Multiple solutions for a critical quasilinear equation with Hardy potential (Q2321751)

Let \(N\ge7\) and let \(\Omega\subset\mathbb R^N\) be a bounded domain containing the origin. Denote by \(2^*=\frac{2N}{N-2}\) the critical Sobolev exponent for the embedding of \(H^1_0(\Omega)\) in \(L^p(\Omega)\). The authors study the quasilinear equation \(-\sum_{i,j=1}^N D_j(a_{ij}(u)D_iu)+\frac12 \sum_{i,j=1}^N a^\prime_{ij}(u)D_iu D_ju - \frac\mu{|x|^2}u = au+|u|^{2^*-2}u\) in \(\Omega\) with the boundary condition \(u=0\) on \(\partial\Omega\), where the coefficients \(a_{ij}\) satisfy certain assumptions, \(a>0\) is a constant and \(\mu\in(0,\mu^*)\) for some constant \(\mu^*\) depending on \(N\) and on the coefficients \(a_{ij}\). To overcome the difficulty caused by the critical exponent, they follow the idea of \textit{G. Devillanova} and \textit{S. Solimini} [Adv. Differ. Equ. 7, No. 10, 1257--1280 (2002; Zbl 1208.35048)] and consider the sequence of perturbed problems where the exponent \(2^*-2\) is replaced with \(2^*-2-\varepsilon_n\), \(\varepsilon_n\to0\). They prove that the sequence of solutions \(u_n\) to the perturbed problems is bounded and has a strong limit in \(H^1_0(\Omega)\). As a consequence they obtain that the original problem has infinitely many solutions.