Boundary Value Problems (Jun 2018)
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