Electron. J. Differential Equations, Vol. 2017 (2017), No. 183, pp. 1-28.

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

Sarika Goyal

Abstract:
In this article, we study the following fractional-Laplacian system with singular nonlinearity
$$\displaylines{ (-\Delta)^s u = \lambda f(x) u^{-q} + \frac{\alpha}{\alpha+\beta}b(x) u^{\alpha-1} w^\beta\quad \text{in }\Omega \cr (-\Delta)^s w = \mu g(x) w^{-q}+ \frac{\beta}{\alpha+\beta} b(x) u^{\alpha} w^{\beta-1}\; \text{in } \Omega \cr u, w>0\text{ in }\Omega, \quad u = w = 0 \text{ in } \mathbb{R}^n \setminus\Omega, }$$
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 with respect to the pair of parameters $(\lambda,\mu)$.

Submitted July 20, 2016. Published July 18, 2017.
Math Subject Classifications: 35A15, 35J75, 35R11.
Key Words: Fractional-Laplacian system; singular nonlinearity; sign-changing weight function.

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Sarika Goyal
Department of Mathematics
Bennett University, Greater Noida
Uttar Pradesh, India
email: sarika1.iitd@gmail.com, sarika.goyal@bennett.edu.in

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