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Generalized linking-type theorem with applications to strongly indefinite problems with sign-changing nonlinearities

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Abstract

We show a linking-type result which allows us to study strongly indefinite problems with sign-changing nonlinearities. We apply the abstract theory to the singular Schrödinger equation

$$\begin{aligned} -\Delta u + V(x)u + \frac{a}{r^2} u = f(u) - \lambda g(u), \quad x = (y,z) \in {\mathbb {R}}^K \times {\mathbb {R}}^{N-K}, \ r = |y|, \end{aligned}$$

where

$$\begin{aligned} 0 \not \in \sigma \left( -\Delta + \frac{a}{r^2} + V(x) \right) . \end{aligned}$$

As a consequence we obtain also the existence of solutions to the nonlinear curl-curl problem.

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References

  1. Agrawal, G.: Nonlinear Fiber Optics, Academic Press, 5th Edition (2013)

  2. Alama, S., Li, Y.Y.: Existence of solutions for semilinear elliptic equations with indefinite linear part. J. Differ. Equat. 96, 299–303 (1992)

    Article  MathSciNet  Google Scholar 

  3. Ambrosetti, A., Rabinowtiz, P.H.: Dual variational methods in critical point theory and applications. J. Funct. Anal. 14, 349–381 (1973)

    Article  MathSciNet  Google Scholar 

  4. Badiale, M., Benci, V., Rolando, S.: A nonlinear elliptic equation with singular potential and applications to nonlinear field equations. J. Eur. Math. Soc. 9, 355–381 (2007)

    Article  MathSciNet  Google Scholar 

  5. Badiale, M., Tarantello, G.: A Sobolev-Hardy inequality with applications to a nonlinear elliptic equation arising in astrophysics. Arch. Ration. Mech. Anal. 163, 259–293 (2002)

    Article  MathSciNet  Google Scholar 

  6. Bartsch, T., Dohnal, T., Plum, M., Reichel, W.: Ground States of a Nonlinear Curl-Curl Problem in Cylindrically Symmetric Media. Nonlin. Diff. Equ. Appl. 23:52(5), 34 (2016)

    MathSciNet  MATH  Google Scholar 

  7. Bartsch, T., Mederski, J.: Nonlinear time-harmonic Maxwell equations in domains. J. fixed point theory appl. 19(1), 959–986 (2017)

    Article  MathSciNet  Google Scholar 

  8. Bartsch, T., Mederski, J.: Nonlinear time-harmonic Maxwell equations in an anisotropic bounded medium. J. Funct. Anal. 272(10), 4304–4333 (2017)

    Article  MathSciNet  Google Scholar 

  9. Bartsch, T., Mederski, J.: Ground and Bound State Solutions of Semilinear Time-Harmonic Maxwell Equations in a Bounded Domain. Arch. Rational Mech. Anal. 215(1), 283–306 (2015)

    Article  MathSciNet  Google Scholar 

  10. Bieganowski, B.: Solutions to a nonlinear Maxwell equation with two competing nonlinearities in \({\mathbb{R} }^3\). Bulletin Polish Acad. Sci. Math. 69, 37–60 (2021)

    Article  MathSciNet  Google Scholar 

  11. Bieganowski, B., Mederski, J.: Nonlinear Schrödinger equations with sum of periodic and vanishing potentials and sign-changing nonlinearities. Commun. Pure Appl. Anal. 17(1), 143–161 (2018)

    Article  MathSciNet  Google Scholar 

  12. Bieganowski, B., Mederski, J.: Bound states for the Schrödinger equation with mixed-type nonlinearites. Indiana Univ. Math. J. 71(1), 65–92 (2022)

    Article  MathSciNet  Google Scholar 

  13. Buryak, A.V., Di Trapani, P., Skryabin, D.V., Trillo, S.: Optical solitons due to quadratic nonlinearities: from basic physic to futuristic applications. Phys. Rep. 370, 63–235 (2002)

    Article  MathSciNet  Google Scholar 

  14. Chen, S., Wang, C.: An infinite-dimensional linking theorem without upper semi-continuous assumptions and its applications. J. Math. Anal. Appl. 420(2), 1552–1567 (2014)

    Article  MathSciNet  Google Scholar 

  15. Dörfler, W., Lechleiter, A., Plum, M., Schneider, G., Wieners, C.: Photonic Crystals: Mathematical Analysis and Numerical Approimation, Springer Basel (2012)

  16. Gaczkowski, M., Mederski, J., Schino, J.: Multiple solutions to cylindrically symmetric curl-curl problems and related Schrödinger equations with singular potentials, arXiv:2006.03565

  17. Hazard, C., Lenoir, M.: On the solution of time-harmonic scattering problems for Maxwell’s equations. SIAM J. Math. Anal. 27(6), 1597–1630 (1996)

    Article  MathSciNet  Google Scholar 

  18. Jeanjean, L.: Solutions in spectral gaps for nonlinear equations of Schrödinger type. J. Differ. Equat. 112, 53–80 (1994)

    Article  Google Scholar 

  19. Kryszewski, W., Szulkin, A.: Generalized linking theorem with an application to a semilinear Schrödinger equation. Adv. Differential Equations 3(3), 441–472 (1998)

    MathSciNet  MATH  Google Scholar 

  20. Kuchment, P.: The mathematics of photonic crystals, Mathematical modeling in optical science, Frontiers Appl. Math. 22, SIAM, Philadelphia 207–272 (2001)

  21. Mederski, J.: Nonradial solutions of nonlinear scalar field equations. Nonlinearity 33, 6349–6380 (2020)

    Article  MathSciNet  Google Scholar 

  22. Mederski, J.: Ground states of a system of nonlinear Schrödinger equations with periodic potentials. Comm. Partial Differential Equations 41(9), 1426–1440 (2016)

    Article  MathSciNet  Google Scholar 

  23. Mederski, J.: Ground states of time-harmonic semilinear Maxwell equations in \({\mathbb{R} }^3\) with vanishing permittivity. Arch. Rational Mech. Anal. 218, 825–861 (2015)

    Article  MathSciNet  Google Scholar 

  24. Mederski, J., Schino, J., Szulkin, A.: Multiple solutions to a nonlinear curl-curl problem in \({\mathbb{R} }^3\). Arch. Rational Mech. Anal. 236, 253–288 (2020)

    Article  MathSciNet  Google Scholar 

  25. Nehari, Z.: On a class of nonlinear second-order differential equations. Trans. Amer. Math. Soc. 95, 101–123 (1960)

    Article  MathSciNet  Google Scholar 

  26. Nie, W.: Optical Nonlinearity: Phenomena, Applications, and Materials. Adv. Mater. 5, 520–545 (1993)

    Article  Google Scholar 

  27. Pankov, A.: Periodic Nonlinear Schrödinger Equation with Application to Photonic Crystals. Milan J. Math. 73, 259–287 (2005)

    Article  MathSciNet  Google Scholar 

  28. Rabinowitz, P.H.: Some critical point theorems and applications to semilinear elliptic partial differential equations. Annali della Scuola Normale Superiore di Pisa, Classe di Scienze 4e série, tome 5(1), 215–223 (1978)

    MathSciNet  MATH  Google Scholar 

  29. Slusher, R. E., Eggleton, B. J.: Nonlinear Photonic Crystals, Springer (2003)

  30. Stuart, C.A.: Self-trapping of an electromagnetic field and bifurcation from the essential spectrum. Arch. Ration. Mech. Anal. 113, 65–96 (1991)

    Article  MathSciNet  Google Scholar 

  31. Stuart, C.A.: Guidance Properties of Nonlinear Planar Waveguides. Arch. Rational Mech. Anal. 125, 145–200 (1993)

    Article  MathSciNet  Google Scholar 

  32. Szulkin, A., Weth, T.: Ground state solutions for some indefinite variational problems. J. Funct. Anal. 257(12), 3802–3822 (2009)

    Article  MathSciNet  Google Scholar 

  33. Troestler, C.: Bifurcation into spectral gaps for a noncompact semilinear Schrödinger equation with nonconvex potential, arXiv:1207.1052

  34. Troestler, C., Willem, M.: Nontrivial solutions of a semilinear Schrödinger equation. Commun. Part. Differ. Equat. 21, 1431–1449 (1996)

    Article  Google Scholar 

  35. Willem, M.: Minimax Theorems, Birkhäuser Verlag (1996)

  36. Yin, H.-M.: An eigenvalue problem for curlcurl operators. Can. Appl. Math. Q. 20(3), 421–434 (2012)

    MathSciNet  MATH  Google Scholar 

  37. Zeng, X.: Cylindrically symmetric ground state solutions for curl-curl equations with critical exponent. Z. Angew. Math. Phys. 68(135), 12 (2017)

    MathSciNet  MATH  Google Scholar 

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Acknowledgements

The authors would like to thank anonymous referees for valuable comments and especially for indicating a gap in the assumption (A3) and providing simplifications of the proof of Theorem 2.1 in the original version of the manuscript.

Bartosz Bieganowski was partially supported by the National Science Centre, Poland (Grant No. 2017/25/N/ST1/00531). Federico Bernini started working on this project during his scientific internship in Nicolaus Copernicus University in Toruń, supported by the Project PROM. "Project PROM - International scolarship exchange of PhD candidates and academic staff" is co-funded from the European Social Fund as part of the Program Knowledge Education Development, a non-contest project entitled "International scolarship exchange of PhD candidates and academic staff", agreement number POWR.03.03.00-00-PN/13/18.

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Authors and Affiliations

  1. Dipartimento di Matematica e Applicazioni, Università degli Studi di Milano-Bicocca, via Roberto Cozzi 55, I-20125, Milano, Italy

    Federico Bernini

  2. Faculty of Mathematics and Computer Science, Nicolaus Copernicus University, ul. Chopina 12/18, 87-100, Toruń, Poland

    Bartosz Bieganowski

  3. Institute of Mathematics, Polish Academy of Sciences, ul. Śniadeckich 8, 00-656, Warsaw, Poland

    Bartosz Bieganowski

Authors
  1. Federico Bernini
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Correspondence to Bartosz Bieganowski.

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Communicated by S. Terracini.

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Bernini, F., Bieganowski, B. Generalized linking-type theorem with applications to strongly indefinite problems with sign-changing nonlinearities. Calc. Var. 61, 182 (2022). https://doi.org/10.1007/s00526-022-02297-2

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