On the Fučik point spectrum for Schrödinger operators on \({\mathbb{R}}^{N}\)

  • Thomas Bartsch
  • Zhi-Qiang Wang
  • Zhitao Zhang


We investigate the Fučik point spectrum of the Schrödinger operator \(S_{\lambda} = - \Delta + V_{\lambda}\, {\rm in}\, L^{2}({\mathbb{R}}^{N})\) when the potential Vλ has a steep potential well for sufficiently large parameter λ > 0. It is allowed that Sλ has essential spectrum with finitely many eigenvalues below the infimum of \(\sigma_{\rm ess}(S_\lambda)\). We construct the first nontrivial curve in the Fučik point spectrum by minimax methods and show some qualitative properties of the curve and the corresponding eigenfunctions. As applications we establish some results on existence of multiple solutions for nonlinear Schrödinger equations with jumping nonlinearity.

Mathematics Subject Classification (2000).

35J60 35B33 


Fučik spectrum Schrödinger operator nonlinear Schrödinger equations jumping nonlinearity critical point theory 


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Copyright information

© Birkhäuser Verlag Basel/Switzerland 2009

Authors and Affiliations

  1. 1.Mathematisches InstitutUniversität GiessenGiessenGermany
  2. 2.Department of Mathematics and StatisticsUtah State UniversityLoganUSA
  3. 3.Academy of Mathematics & Systems ScienceChinese Academy of SciencesBeijingP. R. China

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