Advanced Search
Article Contents
Article Contents

Existence and concentration for Kirchhoff type equations around topologically critical points of the potential

Abstract Full Text(HTML) Related Papers Cited by
  • We consider the existence and concentration of solutions for the following Kirchhoff Type Equations

    $-\varepsilon^2 M \left( \varepsilon^{2-N} \displaystyle \int_{\mathbb{R}^N} |\nabla v|^2dx \right)Δ v+V(x)v=f(v), \mathrm{in} \ \mathbb{R}^N.$

    Under suitable conditions on the continuous functions $M$, $V$ and $f$, we obtain a family of positive solutions concentrating around the local maximum or saddle points of $V$. Moreover with appropriate assumptions on $V$, we also have multiple solutions clustering respectively around three kinds of critical points of $V$.

    Mathematics Subject Classification: 35B05, 35B45.


    \begin{equation} \\ \end{equation}
  • 加载中
  • [1] A. Azzollini, The elliptic Kirchhoff equation in $R^N$ perturbed by a local nonlinearity, Differ. Integ. Equ., 25 (2012), 543-554. 
    [2] F. J. Almgren and E. H. Lieb, Symmetric decreasing rearrangement is sometimes continuous, J. Amer. Math. Soc., 2 (1989), 683-773.  doi: 10.2307/1990893.
    [3] P. AveniaA. Pomponio and D. Ruiz, Semiclassical states for the nonlinear Schrödinger equation on saddle points of the potential via variational methods, J. Func. Anal., 262 (2012), 4600-4633.  doi: 10.1016/j.jfa.2012.03.009.
    [4] H. Brezis and T. Kato, Remarks on the Schrödinger operator with singular complex potentials, J. Math. Pures Appl., 58 (1979), 137-151. 
    [5] H. Berestycki and P. L. Lions, Nonlinear scalar field equations Ⅰ, existence of a ground state, Arch. Rational. Mech. Anal., 82 (1983), 313-345.  doi: 10.1007/BF00250555.
    [6] J. Byeon and L. Jeanjean, Standing waves for nonlinear Schrödinger equations with a general nonlinearity, Arch. Rational. Mech. Anal. , 185 (2007), 185-200; Arch. Rational. Mech. Anal. , 190 (2008), 549-551. doi: 10.1007/s00205-008-0178-5.
    [7] J. Byeon and K. Tanaka, Semi-classical standing waves for nonlinear Schrödinger equations at structurally stable critical points of the potential, J. Eur. Math. Soc., 15 (2013), 1859-1899.  doi: 10.4171/JEMS/407.
    [8] J. Byeon and K. Tanaka, Semiclassical standing waves with clustering peaks for nonlinear Schrödinger equations, Mem. Amer. Math. Soc., 229 (2014). 
    [9] C. ChenY. Kuo and T. Wu, The Nehari manifold for a Kirchhoff type problem involving sign-changing weight functions, J. Differ. Equ., 250 (2011), 1876-1908.  doi: 10.1016/j.jde.2010.11.017.
    [10] V. Coti Zelati and P. H. Rabinowitz, Homoclinic type solutions for a semilinear elliptic PDE on $R^n$, Comm. Pure App. Math., 45 (1992), 1217-1269.  doi: 10.1002/cpa.3160451002.
    [11] M. Del Pino and P. L. Felmer, Local mountain passes for semilinear elliptic problems in unbounded domains, Calc. Var. Par. Differ. Equ., 4 (1996), 121-137.  doi: 10.1007/BF01189950.
    [12] M. Del Pino and P. L. Felmer, Multi-peak bound states for nonlinear Schrödinger equations, Ann. Inst. H. Poincaré Anal. Non Linéaire., 15 (1998), 127-149.  doi: 10.1016/S0294-1449(97)89296-7.
    [13] M. Del Pino and P. L. Felmer, Semi-classical states for nonlinear Schrödinger equations, J. Func. Anal., 149 (1997), 245-265.  doi: 10.1006/jfan.1996.3085.
    [14] M. Del PinoP. L. Felmer and O. H. Miyagaki, Existence of positive bound states of nonlinear Schrödinger equations with saddle-like potential, Nonlinear Anal. Theo. Meth. Appl., 34 (1998), 979-989.  doi: 10.1016/S0362-546X(97)00593-2.
    [15] T. D'Aprile and D. Ruiz, Positive and sign-changing clusters around saddle points of the potential for nonlinear elliptic problems, Math. Zeit., 268 (2011), 605-634.  doi: 10.1007/s00209-010-0686-5.
    [16] G. FigueiredoN. Ikoma and J. R. Júnior, Existence and concentration result for the Kirchhoff type equations with general nonlinearities, Arch. Rational. Mech. Anal., 213 (2014), 931-979.  doi: 10.1007/s00205-014-0747-8.
    [17] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Reprint of the 1998 edition, Springer, Berlin, 2001.
    [18] M. W. Hirsch, Differential Topology, Springer Science and Business Media, 2012.
    [19] X. He and W. Zou, Existence and concentration behavior of positive solutions for a Kirchhoff equation in $R^3$, J. Differ. Equ., 252 (2012), 1813-1834.  doi: 10.1016/j.jde.2011.08.035.
    [20] L. Jeanjean and K. Tanaka, A remark on least energy solutions in $R^N$, Proc. Amer. Math. Soc., 131 (2003), 2399-2408.  doi: 10.1090/S0002-9939-02-06821-1.
    [21] G. Mechanik Kirchhoff, Teubner, Leipzig, 1883.
    [22] J. L. Lions, On some questions in boundary value problems of mathematical physics, North-Holland Mathematics Studies, 30 (1978), 284-346. 
    [23] P. L. Lions, A remark on Bony maximum principle, Proc. Amer. Math. Soc., 88 (1983), 503-508.  doi: 10.2307/2045002.
    [24] P. L. Lions, The concentration-compactness principle in the calculus of variations: The locally compact case, part 2, Ann. Inst. H. Poincaré Anal. Non Linéaire., 1 (1984), 223-283. 
    [25] Z. Liu and S. Guo, Existence and concentration of positive ground states for a Kirchhoff equation involving critical Sobolev exponent, Z. Angew. Math. Phys., 66 (2014), 747-769.  doi: 10.1007/s00033-014-0431-8.
    [26] Z. LiangF. Li and J. Shi, Positive solutions to Kirchhoff type equations with nonlinearity having prescribed asymptotic behavior, Ann. Inst. H. Poincaré Anal. Non Linéaire., 31 (2014), 155-167.  doi: 10.1016/j.anihpc.2013.01.006.
    [27] G. Li and H. Ye, Existence of positive ground state solutions for the nonlinear Kirchhoff type equations in $R^3$, J. Differ. Equ., 257 (2014), 566-600.  doi: 10.1016/j.jde.2014.04.011.
    [28] T. F. Ma, Remarks on an elliptic equation of Kirchhoff type, Nonlinear Anal. Theo. Meth. Appl., 63 (2005), 1967-1977. 
    [29] T. F. Ma and J. E. M. Rivera, Positive solutions for a nonlinear nonlocal elliptic transmission problem, Appl. Math. Let., 16 (2003), 243-248.  doi: 10.1016/S0893-9659(03)80038-1.
    [30] K. Perera and Z. T. Zhang, Nontrivial solutions of Kirchhoff-type problems via the Yang index, J. Differ. Equ., 221 (2006), 246-255.  doi: 10.1016/j.jde.2005.03.006.
    [31] W. A. Strauss, Existence of solitary waves in higher dimensions, Commu. Math. Phy., 55 (1977), 149-162. 
    [32] C. E. Vasconcellos, On a nonlinear stationary problem in unbound domains, Rev. Mat. Complut., 5 (1992), 309-329. 
    [33] J. WangL. Tian and J. Xu, et al., Multiplicity and concentration of positive solutions for a Kirchhoff type problem with critical growth, J. Differ. Equ., 253 (2012), 2314-2351.  doi: 10.1016/j.jde.2012.05.023.
    [34] X. Wu, Existence of nontrivial solutions and high energy solutions for Schrödinger-Kirchhoff-type equations in $R^N$, Nonlinear Anal. Real Wor. Appl., 12 (2011), 1278-1287.  doi: 10.1016/j.nonrwa.2010.09.023.
    [35] X. Wang, On concentration of positive bound states of nonlinear Schrödinger equations, Comm. Math. Phys., 153 (1993), 229-244. 
    [36] X. Wang and B. Zeng, On concentration of positive bound states of nonlinear Schrödinger equations with competing potential functions, SIAM J. Math. Anal., 28 (1997), 633-655.  doi: 10.1137/S0036141095290240.
    [37] Z. T. Zhang and K. Perera, Sign changing solutions of Kirchhoff type problems via invariant sets of descent flow, J. Math. Anal. Appl., 317 (2006), 456-463.  doi: 10.1016/j.jmaa.2005.06.102.
  • 加载中

Article Metrics

HTML views(470) PDF downloads(230) Cited by(0)

Access History

Other Articles By Authors



    DownLoad:  Full-Size Img  PowerPoint