mathbb {R}} ^ N，\\ u，v> 0〜in〜{\ mathbb {R}} ^ N，~~ u（x），v（x）{\ mathop {\ longrightarrow} \ limits ^ {| x | \ rightarrow \ infty}} 0，\ end {array} \ right。\ end {aligned} $$其中\（\ Delta _p \）是\（p- \）具有\（1 <p <N \）（\（3 \ le N \））的Laplacian运算符；\（0 <a，b，c，d \ in C（{\ mathbb {R}} ^ {N}）\） ; ） ; 或者\（\ ETA = \ THETA = 1 \）或\（\ ETA = \拉姆达，\ THETA = \亩\）和\（f_i，的G_i，h_i〜（I = 1,2）\）是正的连续函数满足一些技术条件，这些条件允许\（f_ {i} \）在0和\（g_ih_i \）处以奇异方式表现为\（（p-1）\） -在0和无穷大处的超线性项。解决我们问题的主要困难来自其无变异结构，建立有序的子超解以及缺乏明确定义的光谱理论。使用适当的截断，可以概括\（{\ mathbb {R}} ^ {N} \）中的第一个特征值 和先验估计，我们能够证明我们的主要结果。