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Critical nonhomogeneous fourth-order Schrödinger–Kirchhoff-type equations

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Abstract

In this paper we study the following class of stationary fourth-order Schrödinger–Kirchhoff-type equations:

$$\begin{aligned} \Delta ^{2} u-M\left( \Vert \nabla u\Vert ^2_2 \right) \Delta u+V(x)u=h(x)|u|^{q-2}u+|u|^{2_*-2}u+ g(x)|u|^{\tau -2}u, ~~x \in \mathbb {R}^{N}, \end{aligned}$$

where \(N\ge 8,\) and \(2_*=\frac{2N}{N-4}\) is the critical Sobolev exponent. Under some assumptions on the Kirchhoff function M, the potential V(x) and g(x), by using Ekeland’s Variational Principle and the Mountain Pass Theorem, we obtain the existence of multiple solutions for the above problem. These results are new even for the local case, which corresponds to nonlinear fourth order Schrödinger equations.

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References

  1. Arosio, A., Panizzi, S.: On the well-posedness of the Kirchhoff string. Trans. Am. Math. Soc. 348, 305–330 (1996)

    Article  MathSciNet  Google Scholar 

  2. Bernis, F., Garcia Azorero, J., Peral, I.: Existence and multiplicity of nontrivial solutions in semilinear critical problems of fourth order. Adv. Differ. Equ. 1, 219–240 (1996)

    MathSciNet  Google Scholar 

  3. Bernstein, S.: Sur une classe d’équations fonctionnelles aux dérivées partielles. Bull. Acad. Sci. URSS Sér. Math. (Izv. Akad. Nauk SSSR) 4, 17–26 (1940)

    Google Scholar 

  4. Cabada, A., Figueiredo, G.M.: A generalization of an extensible beam equation with critical growth in \(\mathbb{R} ^N\). Nonlinear Anal. Real World Appl. 20, 134–142 (2014)

    Article  MathSciNet  Google Scholar 

  5. Cavalcanti, M.M., Domingos Cavalcanti, V.N., Soriano, J.A.: Global existence and uniform decay rates for the Kirchhoff–Carrier equation with nonlinear dissipation. Adv. Differ. Equ. 6, 701–730 (2001)

    MathSciNet  Google Scholar 

  6. Chabrowski, J.: On multiple solutions for the nonhomogeneous \(p\)-Laplacian with a critical Sobolev exponent. Differ. Integr. Equ. 8, 705–716 (1995)

    MathSciNet  Google Scholar 

  7. Chabrowski, J., Yang, J.: Nonnegative solutions for semilinear biharmonic equations in \(\mathbb{R} ^N\). Analysis 17, 35–59 (1997)

    Article  MathSciNet  Google Scholar 

  8. Chabrowski, J., Marcos do Ó, J.: On some fourth-order semilinear problems in \(\mathbb{R} ^{N}\). Nonlinear Anal. 49, 861–884 (2002)

    Article  MathSciNet  Google Scholar 

  9. Edmunds, D.E., Fortunato, D., Jannelli, E.: Critical exponent, critical dimensions and the biharmonic operator. Arch. Rat. Mech. Anal. 112, 269–289 (1990)

    Article  MathSciNet  Google Scholar 

  10. Ferrero, A., Warnault, G.: On solutions of second and fourth order elliptic equations with power-type nonlinearities. Nonlinear Anal. 70, 2889–2902 (2009)

    Article  MathSciNet  Google Scholar 

  11. Kirchhoff, G.: Vorlesungen uber Mechanik. Teubner, Leipzig (1883)

    Google Scholar 

  12. Lazer, A., McKenna, P.: Large-amplitude periodic oscillations in suspension bridges: some new connections with nonlinear analysis. SIAM Rev. 32, 537–578 (1990)

    Article  MathSciNet  Google Scholar 

  13. Li, C., Tang, C.L.: Three solutions for a Navier boundary value problem involving the p-biharmonic. Nonlinear Anal. 72, 1339–1347 (2010)

    Article  MathSciNet  Google Scholar 

  14. Lions, J.-L.: On some questions in boundary value problems of mathematical physics. In: Contemporary Developments in Continuum Mechanics and Partial Differential Equations, Proceedings of the International Symposium of Inst. Mat., Univ. Fed. Rio de Janeiro, 1997, North-Holland Mathematical Studies, 30, pp. 284–346. North-Holland, Amsterdam (1978)

  15. Lions, P.-L.: The concentration-compactness principle in the calculus of variation. The limit case—part 1. Revista Matematica Iberoamericana 1, 145–201 (1985)

    Article  MathSciNet  Google Scholar 

  16. Myers, T.G.: Thin films with high surface tension. SIAM Rev. 40, 441–462 (1998)

    Article  MathSciNet  Google Scholar 

  17. Noussair, E.S., Swanson, C.A., Yang, J.: Critical semilinear biharmonic equations in \(\mathbb{R} ^N\). Proc. R. Soc. Ed. 121A, 139–148 (1992)

    Article  Google Scholar 

  18. Noussair, E.S., Swanson, C.A., Yang, J.: Transcritical biharmonic equations in \(\mathbb{R} ^N\). Funkcialaj Ekvacioj 35, 533–543 (1992)

    MathSciNet  Google Scholar 

  19. Pimenta, M.T.O., Soares, S.H.M.: Existence and concentration of solutions for a class of biharmonic equations. J. Math. Anal. Appl. 390, 274–289 (2012)

    Article  MathSciNet  Google Scholar 

  20. Pohozaev, S.: On a class of quasilinear hyperbolic equations. Math. Sborniek 96, 152–166 (1975)

    MathSciNet  Google Scholar 

  21. Pucci, P., Serrin, J.: Critical exponent and critical dimensions for polyharmonic operators. J. Math. Pures Appi. 69, 55–83 (1990)

    MathSciNet  Google Scholar 

  22. Rabinowitz, P.: Minimax methods in critical point theory with applications to differential equations. In: CBMS Regional Conference Series in Mathematics, vol. 65. American Mathematical Society, Providence, RI (1986)

  23. Song, H., Chen, C.: Infinitely many solutions for Schrödinger–Kirchhoff-type fourth-order elliptic equations. Proc. Edinb. Math. Soc. 4, 1–18 (2017)

    Google Scholar 

  24. Tarantello, G.: On nonhomogeneous elliptic involving critical Sobolev exponent. Ann. Inst. H. Poincaré Anal. Non Linéaire 9, 281–304 (1992)

    Article  MathSciNet  Google Scholar 

  25. Xiang, M., Radulescu, V.D., Zhang, B.: Existence results for singular fractional p-Kirchhoff problems. Acta Math. Sci. 42, 1209–1224 (2022)

    Article  MathSciNet  Google Scholar 

  26. Wang, F., Hu, D., Xiang, M.: Combined effects of Choquard and singular nonlinearities in fractional Kirchhoff problems. Adv. Nonlinear Anal. 10, 636–658 (2021)

    Article  MathSciNet  Google Scholar 

  27. Wang, W., Zhao, P.: Nonuniformly nonlinear elliptic equations of p-biharmonic type. J. Math. Anal. Appl. 348, 730–738 (2008)

    Article  MathSciNet  Google Scholar 

  28. Wu, X.: Existence of nontrivial solutions and high energy solutions for Schrödinger–Kirchhoff-type equations in \({{ R}}^N\). Nonlinear Anal. Real World Appl. 12(2), 1278–1287 (2011)

    Article  MathSciNet  Google Scholar 

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  1. Departamento de Ciências Exatas e Naturais, Universidade Federal Rural do Semi-Árido, Pau dos Ferros, RN, 59900-000, Brazil

    Antônio de Pádua Farias de Souza Filho

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Souza Filho, A.d.P.F.d. Critical nonhomogeneous fourth-order Schrödinger–Kirchhoff-type equations. J Elliptic Parabol Equ 10, 1–18 (2024). https://doi.org/10.1007/s41808-023-00249-1

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  • DOI: https://doi.org/10.1007/s41808-023-00249-1

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