The eigenvalue problem and infinitely many sign-changing solutions for an elliptic equation with critical Hardy constant (Q335425)

The paper deals with the following problem NEWLINE\[NEWLINE\begin{aligned} - \Delta u - \frac{(n-2)^2}{4}\frac{u}{|x|^2} & = \lambda u, \text{ in } \Omega, \\ u & = 0, \text{ on } \partial \Omega, \end{aligned}NEWLINE\]NEWLINE where \(\Omega \subset R^n (n \geq 3) \) is an open bounded domain with smooth boundary containing the origin.NEWLINENEWLINEA first result consists of lower bounds for the eigenvalues.NEWLINENEWLINETheorem 1.1. Let \( \lambda_k \) be the \(k\)-the eigenvalue of the problem above, then NEWLINE\[NEWLINE\lambda_k \geq C_2 k^{\frac{2}{n}- \frac{2-q}{q}},NEWLINE\]NEWLINE for any \(k \geq 1 \) and \(q \in (\frac{2n}{n+2},2), \) where \(C_2\) is a known constant.NEWLINENEWLINEThe proof based on \textit{P. Li} and \textit{S.-T. Yau}'s idea [Commun. Math. Phys. 88, 309--318 (1983; Zbl 0554.35029)] and a more precise Hardy's inequality.NEWLINENEWLINEFurthermore the paper deals with the multiplicity of sign-changing solutions for the following nonlinear elliptic equation NEWLINE\[NEWLINE\begin{aligned} - \Delta u - \gamma\frac{u}{|x|^2} & = f(x,u) + g(x,u), \text{ in } \Omega, \\ u & = 0, \text{ on } \partial \Omega,\end{aligned}NEWLINE\]NEWLINE under special assumptions for \(f(x,u), g(x,u)\). The result is that infinitely many sign-changing solutions exist.