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Min-Max Principles with Nonlinear Generalized Rayleigh Quotients for Nonlinear Equations

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We generalize the Poincaré and Courant–Fischer–Weyl min-max principles to nonlinear equations by applying the Lusternik–Schnirelmann theory to nonlinear generalized Rayleigh quotients. Based on this approach, we establish the existence of countably many solutions with prescribed energy to the Dirichlet problem with the p-Laplacian and convex-concave nonlinearity and prove new type asymptotic estimates for the spectral values of the problem.

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References

  1. Y. Ilyasov, “On extreme values of Nehari manifold method via nonlinear Rayleigh’s quotient,” Topol. Methods Nonlinear Anal. 49, No. 2, 683–714 (2017).

    MathSciNet  MATH  Google Scholar 

  2. P. Drábek and J. Milota, Methods of Nonlinear Analysis. Applications to Differential Equations, Springer, Basel (2013).

    MATH  Google Scholar 

  3. M. A. Krasnosel’skii, Topological Methods in the Theory of Nonlinear Integral Equations, Pergamon, Oxford etc. (1964).

  4. A. Ambrosetti and P. H. Rabinowitz, “Dual variational methods in critical point theory and applications,” J. Funct. Anal. 14, 349–381 (1973).

    Article  MathSciNet  Google Scholar 

  5. M. Struwe, Variational Methods. Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems, Springer, Berlin (2008).

  6. Ya. Sh. Il’yasov, “Fundamental frequency solutions with prescribed action value to nonlinear Schrödinger equations,” J. Math. Sci. 259, No. 2, 187–204 (2021).

  7. J. I. Díaz, J. Hernández, and Y. Sh. Ilyasov, “On the exact multiplicity of stable ground states of non-Lipschitz semilinear elliptic equations for some classes of starshaped sets,” Adv. Nonlinear Anal. 9, No. 1, 1046–1065 (2020).

    Article  MathSciNet  Google Scholar 

  8. M. L. Carvalho, Y. Il’yasov, and C. A. Santos, “Separating solutions of nonlinear problems using nonlinear generalized Rayleigh quotients,” Topol. Methods Nonlinear Anal. Advance Publ. 1-28 (2021). https://doi.org/10.12775/TMNA.2020.075

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Authors and Affiliations

  1. Institute of Mathematics, UFRC RAS, 112, Chernyshevskii St, Ufa, 450008, Russia

    Ya. Sh. Il’yasov

  2. Universidade Federal de Goiás, Instituto de Matemática, Goiânia, GO, 74690-900, Brazil

    Ya. Sh. Il’yasov

  3. JSC Concern “Sozvezdie”, 16, Plekhanovskaya St, Voronezh, 394018, Russia

    A. B. Muravnik

Authors
  1. Ya. Sh. Il’yasov
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  2. A. B. Muravnik
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Correspondence to Ya. Sh. Il’yasov.

Additional information

Translated from Problemy Matematicheskogo Analiza 113, 2022, pp. 29-36.

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Il’yasov, Y.S., Muravnik, A.B. Min-Max Principles with Nonlinear Generalized Rayleigh Quotients for Nonlinear Equations. J Math Sci 260, 738–747 (2022). https://doi.org/10.1007/s10958-022-05732-z

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  • DOI: https://doi.org/10.1007/s10958-022-05732-z

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