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

Abstract

Let Δp denote the p-Laplacian operator and Ω be a bounded domain in ℝN. We consider the eigenvalue problem 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. 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. © 2009 Birkhäuser Verlag Basel/Switzerland.