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Multiple Solutions for \({\varvec{(p,2)}}\)-Equations with Resonance and Concave Terms

  • Leszek GasińskiEmail author
  • Nikolaos S. Papageorgiou
Open Access
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Abstract

We consider parametric Dirichlet problems driven by the sum of a p-Laplacian (\(p>2\)) and a Laplacian ((p, 2)-equation) and with a reaction term which exhibits competing nonlinearities. We prove two multiplicity theorems. In the first the competing terms are not decoupled, the dependence on the parameter is not necessarily linear and the reaction term has a general polynomial growth, possibly supercritical. We produce three nontrivial solutions for small values of the parameter. We provide sign information for all solutions (two of constant sign and the third nodal). Then we decouple the competing nonlinearities and allow for resonance to occur at \(\pm \,\infty \). We produce six nontrivial smooth solutions for small values of the parameter. We provide sign information for five of these solutions.

Keywords

Competing nonlinearities concave term constant sign and nodal solutions multiplicity theorems critical groups resonance 

Mathematics Subject Classification

35J20 35J60 35J92 58E05 

Notes

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© The Author(s) 2019

Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  1. 1.Department of MathematicsPedagogical University of CracowKrakówPoland
  2. 2.Faculty of Mathematics and Computer ScienceJagiellonian UniversityKrakówPoland
  3. 3.Department of MathematicsNational Technical UniversityAthensGreece

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