Weak positive solutions to singular quasilinear elliptic equation

Abstract

Abstract In this paper, we study the existence of multiple solutions for the singular problem { a ⁢ ( x , u , ∇ ⁡ u ) - div ⁢ ( b ⁢ ( x , u , ∇ ⁡ u ) ) = u - α + λ ⁢ c ⁢ ( x , u ) in  ⁢ Ω , u > 0 in  ⁢ Ω , u = 0 on  ⁢ ℝ n ∖ Ω , \left\{\begin{aligned} \displaystyle{}a(x,u,\nabla u)-{\rm div}(b(x,u,\nabla u% ))&\displaystyle=u^{-\alpha}+\lambda c(x,u)&&\displaystyle\phantom{}\text{in }% \Omega,\\ \displaystyle u&\displaystyle>0&&\displaystyle\phantom{}\text{in }\Omega,\\ \displaystyle u&\displaystyle=0&&\displaystyle\phantom{}\text{on }{\mathbb{R}}% ^{n}\setminus\Omega,\end{aligned}\right. where Ω ⊂ ℝ n {\Omega\subset\mathbb{R}^{n}} ( n ≥ 3 ) {(n\geq 3)} is a bounded domain with C 1 {C^{1}} boundary, λ is a positive parameter, 0 < α ≤ 1 < p ≤ n {0<\alpha\leq 1<p\leq n} and p * = n ⁢ p n - p {p^{*}=\frac{np}{n-p}} is the critical exponent for Sobolev embedding. Using the fibering maps and the Nehari manifold, we prove the existence of at least two positive solutions for all values of the parameter λ belonging to an open bounded interval of ℝ + {\mathbb{R}_{+}} .