This paper concerns in building extremal curves with respect to the parameters \(\lambda,\mu \geq 0\) for existence and multiplicity of \(D^{1,2}(\mathbb{R}^N)\)-solutions for the multi-parameter elliptic system$$ \left\{\begin{array}{lll} -{\Delta}{u} = {\lambda}{w}(x)f_{1}(u)g_{1}(v)\quad {\rm in}\quad \mathbb{R}^N,\\ -{\Delta}{v} = {\mu}w(x)f_{2}(v)g_{2}(u)\quad {\rm in} \quad \mathbb{R}^N,\\ u,v > 0 \quad {\rm in}\quad \mathbb{R}^{N} {\rm and} \quad u(x), v(x) \mathop{\longrightarrow}\limits^{{|x|\to\infty}} 0, \end{array}\right. $$where \(f_{i}, g_{i} \in {C}(\mathbb{R}, (0,{\infty}))(i = 1,2)\) satisfy some technical conditions, w is a vanishing positive potential at infinity, and \(N \geq 3\). The principal difficulties in approaching our problem come from the fact that it may not have the variational structure and the construction of an associated compact operator. By tanking advantage of the spectral theory of a related problem treated in [2] and introducing appropriated functions spaces, we are able to prove our principal results by using topological arguments.

中文翻译：

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$$ \ mathbb {R} ^ N $$ RN中的多参数椭圆系统正解存在的极值曲线

本文关注于针对参数\（\ lambda，\ mu \ geq 0 \）建立极值曲线，以了解\（D ^ {1,2}（\ mathbb {R} ^ N）\）的存在和多重性-多参数椭圆系统的解$$ \ left \ {\ begin {array} {lll}-{\ Delta} {u} = {\ lambda} {w}（x）f_ {1}（u）g_ { 1}（v）\ quad {\ rm in} \ quad \ mathbb {R} ^ N，\\-{\ Delta} {v} = {\ mu} w（x）f_ {2}（v）g_ { 2}（u）\ quad {\ rm in} \ quad \ mathbb {R} ^ N，\\ u，v> 0 \ quad {\ rm in} \ quad \ mathbb {R} ^ {N} {\ rm和} \ quad u（x），v（x）\ mathop {\ longrightarrow} \ limits ^ {{| x | \ to \ infty}} 0，\ end {array} \ right。$$其中\（f_ {i}，g_ {i} \ in {C}（\ mathbb {R}，（0，{\ infty}））（i = 1,2）\） 满足某些技术条件w在无限远处是消失的正势，并且\（N \ geq 3 \）。解决我们的问题的主要困难来自这样一个事实，即它可能没有可变的结构和关联的紧凑算子的构造。通过利用在[2]中处理的相关问题的频谱理论的优势，并引入适当的函数空间，我们能够通过使用拓扑论证来证明我们的主要结果。