\(p\)-Laplacian problems involving critical Hardy-Sobolev exponents
The authors study the nonlinear eigenvalue problem \[ -\Delta_p u=\mu|u|^{p-2} u+{|u|^{p^*(s)-2}u\over |x|^s}\quad\text{in }\Omega, \] \(u=0\) on \(\partial\Omega\), where \(\Omega\) is a bounded domain in \(\mathbb{R}^N\), \(1<p<N\), \(0<s<p\) and \(p^*(s)= (N-s)p/(N-p)\) is the critical Hardy-Sobolev exponent. The authors prove existence, multiplicity and bifurcation results extending to \(\mu\geq\mu_1\) (where \(\mu_1\) is the first eigenvalue of \(-\Delta_p\)) results which are known for \(\mu<\mu_1\). The methods of proof include mainly several versions of variational arguments, in particular some non-standard linking theorems involving sub-level sets due to Perera and Szulkin. An alternative sequence of eigenvalues for the \(p\)-Laplacian obtained via a cohomological index is also used.
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