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September  2017, 16(5): 1641-1671. doi: 10.3934/cpaa.2017079

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

Yu Chen 1,2, , Yanheng Ding 1, and  Suhong Li 3,

1. 

Institute of Mathematics, Academy of Mathematics and Systems Science, University of Chinese Academy of Science, Chinese Academy of Science, Beijing 100190, China
2. 

Department of Mathematics, Honghe University Mengzi, Yunnan 661100, China
3. 

Department of Mathematics and Information Technology, Hebei Normal University of Science and Technology, Qinhuangdao, Hebei 066004, China

Received  September 2016 Revised  March 2017 Published  May 2017

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
$V$
.
Keywords: Kirchhoff type equations, topologically critical points, concentration, local maximum, saddle Point.
Mathematics Subject Classification: 35B05, 35B45.
Citation: Yu Chen, Yanheng Ding, Suhong Li. Existence and concentration for Kirchhoff type equations around topologically critical points of the potential. Communications on Pure & Applied Analysis, 2017, 16 (5) : 1641-1671. doi: 10.3934/cpaa.2017079
References:
[1]

A. Azzollini, The elliptic Kirchhoff equation in $R^N$ perturbed by a local nonlinearity, Differ. Integ. Equ., 25 (2012), 543-554.   Google Scholar

[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.  Google Scholar

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P. Avenia, A. 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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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J. Byeon and K. Tanaka, Semiclassical standing waves with clustering peaks for nonlinear Schrödinger equations, Mem. Amer. Math. Soc., 229 (2014).   Google Scholar

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C. Chen, Y. 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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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M. Del Pino, P. 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.  Google Scholar

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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.  Google Scholar

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G. Figueiredo, N. 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.  Google Scholar

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D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Reprint of the 1998 edition, Springer, Berlin, 2001.  Google Scholar

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M. W. Hirsch, Differential Topology, Springer Science and Business Media, 2012.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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G. Mechanik Kirchhoff, Teubner, Leipzig, 1883. Google Scholar

[22]

J. L. Lions, On some questions in boundary value problems of mathematical physics, North-Holland Mathematics Studies, 30 (1978), 284-346.   Google Scholar

[23]

P. L. Lions, A remark on Bony maximum principle, Proc. Amer. Math. Soc., 88 (1983), 503-508.  doi: 10.2307/2045002.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[26]

Z. Liang, F. 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.  Google Scholar

[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.  Google Scholar

[28]

T. F. Ma, Remarks on an elliptic equation of Kirchhoff type, Nonlinear Anal. Theo. Meth. Appl., 63 (2005), 1967-1977.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[31]

W. A. Strauss, Existence of solitary waves in higher dimensions, Commu. Math. Phy., 55 (1977), 149-162.   Google Scholar

[32]

C. E. Vasconcellos, On a nonlinear stationary problem in unbound domains, Rev. Mat. Complut., 5 (1992), 309-329.   Google Scholar

[33]

J. Wang, L. Tian and J. Xu, 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.  Google Scholar

[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.  Google Scholar

[35]

X. Wang, On concentration of positive bound states of nonlinear Schrödinger equations, Comm. Math. Phys., 153 (1993), 229-244.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

show all references

References:
[1]

A. Azzollini, The elliptic Kirchhoff equation in $R^N$ perturbed by a local nonlinearity, Differ. Integ. Equ., 25 (2012), 543-554.   Google Scholar

[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.  Google Scholar

[3]

P. Avenia, A. 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.  Google Scholar

[4]

H. Brezis and T. Kato, Remarks on the Schrödinger operator with singular complex potentials, J. Math. Pures Appl., 58 (1979), 137-151.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

J. Byeon and K. Tanaka, Semiclassical standing waves with clustering peaks for nonlinear Schrödinger equations, Mem. Amer. Math. Soc., 229 (2014).   Google Scholar

[9]

C. Chen, Y. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

M. Del Pino, P. 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.  Google Scholar

[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.  Google Scholar

[16]

G. Figueiredo, N. 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.  Google Scholar

[17]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Reprint of the 1998 edition, Springer, Berlin, 2001.  Google Scholar

[18]

M. W. Hirsch, Differential Topology, Springer Science and Business Media, 2012.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[21]

G. Mechanik Kirchhoff, Teubner, Leipzig, 1883. Google Scholar

[22]

J. L. Lions, On some questions in boundary value problems of mathematical physics, North-Holland Mathematics Studies, 30 (1978), 284-346.   Google Scholar

[23]

P. L. Lions, A remark on Bony maximum principle, Proc. Amer. Math. Soc., 88 (1983), 503-508.  doi: 10.2307/2045002.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[26]

Z. Liang, F. 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.  Google Scholar

[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.  Google Scholar

[28]

T. F. Ma, Remarks on an elliptic equation of Kirchhoff type, Nonlinear Anal. Theo. Meth. Appl., 63 (2005), 1967-1977.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[31]

W. A. Strauss, Existence of solitary waves in higher dimensions, Commu. Math. Phy., 55 (1977), 149-162.   Google Scholar

[32]

C. E. Vasconcellos, On a nonlinear stationary problem in unbound domains, Rev. Mat. Complut., 5 (1992), 309-329.   Google Scholar

[33]

J. Wang, L. Tian and J. Xu, 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.  Google Scholar

[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.  Google Scholar

[35]

X. Wang, On concentration of positive bound states of nonlinear Schrödinger equations, Comm. Math. Phys., 153 (1993), 229-244.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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