Annali di Matematica Pura ed Applicata (1923 -)

, Volume 193, Issue 5, pp 1255–1282

Multiple existence results of solutions for quasilinear elliptic equations with a nonlinearity depending on a parameter


DOI: 10.1007/s10231-013-0327-9

Cite this article as:
Motreanu, D. & Tanaka, M. Annali di Matematica (2014) 193: 1255. doi:10.1007/s10231-013-0327-9


We provide existence results of multiple solutions for quasilinear elliptic equations depending on a parameter under the Neumann and Dirichlet boundary condition. Our main result shows the existence of two opposite constant sign solutions and a sign changing solution in the case where we do not impose the subcritical growth condition to the nonlinear term not including derivatives of the solution. The studied equations contain the \(p\)-Laplacian problems as a special case. Our approach is based on variational methods combining super- and sub-solution and the existence of critical points via descending flow.


Quasilinear elliptic equations Nonhomogeneous operator Super-solution and sub-solution Critical point  Invariant sets of descending flow 

Mathematics Subject Classification (2000)

35J20 58E05 35J62 

Copyright information

© Fondazione Annali di Matematica Pura ed Applicata and Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.Départment de MathématiquesUniversité de PerpignanPerpignanFrance
  2. 2.Department of MathematicsTokyo University of ScienceTokyoJapan

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