Infinitely many solutions for an elliptic equation in divergent form with critical Hardy–Sobolev exponent

Abstract

By using concentration estimates, Fountain Theorem and its Dual form we prove the existence of two disjoint and infinite sets of solutions for the following elliptic equation in divergent form with critical Hardy–Sobolev exponent and concave–convex nonlinearity $\left(\begin{array}{cc}-div\left(a\left(x\right)Du\right)=Q\left(x\right)\frac{|u{|}^{2^{\ast }\left(s\right)-2}u}{|x{|}^s}+\lambda |u{|}^{q-2}u& x\in \Omega ,\\ u=0& \text{on}\partial \Omega .\end{array}\right)$The problem is considered in an open bounded domain $\Omega \subset R^N$ under certain assumptions on $a$ and $Q$.