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The Existence of Normalized Solutions to the Kirchhoff Equation with Potential and Sobolev Critical Nonlinearities

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Abstract

In the present paper, we study the existence of normalized solutions \((u_c, \lambda _c)\in H^1(\mathbb {R}^3)\times \mathbb {R}\) to the following Kirchhoff problem

$$\begin{aligned} -\left( a+b\int _{\mathbb {R}^3} |\nabla u|^2 \textrm{d}x\right) \Delta u+V(x)u +\lambda u=g(u) +|u|^{4}u \quad \hbox {in}~\mathbb {R}^3, \end{aligned}$$

satisfying the normalization constraint \( \displaystyle \int _{\mathbb {R}^3}u^2\textrm{d}x=c, \) where \(a,b,c>0\) are prescribed constants, and the nonlinearities g(s) are very general and of mass super-critical. Under some suitable assumptions on V(x) and g(u), we will prove that the above problem has a ground state normalized solutions for any given \(c>0\), by studying a constraint problem on a Nehari–Pohozaev manifold.

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References

  1. Alves, C.O., Crrea, F.J.S.A.: On existence of solutions for a class of problem involving a nonlinear operator. Commun. Appl. Nonlinear Anal. 8, 43–56 (2001)

    MathSciNet  Google Scholar 

  2. Benci, V., Cerami, G.: Existence of positive solutions of the equation \(-\Delta u+a(x)u=u^\frac{N+2}{N-2}\) in \({\mathbb{R} }^N\). J. Funct. Anal. 88, 90–117 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  3. He, X.M., Zou, W.M.: Existence and concentration behavior of positive solutions for a Kirchhoff equation in \(\mathbb{R} ^3\). J. Differ. Equ. 252, 1813–1834 (2012)

    Article  MATH  Google Scholar 

  4. Figueiredo, G.M., Ikoma, N., Santos Junior, J.R.: Existence and concentration result for the Kirchhoff type equations with general nonlinearities. Arch. Ration. Mech. Anal. 213, 931–979 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  5. Guo, Z.J.: Ground states for Kirchhoff equations without compact condition. J. Differ. Equ. 259, 2884–2902 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  6. He, Y., Li, G.B.: Standing waves for a class of Kirchhoff type problems in \(\mathbb{R} ^3\) involving critical Sobolev exponents. Calc. Var. 54, 3067–3106 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  7. He, Y., Li, G.B., Peng, S.J.: Concentrating bound states for Kirchhoff type problems in \({\mathbb{R} }^3\) involving critical Sobolev exponents. Adv. Nonlinear Stud. 14, 441–468 (2014)

    Article  Google Scholar 

  8. Kirchhoff, G.: Mechanik. Teubner, Leipzig (1883)

    MATH  Google Scholar 

  9. Li, G.B., Ye, H.Y.: Existence of positive ground state solutions for the nonlinear Kirchhoff type equations in \({\mathbb{R} }^3\). J. Differ. Equ. 257, 566–600 (2014)

    Article  MATH  Google Scholar 

  10. 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 International Symposium. Inst. Mat., Univ. Fed. Rio de Janeino, 1977, in: North-Holl. Math. Stud., vol. 30, North-Hollad, Amsterdam, pp. 284-346 (1978)

  11. Luo, X., Wang, Q.F.: Existence and asymptotic behavior of high energy normalized solutions for the Kirchhoff type equations in \({\mathbb{R} }^3\). Nonlinear Anal. 33, 19–32 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  12. Wang, J., Tian, L., Xu, J., Zhang, F.: Multiplicity and concentration of positive solutions for a Kirchhoff type problem with critical growth. J. Differ. Equ. 253, 2314–2351 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  13. Wu, X.: Existence of nontrivial solutions and high energy solutions for Schrödinger-Kirchhoff-type equations in \({\mathbb{R} }^3\). Nonlinear Anal. Real World Appl. 12, 1278–1287 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  14. Ye, H.Y.: The sharp existence of constrained minimizers for a class of nonlinear Kirchhoff equations. Math. Methods Appl. Sci. 38, 2663–2679 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  15. Ye, H.Y.: The existence of normalized solutions for \(L^2\)-critical constrained problems related to Kirchhoff equations. Z. Angew. Math. Phys. 66, 1483–1497 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  16. Li, G.B., Luo, X., Yang, T.: Normalized solutions to a class of Kirchhoff equations with Sobolev critical expoent. Ann. Fenn. Math. 47, 895–925 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  17. Willem, M.: Minimax Theorems, Progress in Nonlinear Differential Equations and Their Applications, Vol.24, Birkhäuser Boston, Inc., Boston (1996)

  18. Wang, Z.Z., Zeng, X.Y., Zhang, Y.M.: Multi-peak solutions of Kirchhoff equations involving subcritical or critical Sobolev exponents. Math. Meth. Appl. Sci. 43, 5151–5161 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  19. Zeng, X.Y., Zhang, Y.M.: Existence and uniqueness of normalized solutions for the Kirchhoff equation. Appl. Math. Lett. 74, 52–59 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  20. Zeng, Y.L., Chen, K.S.: Remarks on normalized solutions for \(L^2\)-critical Kirchhoff problems. Taiwan. J. Math. 20, 617–627 (2016)

    Article  MATH  Google Scholar 

  21. Deng, Y.B., Peng, S.J., Shuai, W.: Existence and asymptotic behavior of nodal solutions for the Kirchhoff-type problems in \({\mathbb{R} }^3\). J. Funct. Anal. 269, 3500–3527 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  22. Ding, Y.H., Zhong, X.X.: Normalized solution to the Schödinger equation with potential and general nonlinear term: Mass super-critical case. J. Differ. Equ. 334, 194–215 (2022)

    Article  MATH  Google Scholar 

  23. Figueiredo, G.M., Ikoma, N., Santos Junior, J.R.: Existence and concentration result for the Kirchhoff type equations with general nonlinearities. Arch. Ration. Mech. Anal. 213, 931–979 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  24. He, Q.H., Lv, Z.Y., Zhang, Y.M., Zhong, X.X.: Positive normalized solution to the Kirchhoff equation with general nonlinearities of mass super-critical. J. Differ. Equ. 356, 375–406 (2023)

    Article  MATH  Google Scholar 

  25. Cui, L.L., He, Q. H., Lv, Z.Y., Zhong, X.X.: The existence of normalized solutions for a Kirchhoff type equations with potential in \({\mathbb{R}}^3\). arXiv:2304.07194 (2023)

  26. He, Y.: Concentrating bounded states for a class of singularly perturbed Kirchhoff type equations with a general nonlinearity. J. Differ. Equ. 261, 6178–6220 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  27. Jeanjean, L.: Existence of solutions with prescribed norm for semilinear elliptic equations. Nonlinear Anal. Theory T. M. A. 28, 1633–1659 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  28. Jeanjean, L., Zhang, J.J., Zhong, X.X.: A global branch approach to normalized solutions for the Schrödinger equation. arXiv:2112.05869 (2021)

  29. Kwong, M.K.: Uniqueness of positive solutions of \(\Delta u-u+u^p=0\) in \({\mathbb{R} }^n\). Arch. Ration. Mech. Anal. 105, 243–266 (1989)

    Article  MATH  Google Scholar 

  30. Li, G.B., Luo, P., Peng, S.J., Wang, C.H., Xiang, C.-L.: A singularly perturbed Kirchhoff problem revisited. J. Differ. Equ. 268, 541–589 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  31. Luo, P., Peng, S.J., Wang, C.H., Xiang, C.-L.: Multi-peak positive solutions to a class of Kirchhoff equations. Proc. R. Soc. Edinb. A 149, 1097–1122 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  32. Weinstein, M.I.: Nonlinear Schrödinger equations and sharp interpolation estimates. Commun. Math. Phys. 87, 567–576 (1983)

    Article  MATH  Google Scholar 

  33. Yang, Z.: A new observation for the normalized solution of the Schrödinger equation. Arch. Math. 115, 329–338 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  34. Zeng, X.Y., Zhang, J.J., Zhang, Y.M., Zhong, X.X.: Positive normalized solution to the Kirchhoff equation with general nonlinearities. arXiv:2112.10293 (2021)

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Authors and Affiliations

  1. College of Mathematics and Information Science, Guangxi University, Nanning, 530003, People’s Republic of China

    Qihan He

  2. School of Mathematical Sciences, Beijing Normal University, Beijing, 100875, People’s Republic of China

    Zongyan Lv & Zhongwei Tang

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  1. Qihan He
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  2. Zongyan Lv
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Correspondence to Zongyan Lv.

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The research was supported by the Natural Science Foundation of China (No. 12061012) and the special foundation for Guangxi Ba Gui Scholars.

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He, Q., Lv, Z. & Tang, Z. The Existence of Normalized Solutions to the Kirchhoff Equation with Potential and Sobolev Critical Nonlinearities. J Geom Anal 33, 236 (2023). https://doi.org/10.1007/s12220-023-01298-7

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