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Ground state solution to the biharmonic equation

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Abstract

The biharmonic equation arises in areas of continuum mechanics, including mechanics of elastic plates and the slow flow of viscous fluids. In this paper, we make an effort to establish the generalized versions of Lions-type theorem under various conditions and then apply them to study the existence of ground state solutions for the biharmonic equation.

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References

  1. Alves, C.O., do Ó, J.M.: Positive solutions of a fourth-order semilinear problem involving critical growth. Adv. Nonlinear Stud. 2(4), 437–458 (2002)

    Article  MathSciNet  Google Scholar 

  2. Alves, C.O., do Ó, J.M., Miyagaki, O.H.: On a class of singular biharmonic problems involving critical exponents. J. Math. Anal. Appl. 277(1), 12–26 (2003)

    Article  MathSciNet  Google Scholar 

  3. Bhakta, M.: Semilinear elliptic equation with biharmonic operator and multiple critical nonlinearities. Adv. Nonlinear Stud. 15(4), 835–848 (2015)

    Article  MathSciNet  Google Scholar 

  4. Bhakta, M., Musina, R.: Entire solutions for a class of variational problems involving the biharmonic operator and Rellich potentials. Nonlinear Anal. 75(9), 3836–3848 (2012)

    Article  MathSciNet  Google Scholar 

  5. Brézis, H., Lieb, E.H.: A relation between pointwise convergence of functions and convergence of functionals. Proc. Am. Math. Soc. 88(3), 486–490 (1983)

    Article  MathSciNet  Google Scholar 

  6. Chabrowski, J., do Ó, J.M.: On some fourth-order semilinear elliptic problems in \({\mathbb{R}}^N\). Nonlinear Anal. 49(6), 861–884 (2002)

    Article  MathSciNet  Google Scholar 

  7. Cotsiolis, A., Tavoularis, N.K.: Best constants for Sobolev inequalities for higher order fractional derivatives. J. Math. Anal. Appl. 295(1), 225–236 (2004)

    Article  MathSciNet  Google Scholar 

  8. D’Ambrosio, L., Jannelli, E.: Nonlinear critical problems for the biharmonic operator with Hardy potential. Calc. Var. Partial Differ. Equ. 54(1), 365–396 (2015)

  9. Deng, Y., Gao, Q., Jin, L.: On the existence of nontrivial solutions for p-harmonic equations on unbounded domains. Nonlinear Anal. 69(12), 4713–4731 (2008)

    Article  MathSciNet  Google Scholar 

  10. Edmunds, D.E., Fortunato, D., Jannelli, E.: Critical exponents, critical dimensions and the biharmonic operator. Arch. Ration. Mech. Anal. 112(3), 269–289 (1990)

    Article  MathSciNet  Google Scholar 

  11. Ercole, G.: Absolute continuity of the best Sobolev constant. J. Math. Anal. Appl. 404(2), 420–428 (2013)

    Article  MathSciNet  Google Scholar 

  12. Frank, R.L., König, T.: Classification of positive singular solutions to a nonlinear biharmonic equation with critical exponent. Anal. PDE 12(4), 1101–1113 (2019)

    Article  MathSciNet  Google Scholar 

  13. Gazzola, F., Grunau, H.C., Sweers, G.: Polyharmonic Boundary Value Problems. Positivity Preserving and Nonlinear Higher Order Elliptic Equations in Bounded Domains. Lecture Notes in Mathematics, vol. 1991. Springer, Berlin (2010)

    MATH  Google Scholar 

  14. Guo, Z., Wei, L.: Radial symmetry of entire solutions of a biharmonic equation with supercritical exponent. Adv. Nonlinear Stud. 19(2), 291–316 (2019)

    Article  MathSciNet  Google Scholar 

  15. Guzmán, C.M., Pastor, A.: On the inhomogeneous biharmonic nonlinear Schrödinger equation: local, global and stability results. Nonlinear Anal. Real World Appl. 56, 103174 (2020). (35 pp)

    Article  MathSciNet  Google Scholar 

  16. Hayek, S.I.: Advanced Mathematical Methods in Science and Engineering. Marcel Dekker, New York (2000)

    MATH  Google Scholar 

  17. Jannelli, E.: Critical behavior for the polyharmonic operator with Hardy potential. Nonlinear Anal. 119, 443–456 (2015)

    Article  MathSciNet  Google Scholar 

  18. Kefi, K., Radulescu, V.D.: On a \(p(x)\)-biharmonic problem with singular weights. Z. Angew. Math. Phys. 68:80, 1–13 (2017)

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  20. Lions, P.L.: The concentration-compactness principle in the calculus of variations. The locally compact case, I. Ann. Inst. H. Poincarè Anal. Non Linéaire 1, 109–145 (1984)

    Article  MathSciNet  Google Scholar 

  21. Lions, P.L.: The concentration-compactness principle in the calculus of variations. The locally compact case, II. Ann. Inst. H. Poincarè Anal. Non Linéaire 1, 223–283 (1984)

    Article  MathSciNet  Google Scholar 

  22. Liu, H.L., Chen, H.B.: Ground-state solution for a class of biharmonic equations including critical exponent. Z. Angew. Math. Phys. 66, 3333–3343 (2015)

    Article  MathSciNet  Google Scholar 

  23. Noussair, E.S., Swanson, C.A., Yang, J.: Transcritical biharmonic equations in \({\mathbb{R}}^{N}\). Funkcial. Ekvac. 35(3), 533–543 (1992)

    MathSciNet  MATH  Google Scholar 

  24. Palatucci, G., Pisante, A.: Improved Sobolev embeddings, profile decomposition, and concentration-compactness for fractional Sobolev spaces. Calc. Var. Partial Differ. Equ. 50(3–4), 799–829 (2014)

    Article  MathSciNet  Google Scholar 

  25. Sun, J., Chu, J., Wu, T.: Existence and multiplicity of nontrivial solutions for some biharmonic equations with p-Laplacian. J. Differ. Equ. 262(2), 945–977 (2017)

    Article  MathSciNet  Google Scholar 

  26. Sun, J., Wu, T.: Existence of nontrivial solutions for a biharmonic equation with p-Laplacian and singular sign-changing potential. Appl. Math. Lett. 66, 61–667 (2017)

    Article  MathSciNet  Google Scholar 

  27. Ye, Y., Tang, C.: Infinitely many solutions for fourth-order elliptic equations. J. Math. Anal. Appl. 394(2), 841–854 (2012)

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. School of Mathematical and Statistical Sciences, University of Texas Rio Grande Valley, Edinburg, TX, 78539, USA

    Zhaosheng Feng

  2. School of Mathematics and Big Data, Anhui University of Science and Technology, Huainan, 232001, Anhui, China

    Yu Su

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  1. Zhaosheng Feng
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  2. Yu Su
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Correspondence to Zhaosheng Feng.

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Feng, Z., Su, Y. Ground state solution to the biharmonic equation. Z. Angew. Math. Phys. 73, 15 (2022). https://doi.org/10.1007/s00033-021-01643-2

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  • DOI: https://doi.org/10.1007/s00033-021-01643-2

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