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Nehari–Pohožaev-type ground state solutions of Kirchhoff-type equation with singular potential and critical exponent

Part of: Elliptic equations and systems Equations of mathematical physics and other areas of application

Published online by Cambridge University Press:  21 June 2021

Yu Su  and
Senli Liu
Yu Su
Affiliation:
School of Mathematics and Big Data, Anhui University of Science and Technology, Huainan, Anhui232001, P.R. China e-mail: yusumath@aust.edu.cn
Senli Liu*
Affiliation:
School of Mathematics and Statistics, Central South University, Changsha, Hunan410083, P.R. China
*
e-mail: mathliusl@csu.edu.cn
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Abstract

In this paper, we focus on a Kirchhoff-type equation with singular potential and critical exponent. By virtue of the generalized version of Lions-type theorem and the Nehari–Pohožaev manifold, we established the existence of Nehari–Pohožaev-type ground state solutions to the mentioned equation. Some recent results from the literature are generally improved and extended.

Keywords

Kirchhoff-type equationLions-type theoremNehari–Pohožaev manifoldsingular potentialground state solution

MSC classification

Primary: 35J20: Variational methods for second-order elliptic equations 35J75: Singular elliptic equations 35Q55: NLS-like equations (nonlinear Schrödinger)
Type
Article
Information
Canadian Mathematical Bulletin , Volume 65 , Issue 2 , June 2022 , pp. 473 - 495
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Copyright
© Canadian Mathematical Society 2021

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Footnotes

Y. Su is supported by the Key Program of University Natural Science Research Fund of Anhui Province (KJ2020A0292). S. Liu is supported by the Fundamental Research Funds for the Central Universities of Central South University 2019zzts210.

References

Alves, C., Corrêa, F., and Ma, T., Positive solutions for a quasilinear elliptic equation of Kirchhoff type. Comput. Math. Appl. 49(2005), no. 1, 85–93.CrossRefGoogle Scholar
Arosio, A., Averaged evolution equations. The Kirchhoff string and its treatment in scales of Banach spaces . In: Functional analytic methods in complex analysis and applications to partial differential equations (Trieste, 1993), World Scientific Publishing, River Edge, NJ, 1995, 220–254.Google Scholar
Badiale, M., Benci, V., and Rolando, S., A nonlinear elliptic equation with singular potential and applications to nonlinear field equations. J. Eur. Math. Soc. 9(2007), no. 3, 355–381.CrossRefGoogle Scholar
Badiale, M., Greco, S., and Rolando, S., Radial solutions of a biharmonic equation with vanishing or singular radial potentials. Nonlinear Anal. 185(2019), 97–122.CrossRefGoogle Scholar
Badiale, M., Guida, M., and Rolando, S., A nonexistence result for a nonlinear elliptic equation with singular and decaying potential. Commun. Contemp. Math. 17(2015), no. 4, 1450024, 21.CrossRefGoogle Scholar
Badiale, M. and Rolando, S., A note on nonlinear elliptic problems with singular potentials. Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl. 17(2006), no. 1, 1–13.Google Scholar
Bernstein, S., Sur une classe d’équations fonctionnelles aux dérivées partielles. Bull. Acad. Sci. URSS. Sér. Math. [Izvestia Akad. Nauk SSSR] 4(1940), 17–26.Google Scholar
Brézis, H. and Lieb, E., A relation between pointwise convergence of functions and convergence of functionals. Proc. Amer. Math. Soc. 88(1983), no. 3, 486–490.CrossRefGoogle Scholar
Cao, D. and Yan, S., Infinitely many solutions for an elliptic problem involving critical Sobolev growth and Hardy potential. Calc. Var. 38(2010), nos. 3–4, 471–501.CrossRefGoogle Scholar
Carrião, P., Demarque, R., and Miyagaki, O., Nonlinear biharmonic problems with singular potentials. Commun. Pure Appl. Anal. 13(2014), no. 6, 2141–2154.CrossRefGoogle Scholar
Cassani, D., Liu, Z., Tarsi, C., and Zhang, J., Multiplicity of sign-changing solutions for Kirchhoff-type equations. Nonlinear Anal. 186(2019), 145–161.CrossRefGoogle Scholar
Chen, Z. and Zou, W., On an elliptic problem with critical exponent and Hardy potential. J. Differential Equations 252(2012), no. 2, 969–987.CrossRefGoogle Scholar
Conti, M., Crotti, S., and Pardo, D., On the existence of positive solutions for a class of singular elliptic equations. Adv. Differential Equations 3(1998), no. 1, 111–132.Google Scholar
Figueiredo, G. and Santos, J., Multiplicity of solutions for a Kirchhoff equation with subcritical or critical growth. Differential Integral Equations 25(2012), nos. 9–10, 853–868.Google Scholar
Kirchhoff, G., Mechanik. Teubner, Leipzig, 1883.Google Scholar
Li, A. and Su, J., Existence and multiplicity of solutions for Kirchhoff-type equation with radial potentials in ℝ 3 . Z. Angew. Math. Phys. 66(2015), no. 6, 3147–3158.CrossRefGoogle Scholar
Lions, J., On some questions in boundary value problems of mathematical physics . In: Contemporary developments in continuum mechanics and partial differential equations, North-Holland Math. Stud., 30, North-Holland, Amsterdam and New York, 1978, 284–346.Google Scholar
Naimen, D., The critical problem of Kirchhoff type elliptic equations in dimension four. J. Differential Equations 257(2014), no. 4, 1168–1193.CrossRefGoogle Scholar
Naimen, D., Positive solutions of Kirchhoff type elliptic equations involving a critical Sobolev exponent. NoDEA Nonlinear Differential Equations Appl. 21(2014), no. 6, 885–914.CrossRefGoogle Scholar
Perera, K. and Zhang, Z., Nontrivial solutions of Kirchhoff-type problems via the Yang index. J. Differential Equations 221(2006), no. 1, 246–255.CrossRefGoogle Scholar
Pohožaev, S., A certain class of quasilinear hyperbolic equations. Mat. Sb. (N.S.) 96(1975), 152–166, 168 (in Russian).Google Scholar
Rolando, S., Multiple nonradial solutions for a nonlinear elliptic problem with singular and decaying radial potential. Adv. Nonlinear Anal. 8(2019), no. 1, 885–901.CrossRefGoogle Scholar
Strauss, A., Existence of solitary waves in higher dimensions. Comm. Math. Phys. 55(1977), no. 2, 149–162.CrossRefGoogle Scholar
Su, J., Wang, Z., and Willem, M., Nonlinear Schrödinger equations with unbounded and decaying radial potentials. Commun. Contemp. Math. 9(2007), no. 4, 571–583.CrossRefGoogle Scholar
Su, Y., Positive solution to Schrödinger equation with singular potential and double critical exponents. Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl. 31(2020), no. 4, 667–698.CrossRefGoogle Scholar
Willem, M., Minimax theorems. Progress in Nonlinear Differential Equations and Their Applications, 24, Birkhäuser Boston, Inc., Boston, MA, 1996.CrossRefGoogle Scholar