Multiplicity results of fractional-Laplace system with sign-changing and singular nonlinearity

TL;DR

This paper investigates the existence and multiplicity of positive solutions for a fractional-Laplacian system with sign-changing and singular nonlinearities, employing variational methods to analyze parameter-dependent solutions.

Contribution

It introduces new results on positive solutions for a fractional system with singular and sign-changing nonlinearities, extending previous work to more complex nonlinearities and parameter regimes.

Findings

Existence of positive solutions for certain parameter ranges.

Multiple solutions established under specific conditions.

Solutions depend continuously on parameters (mbda,mu).

Abstract

In this article, we study the following fractional-Laplacian system with singular nonlinearity \begin{equation*} (P_{\lambda,\mu}) \left\{ \begin{array}{lr} (-\Delta)^s u = \lambda f(x) u^{-q}+ \frac{\alpha}{\alpha+\beta}b(x) u^{\alpha-1} w^\beta\; \text{in}\;\Omega \\ (-\Delta)^s w = \mu g(x) w^{-q}+ \frac{\beta}{\alpha+\beta} b(x) u^{\alpha} w^{\beta-1}\; \text{in}\;\Omega \\ \quad \quad u, w>0\;\text{in}\;\Omega, \quad u = w = 0 \; \mbox{in}\; \mathbb{R}^n \setminus\Omega, \end{array} \quad \right. \end{equation*} where $\Omega$ is a bounded domain in $\mathbb{R}^n$ with smooth boundary $\partial \Omega$, $n>2s$, $s\in(0,1)$, $0<q<1$, $\alpha>1$, $\beta>1$ satisfy $2<\alpha+\beta< 2_{s}^*-1$ with $2_{s}^*=\frac{2n}{n-2s}$, the pair of parameters $(\lambda,\mu) \in \mathbb{R}^2\setminus\{(0,0)\}$. The weight functions $f,g: \Omega\subset\mathbb{R}^n \to \mathbb{R}$ such that $0<f$, $g\in L^{\frac{\alpha+\beta}{\alpha+\beta-1+q}}(\Omega)$, and $b:\Omega\subset \mathbb{R}^n \to \mathbb{R}$ is a sign-changing function such that $b(x)\in L^{\infty}(\Omega)$. Using variational methods, we show existence and multiplicity of positive solutions of $(P_{\lambda,\mu})$ with respect to the pair of parameters $(\lambda,\mu)$.