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Open AccessArticle

Resonant Anisotropic (p,q)-Equations

1
Department of Mathematics, Pedagogical University of Cracow, Podchorazych 2, 30084 Cracow, Poland
2
Department of Mathematics, National Technical University, Zografou Campus, 15780 Athens, Greece
*
Author to whom correspondence should be addressed.
Mathematics 2020, 8(8), 1332; https://doi.org/10.3390/math8081332
Received: 14 July 2020 / Revised: 1 August 2020 / Accepted: 6 August 2020 / Published: 10 August 2020
(This article belongs to the Special Issue New Trends in Variational Methods in Nonlinear Analysis)
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 (Δp(z),W01,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. View Full-Text
Keywords: anisotropic (p,q)-laplacian; resonance; principal eigenvalue; critical group; constant sign; nodal solutions anisotropic (p,q)-laplacian; resonance; principal eigenvalue; critical group; constant sign; nodal solutions
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MDPI and ACS Style

Gasiński, L.; Papageorgiou, N.S. Resonant Anisotropic (p,q)-Equations. Mathematics 2020, 8, 1332. https://doi.org/10.3390/math8081332

AMA Style

Gasiński L, Papageorgiou NS. Resonant Anisotropic (p,q)-Equations. Mathematics. 2020; 8(8):1332. https://doi.org/10.3390/math8081332

Chicago/Turabian Style

Gasiński, Leszek; Papageorgiou, Nikolaos S. 2020. "Resonant Anisotropic (p,q)-Equations" Mathematics 8, no. 8: 1332. https://doi.org/10.3390/math8081332

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