Some Results for a Class of Kirchhoff-Type Problems with Hardy–Sobolev Critical Exponent

Abstract

We study a class of Kirchhoff equations $$\begin{aligned} {\left\{ \begin{array}{ll} -\left( a+b\displaystyle \int _{\Omega }|\nabla u|^2\mathrm{d}x\right) \Delta u=\displaystyle \frac{u^{3}}{|x|}+\lambda u^{q},&{}\hbox {in } \Omega , \\ u=0, &{}\hbox {on } \partial \Omega , \end{array}\right. } \end{aligned}$$ where $$\Omega \subset {\mathbb {R}}^{3}$$ is a bounded domain with smooth boundary and $$0\in \Omega $$ , $$a,b,\lambda >0,0<q<1.$$ By the variational method, two positive solutions are obtained. Moreover, when $$b>\frac{1}{A_{1}^{2}}$$ ( $$A_{1}>0$$ is the best Sobolev–Hardy constant), using the critical point theorem, infinitely many pairs of distinct solutions are obtained for any $$\lambda >0.$$