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

Abstract

In this article, we study the following fractional-Laplacian system with singular nonlinearity \begin{equation*} (P_{λ,μ}) \left\{ \begin{array}{lr} (-Δ)^s u = λf(x) u^{-q}+ \fracα{α+β}b(x) u^{α-1} w^β\; \text{in}\;Ω\\ (-Δ)^s w = μg(x) w^{-q}+ \fracβ{α+β} b(x) u^α w^{β-1}\; \text{in}\;Ω\\ \quad \quad u, w>0\;\text{in}\;Ω, \quad u = w = 0 \; \mbox{in}\; \mathbb{R}^n \setminusΩ, \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 $2<\alpha +\beta <2_s^{\ast }-1$ with $2_s^{\ast }=\frac{2n}{n-2s}$, 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 $(\lambda ,\mu )$.

27 pages. arXiv admin note: substantial text overlap with arXiv:1604.00801

References in corpus (2)

• Existence, Uniqueness of Positive Solution to a Fractional Laplacians with Singular Nonlinearity
• Multiplicity of positive solutions for a fractional Laplacian equations involving critical nonlinearity