Resonant Anisotropic ( p, q )-Equations

Leszek Gasiński and Nikolaos S. Papageorgiou
Additional contact information
Leszek Gasiński: Department of Mathematics, Pedagogical University of Cracow, Podchorazych 2, 30084 Cracow, Poland
Nikolaos S. Papageorgiou: Department of Mathematics, National Technical University, Zografou Campus, 15780 Athens, Greece

Mathematics, 2020, vol. 8, issue 8, 1-21

Abstract: We consider an anisotropic Dirichlet problem which is driven by the ( p ( z ) , q ( z ) ) -Laplacian (that is, the sum of a p ( z ) -Laplacian and a q ( z ) -Laplacian), The reaction (source) term, is a Carathéodory function which asymptotically as x ± ∞ can be resonant with respect to the principal eigenvalue of ( − Δ p ( z ) , W 0 1 , p ( z ) ( Ω ) ) . First using truncation techniques and the direct method of the calculus of variations, we produce two smooth solutions of constant sign. In fact we show that there exist a smallest positive solution and a biggest negative solution. Then by combining variational tools, with suitable truncation techniques and the theory of critical groups, we show the existence of a nodal (sign changing) solution, located between the two extremal ones.

Keywords: anisotropic ( p , q )-laplacian; resonance; principal eigenvalue; critical group; constant sign; nodal solutions (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2020
References: View references in EconPapers View complete reference list from CitEc
Citations:

Downloads: (external link)
https://www.mdpi.com/2227-7390/8/8/1332/pdf (application/pdf)
https://www.mdpi.com/2227-7390/8/8/1332/ (text/html)

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:gam:jmathe:v:8:y:2020:i:8:p:1332-:d:396938

Access Statistics for this article

Mathematics is currently edited by Ms. Emma He

More articles in Mathematics from MDPI
Bibliographic data for series maintained by MDPI Indexing Manager ().