Infinitely many sign-changing solutions of an elliptic problem involving critical Sobolev and Hardy–Sobolev exponent

Abstract

We study the existence and multiplicity of sign-changing solutions of the following equation

$$\begin{array}{@{}rcl@{}} \left\{\begin{array}{lllllllll} -{\Delta} u = \mu |u|^{2^{\star}-2}u+\frac{|u|^{2^{*}(t)-2}u}{|x|^{t}}+a(x)u \quad\text{in}\, {\Omega}, \\ u=0 \quad\text{on}\quad\partial{\Omega}, \end{array}\right. \end{array} $$

where Ω is a bounded domain in $\mathbb {R}^{N}$, 0∈∂Ω, all the principal curvatures of ∂Ω at 0 are negative and μ≥0, a>0, N≥7, 0<t<2, $2^{\star }=\frac {2N}{N-2}$ and $2^{\star }(t)=\frac {2(N-t)}{N-2}$.