Abstract
This paper concerns with the existence of solutions for the following fractional Kirchhoff problem with critical nonlinearity:
where (−Δ)s is the fractional Laplacian operator with 0 < s < 1, 2 s * = 2N/(N − 2s), N > 2s, p ∈ (1, 2 s *), θ ∈ [1, 2 s */2), h is a nonnegative function and λ a real positive parameter. Using the Ekeland variational principle and the mountain pass theorem, we obtain the existence and multiplicity of solutions for the above problem for suitable parameter λ > 0. Furthermore, under some appropriate assumptions, our result can be extended to the setting of a class of nonlocal integro-differential equations. The remarkable feature of this paper is the fact that the coefficient of fractional Laplace operator could be zero at zero, which implies that the above Kirchhoff problem is degenerate. Hence our results are new even in the Laplacian case.
Similar content being viewed by others
References
Alves C O. Multiple positive solutions for equations involving critical Sobolev exponent in RN. Electron J Differential Equations, 1997, 13: 1–10
Alves C O, Goncalves J V, Miyagaki O H. Multiple positive solutions for semilinear elliptic equations in RN involving critical exponents. Nonlinear Anal, 1998, 32: 41–51
Ambrosetti A, Rabinowitz P. Dual variational methods in critical point theory and applications. J Funct Anal, 1973, 14: 349–381
Applebaum D. Lévy processes—from probability to finance quantum groups. Notices Amer Math Soc, 2004, 51: 1336–1347
Autuori G, Fiscella A, Pucci P. Stationary Kirchhoff problems involving a fractional elliptic operator and a critical nonlinearity. Nonlinear Anal, 2015, 125: 699–714
Autuori G, Pucci P. Elliptic problems involving the fractional Laplacian in RN. J Differential Equations, 2013, 255: 2340–2362
Barrios B, Colorado E, de Pablo A, et al. On some critical problems for the fractional Laplacian operator. J Differential Equations, 2012, 252: 6133–6162
Bisci G M, Rădulescu V. Ground state solutions of scalar field fractional for Schrödinger equations. Calc Var Partial Differential Equations, 2015, 54: 2985–3008
Bisci G M, Rădulescu V, Servadei R. Variational Methods for Nonlocal Fractional Problems. Cambridge: Cambridge University Press, 2016
Bisci G M, Servadei R. A bifurcation result for nonlocal fractional equations. Anal Appl, 2015, 13: 371–394
Bisci G M, Servadei R. Lower semicontinuity of functionals of fractional type and applications to nonlocal equations with critical Sobolev exponent. Adv Difference Equ, 2015, 20: 635–660
Brézis H, Lieb E. A relation between pointwise convergence of functionals and convergence of functionals. Proc Amer Math Soc, 1983, 88: 486–490
Caffarelli L. Non-local diffusions, drifts and games. In: Nonlinear Partial Differential Equations. Abel Symposia, vol. 7. Berlin-Heidelberg: Springer, 2012, 37–52
Caffarelli L, Silvestre L. An extension problem related to the fractional Laplacian. Comm Partial Differential Equations, 2007, 32: 1245–1260
Cao D M, Li G B, Zhou H S. Multiple solutions for nonhomogeneous elliptic equations with critical Sobolev exponent. Proc Roy Soc Edinburgh Sect A, 1994, 124: 1177–1191
Chabrowski J. On multiple solutions for the non-homogeneous p-Laplacian with a critical Sobolev exponent. Differ Integral Equ Appl, 1995, 8: 705–716
Chang X, Wang Z Q. Ground state of scalar field equations involving a fractional Laplacian with general nonlinearity. Nonlinearity, 2013, 26: 479–494
Cotsiolis A, Tavoularis N K. Best constants for Sobolev inequalities for higher order fractional derivatives. J Math Anal Appl, 2004, 295: 225–236
Di Blasio G, Volzone B. Comparison and regularity results for the fractional Laplacian via symmetrization methods. J Differential Equations, 2012, 253: 2593–2615
Di Nezza E, Palatucci G, Valdinoci E. Hitchhiker’s guide to the fractional Sobolev spaces. Bull Sci Math, 2012, 136: 521–573
Ekeland I. On the variational principle. J Math Anal Appl, 1974, 47: 324–353
Fiscella A, Valdinoci E. A critical Kirchhoff type problem involving a nonlocal operator. Nonliear Anal, 2014, 94: 156–170
Goncalves J V, Alves C O. Existence of positive solutions for m-Laplacian equations in RN involving critical exponents. Nonlinear Anal, 1998, 32: 53–70
Laskin N. Fractional quantum mechanics and Lévy path integrals. Phys Lett A, 2000, 268: 298–305
Laskin N. Fractional Schrödinger equation. Phys Rev E, 2002, 66: 056108
Mawhin J, Willem M. Critical Point Theory and Hamilton Systems. Berlin: Springer-Verlag, 1989
Pucci P, Saldi S. Critical stationary Kirchhoff equations in RN involving nonlocal operators. Rev Mat Iberoam, 2016, 32: 1–22
Pucci P, Xiang M Q, Zhang B L. Multiple solutions for nonhomogeneous Schrödinger-Kirchhoff type equations involving the fractional p-Laplacian in RN. Calc Var Partial Differential Equations, 2015, 54: 2785–2806
Pucci P, Xiang M Q, Zhang B L. Existence and multiplicity of entire solutions for fractional p-Kirchhoff equations. Adv Nonlinear Anal, 2016, 5: 27–55
Ros-Oton X, Serra J. Nonexistence results for nonlocal equations with critical and supercritical nonlinearities. Comm Partial Differential Equations, 2015, 40: 115–133
Ros-Oton X, Serra J, Valdinoci E. Pohozaev identies for anisotropic integro-differential operators. ArXiv:1502.01431v1, 2015
Secchi S. Ground state solutions for nonlinear fractional Schrödinger in RN. J Math Phys, 2013, 54: 031501
Servadei R, Valdinoci E. Mountain Pass solutions for non-local elliptic operators. J Math Anal Appl, 2012, 389: 887–898
Servadei R, Valdinoci E. A Brézis-Nirenberg result for non-local critical equations in low dimension. Commun Pure Appl Anal, 2013, 12: 2445–2464
Servadei R, Valdinoci E. The Brézis-Nirenberg result for the fractional Laplacian. Trans Amer Math Soc, 2015, 367: 67–102
Servadei R, Valdinoci E. Fractional Laplacian equations with critical Sobolev exponent. Rev Mat Complut, 2015, 28: 655–676
Xiang M Q, Bisci G M, Tian G H, et al. Infinitely many solutions for the stationary Kirchhoff problems involving the fractional p-Laplacian. Nonlinearity, 2016, 29: 357–374
Xiang M Q, Zhang B L. Degenerate Kirchhoff problems involving the fractional p-Laplacian without the (AR) condition. Complex Var Elliptic Equ, 2015, 60: 1277–1287
Xiang M Q, Zhang B L, Ferrara M. Multiplicity results for the nonhomogeneous fractional p-Kirchhoff equations with concave-convex nonlinearities. Proc Roy Soc A, 2015, 471: 1–14
Xiang M Q, Zhang B L, Rădulescu V. Existence of solutions for perturbed fractional p-Laplacian equations. J Differential Equations, 2016, 260: 1392–1413
Xiang M Q, Zhang B L, Rădulescu V. Multiplicity of solutions for a class of quasilinear Kirchhoff system involving the fractional p-Laplacian. Nonlinearity, 2016, 29: 3186–3205
Acknowledgements
This work was supported by National Natural Science Foundation of China (Grant Nos. 11601515 and 11401574), the Fundamental Research Funds for the Central Universities (Grant No. 3122015L014), and the Doctoral Research Foundation of Heilongjiang Institute of Technology (Grant No. 2013BJ15). The authors thank Professor Giovanni Molica Bisci for useful suggestions and comments.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Xiang, M., Zhang, B. & Qiu, H. Existence of solutions for a critical fractional Kirchhoff type problem in ℝN . Sci. China Math. 60, 1647–1660 (2017). https://doi.org/10.1007/s11425-015-0792-2
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11425-015-0792-2