Title: Singular nonlocal elliptic systems via nonlinear Rayleigh quotient Show Chinese title

Title: 奇异非局部椭圆系统通过非线性瑞利商

Abstract: In the present work, we establish the existence of two positive solutions for singular nonlocal elliptic systems. More precisely, we consider the following nonlocal elliptic problem: $$\left\{\begin{array}{lll} (-\Delta)^su +V_1(x)u = \lambda\frac{a(x)}{u^p} + \frac{\alpha}{\alpha+\beta}\theta |u|^{\alpha - 2}u|v|^{\beta}, \,\,\, \mbox{in} \,\,\, \mathbb{R}^N,\\ (-\Delta)^sv +V_2(x)v= \lambda \frac{b(x)}{v^q}+ \frac{\beta}{\alpha+\beta}\theta |u|^{\alpha}|v|^{\beta-2}v, \,\,\, \mbox{in} \,\,\, \mathbb{R}^N, \end{array}\right. \;\;\;(u, v) \in H^s(\mathbb{R}^N) \times H^s(\mathbb{R}^N),$$ where $ 0<p \leq q < 1<\;\alpha, \beta \;,\;2<\alpha + \beta < 2^*_s$, $\theta > 0, \lambda > 0, N > 2s$, and $s \in (0,1)$. The potentials $V_1, V_2: \mathbb{R}^N \to \mathbb{R}$ are continuous functions which are bounded from below. Under our assumptions, we prove that there exists the largest positive number $\lambda^* > 0$ such that our main problem admits at least two positive solutions for each $\lambda \in (0, \lambda^*)$. Here we apply the nonlinear Rayleigh quotient together with the Nehari method. The main feature is to minimize the energy functional in Nehari set which allows us to prove our results without any restriction on the size of parameter $\theta > 0$. Moreover, we shall consider the multiplicity of solutions for the case $\lambda = \lambda^*$ where degenerated points are allowed. Show Chinese abstract

Abstract: 在本文中，我们建立了奇异非局部椭圆系统存在两个正解的结论。更准确地说，我们考虑以下非局部椭圆问题：$$\left\{ \begin{array}{lll} (-\Delta)^su +V_1(x)u = \lambda\frac{a(x)}{u^p} + \frac{\alpha}{\alpha+\beta}\theta |u|^{\alpha - 2}u|v|^{\beta}, \,\,\, \mbox{in} \,\,\, \mathbb{R}^N,\\ (-\Delta)^sv +V_2(x)v= \lambda \frac{b(x)}{v^q}+ \frac{\beta}{\alpha+\beta}\theta |u|^{\alpha}|v|^{\beta-2}v, \,\,\, \mbox{in} \,\,\, \mathbb{R}^N, \end{array}\right. \;\;\;(u, v) \in H^s(\mathbb{R}^N) \times H^s(\mathbb{R}^N),$$其中$ 0<p \leq q < 1<\;\alpha, \beta \;,\;2<\alpha + \beta < 2^*_s$，$\theta > 0, \lambda > 0, N > 2s$，和$s \in (0,1)$。势函数$V_1, V_2: \mathbb{R}^N \to \mathbb{R}$是从下方有界的连续函数。在我们的假设下，我们证明存在最大的正数$\lambda^* > 0$，使得对于每个$\lambda \in (0, \lambda^*)$，我们的主要问题至少有两个正解。在这里我们应用非线性Rayleigh商结合Nehari方法。 主要特点是通过在Nehari集上最小化能量泛函，这使我们能够在不施加参数$\theta > 0$大小限制的情况下证明我们的结果。 此外，我们将考虑当允许退化点的情况$\lambda = \lambda^*$时解的多重性。