\(H^{1,p}\)-eigenvalues and \(L^\infty\)-estimates in quasicylindrical domains (Q652039)

Let \(\Omega\subset{\mathbb R}^n\) be a quasi-cylindrical domain, i.e. an open connected set with in-radius \(r_\Omega<\infty\), where \(r_\Omega=\sup\{r>0: B_r\subset\Omega\}\), \(B_r\) is a ball of \({\mathbb R}^n\). The author considers the \(p\)-Raleigh quotient \[ \lambda_{\Omega,p}=\inf_{u\in H^{1,p}_0(\Omega)} {\|\nabla u\|^p_{p,\Omega}\over \|u\|^p_{p,\Omega}}. \] It is proved that there exists a positive constant \(C_{n,p}\) such that \[ \lambda_{\Omega,p}\geq {C_{n,p} \over r^p_\Omega},\quad \forall p>n. \] By a counterexample it is shown that this inequality no longer holds for \(p\leq n\). The author applies this result to obtain, by variational methods, existence and uniqueness of weak solutions of the Dirichlet problem for second-order quasilinear elliptic equations in divergent form in quasicylindrical domains. For such solutions global boundedness estimates have been also established.