Two positive solutions for quasilinear elliptic equations with singularity and critical exponents

Abstract

Abstract In this paper, we consider the quasilinear elliptic equation with singularity and critical exponents {−Δpu−μ|u|p−2u|x|p=Q(x)|u|p∗(t)−2u|x|t+λu−s,in Ω,u>0,in Ω,u=0,on ∂Ω, $$ \textstyle\begin{cases} -\Delta_{p}u-\mu \frac{ \vert u \vert ^{p-2}u}{ \vert x \vert ^{p}}=Q(x) \frac{ \vert u \vert ^{p^{*}(t)-2}u}{ \vert x \vert ^{t}}+\lambda u^{-s}, &\text{in }\Omega , \\ u>0, & \text{in }\Omega , \\ u=0, &\text{on }\partial \Omega , \end{cases} $$ where Δp=div(|∇u|p−2∇u) $\Delta_{p}= \operatorname {div}(|\nabla u|^{p-2}\nabla u)$ is a p-Laplace operator with 1<p<N $1< p< N$. p∗(t):=p(N−t)N−p $p^{*}(t):=\frac{p(N-t)}{N-p}$ is a critical Sobolev–Hardy exponent. We deal with the existence of multiple solutions for the above problem by means of variational and perturbation methods.

Keywords

• Quasilinear
• Singularity
• Critical
• Sobolev–Hardy exponent