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Results in Mathematics

, Volume 70, Issue 1–2, pp 31–79 | Cite as

Nonlinear Nonhomogeneous Dirichlet Equations Involving a Superlinear Nonlinearity

  • Nikolaos S. Papageorgiou
  • Patrick Winkert
Article

Abstract

We consider a nonlinear elliptic Dirichlet equation driven by a nonlinear nonhomogeneous differential operator involving a Carathéodory function which is (p−1)-superlinear but does not satisfy the Ambrosetti–Rabinowitz condition. First we prove a three-solutions-theorem extending an earlier classical result of Wang (Ann Inst H Poincaré Anal Non Linéaire 8(1):43–57, 1991). Subsequently, by imposing additional conditions on the nonlinearity \({f(x,\cdot)}\), we produce two more nontrivial constant sign solutions and a nodal solution for a total of five nontrivial solutions. In the special case of (p, 2)-equations we prove the existence of a second nodal solution for a total of six nontrivial solutions given with complete sign information. Finally, we study a nonlinear eigenvalue problem and we show that the problem has at least two nontrivial positive solutions for all parameters \({\lambda > 0}\) sufficiently small where one solution vanishes in the Sobolev norm as \({\lambda \to 0^+}\) and the other one blows up (again in the Sobolev norm) as \({\lambda \to 0^+}\).

Mathematics Subject Classification

35J20 35J60 35J92 58E05 

Keywords

Superlinear nonlinearity Ambrosetti–Rabinowitzcondition nonlinear regularity nodal solutions tangencyprinciple critical groups nonlinear eigenvalue problem 

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Copyright information

© Springer Basel 2015

Authors and Affiliations

  1. 1.Department of MathematicsNational Technical UniversityZografou CampusGreece
  2. 2.Institut für MathematikTechnische Universität BerlinBerlinGermany

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