Infinitely Many Solutions for a Nonlinear Elliptic PDE with Multiple Hardy–Sobolev Critical Exponents

Abstract

In this paper, by an approximating argument, we obtain two disjoint and infinite sets of solutions for the following elliptic equation with multiple Hardy–Sobolev critical exponents

$$\begin{aligned} \left\{ \begin{array}{ll} -\Delta u=\mu \vert u \vert ^{2^{*}-2} u + \sum _{i=1}^{l} \frac{ \vert u \vert ^{2^{*}(s_{i})-2}u}{ \vert x \vert ^{s_{i}}}+ a(x) \vert u \vert ^{q-2} u &{} \; in \; \Omega , \\ u=0 &{} \; on \; \partial \Omega , \end{array}\right. \end{aligned}$$

where $\Omega$ is a smooth bounded domain in ${\mathbb {R}}^{N}$ with $0\in \partial \Omega$ and all the principle curvatures of $\partial \Omega$ at 0 are negative, $a \in {\mathcal {C}}^{1}({\bar{\Omega }}, \mathbb {R^{*}}^{+}),$ $\mu > 0,$ $0<s_{1}<s_{2}<...<s_{l}<2,$ $1<q<2$ and $N > 2\frac{q+1}{q -1}.$ By $2^{*}:=\frac{2 N}{N-2}$ and $2^{*}(s_{i}):=\frac{2 (N-s_{i})}{N-2}$ we denote the critical Sobolev exponent and Hardy–Sobolev exponents, respectively.