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Resonant Anisotropic ( p , q )-Equations

Author

Listed:
  • 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)

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.

Suggested Citation

  • Leszek Gasiński & Nikolaos S. Papageorgiou, 2020. "Resonant Anisotropic ( p , q )-Equations," Mathematics, MDPI, vol. 8(8), pages 1-21, August.
  • Handle: RePEc:gam:jmathe:v:8:y:2020:i:8:p:1332-:d:396938
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    References listed on IDEAS

    as
    1. ,, 2001. "Problems And Solutions," Econometric Theory, Cambridge University Press, vol. 17(6), pages 1157-1160, December.
    2. ,, 2001. "Problems And Solutions," Econometric Theory, Cambridge University Press, vol. 17(5), pages 1025-1031, October.
    3. Leszek Gasiński & Nikolaos Papageorgiou, 2013. "A pair of positive solutions for the Dirichlet p(z)-Laplacian with concave and convex nonlinearities," Journal of Global Optimization, Springer, vol. 56(4), pages 1347-1360, August.
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