Extremal Regions and Multiplicity of Positive Solutions for Singular Superlinear Elliptic Systems with Indefinite-Sign Potential

Abstract

In the present paper we deal with the existence, nonexistence and multiplicity of positive solutions for the singular superlinear and subcritical multi-parameter elliptic system

$$\begin{aligned} {\left\{ \begin{array}{ll} -\Delta u = \lambda a(x)u^{-\gamma }+\frac{\alpha }{\alpha +\beta } b(x)u^{\alpha -1}v^{\beta }~in ~ \Omega ,\\ -\Delta v = \mu c(x)v^{-\gamma }+\frac{\beta }{\alpha +\beta }b(x)u^{\alpha }v^{\beta -1}~in~ \Omega ,\\ u=v=0~\text{ on }~\partial \Omega , \end{array}\right. } \end{aligned}$$

where $\Omega \subset {\mathbb {R}}^{N}$ $(N\ge 3)$ is a bounded domain with smooth boundary $\partial \Omega$, $0<a,c$ in $\Omega$, $b\in L^{\infty }(\Omega )$ may change its sign and satisfy some technical conditions, which will be mentioned later on; $\lambda ,\mu \ge 0$ are real parameters. Our main objective is to establish the existence of two extremal regions, one of which is optimal for the applicability of the Nehari manifold method for non-differentiable functionals, and the other one is optimal for the existence of positive solutions when b is positive. By using the idea of extremal values for the applicability of the Nehari manifold method, we also obtain multiplicity of positive solutions beyond this extremal region. To show the existence of the second region we prove a super-solution theorem. The results obtained are new and improve the existing results in the literature.