Date: 19 Jun 2009

A weighted eigenvalue problem for the p-Laplacian plus a potential

Abstract

Let Δ p denote the p-Laplacian operator and Ω be a bounded domain in \({\mathbb{R}^N}\) . We consider the eigenvalue problem $$-\Delta_p u +V(x) |u|^{p-2}u=\lambda m(x) |u|^{p-2} u, \, \quad u \in W_0^{1,p} (\Omega)$$ for a potential V and a weight function m that may change sign and be unbounded. Therefore the functional to be minimized is indefinite and may be unbounded from below. The main feature here is the introduction of a value α(V, m) that guarantees the boundedness of the energy over the weighted sphere \({M=\{u \in W_0^{1,p}(\Omega); \int_{\Omega}m|u|^p\, dx= 1\}}\) . We show that the above equation has a principal eigenvalue if and only if either m ≥ 0 and α(V, m) > 0 or m changes sign and α(V, m) ≥ 0. The existence of further eigenvalues is also treated here, mainly a second eigenvalue (to the right) and their dependence with respect to V and m.