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



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.

Mathematics Subject Classification (2000)

35J20 35J70 35P05 35P30 


Nonlinear eigenvalue problem p-Laplacian plus a potential Indefinite weight 


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© Birkhäuser Verlag Basel/Switzerland 2009

Authors and Affiliations

  1. 1.LMPA, Université du Littoral Côte d’Opale (ULCO)CalaisFrance
  2. 2.Université Libre de BruxellesBruxellesBelgium

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