Extremal Curves for Existence of Positive Solutions for Multi-parameter Elliptic Systems in $\mathbb{R}^N$

Abstract

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.