Abstract
In this paper, we study the following nonlinear problem of Kirchhoff type with critical Sobolev exponent
where a, b > 0, 4 < q < 6, and \({\lambda}\) is a positive parameter. Under certain assumptions on f(x) and g(x) and \({\lambda}\) is small enough, we obtain a relationship between the number of positive solutions and the topology of the global maximum set of g. The Nehari manifold and Ljusternik–Schnirelmann category are the main tools in our study. Moreover, using the Mountain Pass Theorem, we give an existence result about \({\lambda}\) large.
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Brézis H., Nirenberg L.: Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents. Commun. Pure Appl. Math. 36, 437–477 (1983)
Alves C.O., Correa F.J.S.A., Ma T.F.: Positive solutions for a quasilinear elliptic equation of Kirchhoff type. Comput. Math. Appl. 49(1), 85–93 (2005)
Fan H.: Existence and concentration of ground state solutions for a Kirchhoff type problem. Electron. J. Differ. Equ. 5, 1–18 (2016)
Yang Y., Zhang J.: Positive and negative solutions of a class of nonlocal problems. Nonlinear Anal. 73, 25–30 (2010)
Mao A., Zhang Z.: Sign-changing and multiple solutions of Kirchhoff type problems without the P.S. condition. Nonlinear Anal. 70, 1275–1287 (2009)
Zhang Z., Perera K.: Sign-changing solutions of Kirchhoff type problems via invariant sets of descent flow. J. Math. Anal. Appl. 317, 456–463 (2006)
Jin J., Wu X.: Infinitely many radial solutions for Kirchhoff-type problems in \({\mathbb{R}^3}\). J. Math. Anal. Appl. 369, 564–574 (2010)
Fan H., Liu X.: Multiple positive solutions of degenerate nonlocal problems on unbounded domain. Math. Methods Appl. Sci. 38(7), 1282–1291 (2015)
He X., Zou W.: Existence and concentration behavior of positive solutions for a Kirchhoff equation in \({\mathbb{R}^3}\). J. Differ. Equ. 252, 1813–1834 (2012)
Figueiredo G., Junior J.: Multiplicity and concentration of positive solutions for a Schrödinger–Kirchhoff-type problem via penalization method. ESAIM Control Optim. Calc. Var. 20, 389–415 (2014)
He, Y., Li, G., Peng, S.: Concentrating bound states for Kirchhoff type problems in \({\mathbb{R}^3}\) involving critical Sobolev exponents. arXiv:1306.0122 [math.AP]
Liu, Z., Guo, S.: Existence and concentration of positive ground states for a Kirchhoff equation involving critical Sobolev exponent. Z. Angew. Math. Phys. doi:10.1007/s0033-014-0431-8 (2015)
Li Y., Li F., Shi J.: Existence of positive solutions to Kirchhoff type problems with zero mass. J. Math. Anal. Appl. 410, 361–374 (2014)
Fan H., Liu X.: Positive and negative solutions for a class of Kirchhoff type problems on unbounded domain. Nonlinear Anal. Theory Methods Appl. 114, 186–196 (2015)
Fan H.: Multiple positive solutions for a critical elliptic problem with concave and convex nonlinearities. Electron. J. Differ. Equ. 82, 1–14 (2014)
Lin H.: Positive solutions for nonhomogeneous elliptic equations involving critical Sobolev exponent. Nonlinear Anal. 75, 2660–2671 (2012)
Li T., Wu T.F.: Multiple positive solutions for a Dirichlet problem involving critical Sobolev exponent. J. Math. Anal. Appl. 369, 245–257 (2010)
Fan H.: Multiple positive solutions for semi-linear elliptic systems with sign-changing weight. J. Math. Anal. Appl. 409(1), 399–408 (2014)
Li, Q., Yang, Z.: Multiple positive solutions for a quasilinear elliptic system with critical exponent and sign-changing weight. Comput. Math. Appl. 67(10), 1848–1863 (2014). doi:10.1016/j.camwa.2014.03.018
Fan H.: Multiple positive solutions for a critical elliptic system with concave and convex nonlinearities. Nonlinear Anal. Real World Appl. 18, 14–22 (2014)
Fan H.: Multiple positive solutions for a class of Kirchhoff type problems involving critical Sobolev exponents. J. Math. Anal. Appl. 431(1), 150–168 (2015)
Lions P.L.: The concentration-compactness principle in the calculus of variations: the locally compact case I, II. Ann. Inst. H.Poincaré Anal. Non Linéaire 1, 223–283 (1984)
Struwe M.: Variational Methods, 2nd edn. Springer, Berlin (1996)
Cingolani S., Lazzo M.: Multiple positive solutions to nonlinear Schrödinger equations with competing potential functions. J. Differ. Equ. 160, 118–138 (2000)
Benci V., Cerami G.: Existence of positive solutions of the equation \({-\Delta u+a(x)u=u^\frac{N+2}{N-2}}\). J. Funct. Anal. 88, 90–117 (1990)
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This work is supported by the Natural Science Foundation of Jiangsu Province (Grant No. BK20150168).
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Fan, H. Multiple positive solutions for Kirchhoff-type problems in \({\mathbb{R}^3}\) involving critical Sobolev exponents. Z. Angew. Math. Phys. 67, 129 (2016). https://doi.org/10.1007/s00033-016-0723-2
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DOI: https://doi.org/10.1007/s00033-016-0723-2