Infinitely many sign-changing solutions for nonlinear fractional Kirchhoff equations

References

• Kirchhoff G. Vorlesungen über mechanik. Basel: Birkhäuser ; 1883.
   Google Scholar
• Bernstein S. Sur une classe d'équations fonctionnelles aux dérivées partielles. Bull Acad Sci URSS Sér Math [Izvestia Akad Nauk SSSR]. 1940;4:17–26.
   Google Scholar
• Pohozaev SI. A certain class of quasilinear hyperbolic equations. Mat Sb (NS). 1975;96(138):152–166.
   Google Scholar
• Lions JL. On some questions in boundary value problems of mathematical physics. Contemp Dev Continuum Mech Partial Differ Equ. 1978;30:284–346.(Proc. Internat. Sympos., Inst. Mat., Univ. Fed. Rio de Janeiro, Rio de Janeiro, 1977).
   Google Scholar
• Arosio A, Panizzi S. On the well-posedness of the Kirchhoff string. Trans Amer Math Soc. 1996;348(1):305–330.
   Web of Science ®Google Scholar
• D'Ancona P, Spagnolo S. Global solvability for the degenerate Kirchhoff equation with real analytic data. Invent Math. 1992;108(2):247–262.
   Web of Science ®Google Scholar
• Chen C, Kuo Y, Wu T. The nehari manifold for a Kirchhoff type problem involving sign-changing weight functions. J Differ Equ. 2011;250(4):1876–1908.
   Web of Science ®Google Scholar
• Mao A, Zhang Z. Sign-changing and multiple solutions of Kirchhoff type problems without the P.S. condition. Nonlinear Anal.. 2009;70(3):1275–1287.
   Web of Science ®Google Scholar
• Perera K, Zhang Z. Nontrivial solutions of Kirchhoff-type problems via the Yang index. J Differ Equ. 2006;221(1):246–255.
   Web of Science ®Google Scholar
• Shuai W. Sign-changing solutions for a class of kirchhoff-type problem in bounded domains. J Differ Equ. 2015;259(4):1256–1274.
   Web of Science ®Google Scholar
• Tang X, Cheng B. Ground state sign-changing solutions for Kirchhoff type problems in bounded domains. J Differ Equ. 2016;261(4):2384–2402.
   Web of Science ®Google Scholar
• Zhang Z, Perera K. Sign changing solutions of Kirchhoff type problems via invariant sets of descent flow. J Math Anal Appl. 2006;317(2):456–463.
   Web of Science ®Google Scholar
• Li G, Ye H. Existence of positive ground state solutions for the nonlinear Kirchhoff type equations in R3. J Differ Equ. 2014;257(2):566–600.
   Web of Science ®Google Scholar
• Deng Y, Peng S, Shuai W. Existence and asymptotic behavior of nodal solutions for the kirchhoff-type problems in R3. J Funct Anal. 2015;269(11):3500–3527.
   Web of Science ®Google Scholar
• Li G, Luo P, Peng S, et al. A singularly perturbed Kirchhoff problem revisited. J Differ Equ. 2020;268(2):541–589.
   Web of Science ®Google Scholar
• Chen J, Tang X, Cheng B. Existence and concentration of ground state sign-changing solutions for Kirchhoff type equations with steep potential well and nonlinearity. Topol Methods Nonlinear Anal. 2018;51(1):111–133.
   Web of Science ®Google Scholar
• Chen S, Zhang B, Tang X. Existence and non-existence results for kirchhoff-type problems with convolution nonlinearity. Adv Nonlinear Anal. 2020;9(1):148–167.
   Web of Science ®Google Scholar
• Cheng B, Li G, Tang X. Nehari-type ground state solutions for Kirchhoff type problems in RN. Appl Anal. 2019;98(7):1255–1266.
   Web of Science ®Google Scholar
• Tang X, Chen S.Ground state solutions of Nehari-Pohozaev type for kirchhoff-type problems with general potentials. Calc Var Partial Differ Equ. 2017;56:25. Art. 110.
   Web of Science ®Google Scholar
• Laskin N. Fractional quantum mechanics and Lévy path integrals. Phys Lett A. 2000;268(4-6):298–305.
   Web of Science ®Google Scholar
• Caffarelli L, Silvestre L. An extension problem related to the fractional Laplacian. Comm Partial Differ Equ. 2007;32(7-9):1245–1260.
   Web of Science ®Google Scholar
• Di Nezza E, Palatucci G, Valdinoci E. Hitchhiker's guide to the fractional sobolev spaces. Bull Sci Math. 2012;136(5):521–573.
   Web of Science ®Google Scholar
• Yang X, Yang Z. Existence of multiple non-trivial solutions for a nonlocal problem. AIMS Math. 2019;4(2):299–307.
   Web of Science ®Google Scholar
• Yang Z, Zhao F. Three solutions for a fractional Schrödinger equation with vanishing potentials. Appl Math Lett. 2018;76:90–95.
   Web of Science ®Google Scholar
• Fiscella A, Valdinoci E. A critical Kirchhoff type problem involving a nonlocal operator. Nonlinear Anal. 2014;94:156–170.
   Web of Science ®Google Scholar
• Autuori G, Fiscella A, Pucci P. Stationary Kirchhoff problems involving a fractional elliptic operator and a critical nonlinearity. Nonlinear Anal. 2015;125:699–714.
   Web of Science ®Google Scholar
• Pucci P, Xiang M, Zhang B. Multiple solutions for nonhomogeneous Schrödinger–Kirchhoff type equations involving the fractional p-Laplacian in RN. Calc Var Partial Differ Equ. 2015;54(3):2785–2806.
   Web of Science ®Google Scholar
• Caponi M, Pucci P. Existence theorems for entire solutions of stationary Kirchhoff fractional p-Laplacian equations. Ann Mat Pura Appl (4). 2016;195(6):2099–2129.
   Web of Science ®Google Scholar
• Pucci P, Saldi S. Critical stationary Kirchhoff equations in RN involving nonlocal operators. Rev Mat Iberoam. 2016;32(1):1–22.
   Web of Science ®Google Scholar
• Pucci P, Xiang M, Zhang B. Existence and multiplicity of entire solutions for fractional p-Kirchhoff equations. Adv Nonlinear Anal. 2016;5(1):27–55.
   Web of Science ®Google Scholar
• Molica Bisci G, Radulescu VD, Servadei R. Variational methods for nonlocal fractional problems. Vol. 162, Encyclopedia of Mathematics and its Applications. Cambridge: Cambridge University Press; 2016. (With a foreword by Jean Mawhin).
   Google Scholar
• Zhao L, Liu H, Zhao F. Existence and concentration of solutions for the Schrödinger–Poisson equations with steep well potential. J Differ Equ. 2013;255(1):1–23.
   Web of Science ®Google Scholar
• Zhao L, Zhao F. On the existence of solutions for the Schrödinger–Poisson equations. J Math Anal Appl. 2008;346(1):155–169.
   Web of Science ®Google Scholar
• Jeanjean L. On the existence of bounded Palais-Smale sequences and application to a Landesman–Lazer-type problem set on RN. Proc Roy Soc Edinburgh Sect A. 1999;129(4):787–809.
   Google Scholar
• Wang Y, Liu Y, Cui Y. Multiple sign-changing solutions for nonlinear fractional Kirchhoff equations. Bound Value Probl. 2018;193. 2018.
   Google Scholar
• Liu J, Liu X, Wang Z. Multiple mixed states of nodal solutions for nonlinear Schrödinger systems. Calc Var Partial Differ Equ. 2015;52(3-4):565–586.
   Web of Science ®Google Scholar
• Liu Z, Wang Z, Zhang J. Infinitely many sign-changing solutions for the nonlinear Schrödinger–Poisson system. Ann Mat Pura Appl (4). 2016;195(3):775–794.
   Web of Science ®Google Scholar
• Secchi S. Ground state solutions for nonlinear fractional Schrödinger equations in RN. J Math Phys. 2013;54(3):031501.17.
   Web of Science ®Google Scholar
• Willem M. Minimax theorems, Vol. 24, Progress in Nonlinear Differential Equations and their Applications. Boston Inc., Boston, MA: Birkhäuser, 1996.
   Google Scholar
• Chang X, Wang Z. Ground state of scalar field equations involving a fractional Laplacian with general nonlinearity. Nonlinearity. 2013;26(2):479–494.
   Web of Science ®Google Scholar
• Teng K. Ground state solutions for the non-linear fractional Schrödinger–Poisson system. Appl Anal. 2019;98(11):1959–1996.
   Web of Science ®Google Scholar
• Zhang J, Lou Z, Ji Y, et al. Ground state of Kirchhoff type fractional Schrödinger equations with critical growth. J Math Anal Appl. 2018;462(1):57–83.
   Web of Science ®Google Scholar
• Bartsch T, Liu Z. On a superlinear elliptic p-Laplacian equation. J Differ Equ. 2004;198(1):149–175.
   Web of Science ®Google Scholar