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

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
$V$
.
Citation: Yu Chen, Yanheng Ding, Suhong Li. Existence and concentration for Kirchhoff type equations around topologically critical points of the potential. Communications on Pure and 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. 

[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, 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.

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. 

[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, 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.

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