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Multiple ground-state solutions with sign information for double-phase robin problems

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Abstract

We consider a nonlinear unbalanced double-phase problem with a superlinear reaction and Robin boundary condition. Using suitable variants of the Nehari method, we show that the problem has three nontrivial solutions all with sign information (positive, negative and nodal).

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References

  1. R. A. Adams, Sobolev Spaces, Pure and Applied Mathematics, Vol. 65, Academic Press, New York-London, 1975.

    MATH  Google Scholar 

  2. P. Baroni, M. Colombo and G. Mingione, Harnack inequalities for double-phase functionals, Nonlinear Analysis 121 (2015), 206–222.

    Article  MathSciNet  MATH  Google Scholar 

  3. S.-S. Byun and J. Oh, Global gradient estimates for non-uniformly elliptic equations, Calculus of Variations and Partial Differential Equations 56 (2017), Article no. 46.

  4. S.-S. Byun and Y. Youn, Riesz potential estimates for a class of double-phase problems, Journal of Differential Equations 264 (2018), 1263–1316.

    Article  MathSciNet  MATH  Google Scholar 

  5. M. Cencelj, V. D. Radulescu and D. Repovs, Double phase problems with variable growth, Nonlinear Analysis 177 (2018), 270–287.

    Article  MathSciNet  MATH  Google Scholar 

  6. F. Clarke, Functional Analysis, Calculus of Variation and Optimal Control, Graduate Texts in Mathematics, Vol. 264, Springer, London, 2013.

    Book  MATH  Google Scholar 

  7. F. Colasuonno and M. Squassina, Eigenvalues for double-phase variational integrals, Annali di Matematica Pura ed Applicata 195 (2016), 1917–1959.

    Article  MathSciNet  MATH  Google Scholar 

  8. M. Colombo and G. Mingione, Regularity for double-phase variational problems, Archive for Rational Mechanics and Analysis 215 (2015), 443–496.

    Article  MathSciNet  MATH  Google Scholar 

  9. M. Colombo and G. Mingione, Bounded minimisers of double-phase variational integrals, Archive for Rational Mechanics and Analysis 218 (2015), 219–273.

    Article  MathSciNet  MATH  Google Scholar 

  10. M. Colombo and G. Mingione, Calderón-Zygmund estimates and non-uniformly elliptic operators, Journal of Functional Analysis 270 (2016), 1416–1478.

    Article  MathSciNet  MATH  Google Scholar 

  11. C. De Filippis and G. Mingione, A borderline case of Calderón-Zygmund estimates for nonuniformly elliptic problems, St. Petersburg Mathematical Journal 31 (2020), 455–477.

    Article  MathSciNet  MATH  Google Scholar 

  12. C. De Filippis and G. Mingione, Lipschitz bounds and nonautonomous integrals, Archive for Rational Mechanics and Analysis 242 (2021), 973–1057.

    Article  MathSciNet  MATH  Google Scholar 

  13. L. Diening, P. Harjulehto, P. Hästö and M. Růžička, Lebesgue and Sobolev Spaces with Variable Exponents, Lecture Notes in Mathematics, Vol. 2017, Springer, Heidelberg, 2011.

    MATH  Google Scholar 

  14. L. Gasinski and N. S. Papageorgiou, Nonlinear Analysis, Series in Mathematical Analysis and Applications, Vol. 9, Chapman & Hall/CRC Press, Boca Raton, FL, 2006.

    MATH  Google Scholar 

  15. L. Gasinski and N. S. Papageorgiou, Positive solutions for the Robin p-Laplacian problem with competing nonlinearities, Advances in Calculus of Variations 12 (2019), 31–56.

    Article  MathSciNet  MATH  Google Scholar 

  16. G. Lieberman, The natrual generalization of the natural conditions of Ladyzhenskaya and Uraltseva for elliptic equations, Communiations in Partial Differential Equations 16 (1991), 311–361.

    Article  MATH  Google Scholar 

  17. W. Liu and G. Dai, Existence and multiplicity results for double-phase problems, Journal of Differential Equations 265 (2018), 4311–4334.

    Article  MathSciNet  MATH  Google Scholar 

  18. W. Liu and G. Dai, Three ground-state solutions for double-phase problem, Journal of Mathematical Physics 59 (2018), Article no. 121503.

  19. J. Liu, Y. Wang and Z.-Q. Wang, Solutions for quasilinear Schrödinger equations via the Nehari manifold, Communications in Partial Differential Equations 29 (2004), 879–901.

    Article  MathSciNet  MATH  Google Scholar 

  20. W. A. Luxemburg, Banach Function Spaces, Ph.D. Thesis, Technische Hogeschool te Delft, Delft, 1955.

    MATH  Google Scholar 

  21. P. Marcellini, Regularity and existence of solutions of elliptic equations with p, q-growth conditions, Journal of Differential Equations 90 (1991), 1–30.

    Article  MathSciNet  MATH  Google Scholar 

  22. G. Mingione and V. D. Radulescu, Recent developments in problems with nonstandard growth and nonuniform ellipticity, Journal of Mathematical Analysis and Applications 501 (2021), Article no. 125197.

  23. D. Mugnai and N. S. Papageorgiou, Resonant nonlinear Neumann problems with indefinite weight, Annali della Scuola Normale Superiore di Pisa. Classe di Scienze. Serie V 12 (2012), 729–788.

    MathSciNet  MATH  Google Scholar 

  24. J. Musielak, Orlicz Space and Modular Spaces, Lecture Notes in Mathematics, Vol. 1034, Springer, Berlin, 1983.

    Book  MATH  Google Scholar 

  25. N. S. Papageorgiou and V. D. Radulescu, Nonlinear nonhomogeneous Robin problems with superlinear reaction, Advanced Nonlinear Studies 16 (2016), 737–764.

    Article  MathSciNet  MATH  Google Scholar 

  26. N. S. Papageorgiou, V. D. Radulescu and D. D. Repovs, Nonlinear Analysis-Theory and Methods, Springer Monographs in Mathematics, Springer, Cham, 2019.

    Book  MATH  Google Scholar 

  27. P. Pucci and J. Serrin, The Maximum Principle, Progress in Nonlinear Differential Equations and their Applications, Vol. 73, Birkhäser, Basel, 2007.

    MATH  Google Scholar 

  28. V. D. Radulescu, Isotropic and anisotropic double-phase problems: old and new, Opuscula Mathematica 39 (2019), 259–279.

    Article  MathSciNet  MATH  Google Scholar 

  29. A. Szulkin and T. Weth, The method of Nehari manifold in Handbook of Nonconvex Analysis and Applications, International Press, Somerville, MA, 2010, pp. 597–632.

    MATH  Google Scholar 

  30. V. V. Zhikov, Averaging of functionals of the calculus of variations and elasticity, Mathematics of the USSR—Izvestiya 29 (1987), 33–66.

    Article  MATH  Google Scholar 

  31. V. V. Zhikov, On variational problems and nonlinear elliptic equations with nonstandard growth conditions, Journal of Mathematical Sciences (New York) 173 (2011), 463–570.

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgments

This work was supported by the National Natural Science Foundation of China (No. 12071098). The authors wish to thank a knowledgeable referee for his/her remarks and constructive criticism and for providing additional relevant references.

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Correspondence to Chao Zhang.

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Papageorgiou, N.S., Zhang, C. Multiple ground-state solutions with sign information for double-phase robin problems. Isr. J. Math. 253, 419–443 (2023). https://doi.org/10.1007/s11856-022-2370-y

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  • DOI: https://doi.org/10.1007/s11856-022-2370-y

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