Singular nonlocal elliptic systems via nonlinear Rayleigh quotient
Edcarlos D Silva, Elaine A. F. Leite, Maxwell L. Silva
TL;DR
This work addresses the existence and multiplicity of positive solutions for a singular nonlocal elliptic system driven by the fractional Laplacian on $\mathbb{R}^N$. The authors develop and combine a nonlinear Rayleigh quotient with the Nehari manifold to minimize the energy on a constrained set, obtaining at least two positive weak solutions for each $\lambda$ in $(0,\lambda^*)$, where $\lambda^*$ is the threshold defined via fibering arguments. They prove $0<\lambda_*<\lambda^*<\infty$, show the existence of a ground state in $\mathcal{N}_{\lambda}^+$ and a second solution in $\mathcal{N}_{\lambda}^-$, and treat the degenerate case at $\lambda^*$ by constructing convergent minimizing sequences and proving positivity of the limits. The results hold without restrictions on the size of the coupling parameter $\theta$, and the paper provides a thorough variational framework with detailed continuity and compactness arguments that extend to the boundary case $\lambda=\lambda^*$.
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: $$\left\{\begin{array}{lll} (-Δ)^su +V_1(x)u = λ\frac{a(x)}{u^p} + \fracα{α+β}θ|u|^{α- 2}u|v|^β, \,\,\, \mbox{in} \,\,\, \mathbb{R}^N,\\ (-Δ)^sv +V_2(x)v= λ\frac{b(x)}{v^q}+ \fracβ{α+β}θ|u|^α|v|^{β-2}v, \,\,\, \mbox{in} \,\,\, \mathbb{R}^N, \end{array}\right. \;\;\;(u, v) \in H^s(\mathbb{R}^N) \times H^s(\mathbb{R}^N),$$ where $ 0<p \leq q < 1<\;α, β\;,\;2<α+ β< 2^*_s$, $θ> 0, λ> 0, N > 2s$, 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 $λ^* > 0$ such that our main problem admits at least two positive solutions for each $λ\in (0, λ^*)$. 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 $θ> 0$. Moreover, we shall consider the multiplicity of solutions for the case $λ= λ^*$ where degenerated points are allowed.