In this article we study a nonlocal equation involving singular and critical Hardy-Sobolev non-linearities, \[\displaylines{(-\Delta_p)^su-\mu \frac{|u|^{p-2}u}{|x|^{sp}}=\lambda u^{-\alpha}+\frac{|u|^{p_s^*(t)-2}u}{|x|^t}, \quad\hbox{in }\Omega, \\ u>0,\quad\text{in }\Omega,\\ \quad u=0, \quad\text{in } \mathbb{R}^N \setminus\Omega }\] where \(\Omega \subset \mathbb{R}^N\) is bounded domain...

We study positive solutions to the boundary value problem $$ \displaylines{ -\Delta_p u - \Delta_q = \lambda f(u) \quad \text{in } \Omega, \cr 0 \text{on \partial\Omega, }$$ where \(q \in (1,p)\) and Ω is a bounded domain in \(\mathbb{R}^N\), \(N>1\) with smooth boundary, \(\lambda\) parameter, \(f:[0,\infty) \to (0,\infty)\) \(C_1\), nondecreasing, p-sublinear at infinity i.e. \(\...

In this paper, we investigate the following quasilinear and singular problem : { −∆pu = λ uδ + u q in Ω u|∂Ω = 0 , u > 0 in Ω (1) where Ω is an open bounded domain with smooth boundary, 1 < p, p− 1 < q and λ, δ > 0. We first prove that there exist weak solutions for λ > 0 small in W 1,p 0 (Ω)∩C(Ω̄) if and only if δ < 2+ 1 p−1 . Investigating the radial symmetric case (Ω = BR(0)), we prove by a s...