December  2016, 36(12): 7137-7168. doi: 10.3934/dcds.2016111

Existence and concentration of solutions for a Kirchhoff type problem with potentials

1. 

Faculty of Science, Jiangsu University, Zhenjiang, Jiangsu 212013, China

2. 

School of management, Jiangsu University, Zhenjiang, Jiangsu 212013, China

Received  February 2016 Revised  June 2016 Published  October 2016

In this paper, we concern with the following semilinear Kirchhoff type equation \begin{equation*} \begin{cases} -\left(\varepsilon^{2}a+b\varepsilon\int_{\mathbb{R}^{3}}|\nabla u|^{2}\right)\Delta u+V(x)u=K(x)f(u)+Q(x)|u|^{p-2}u, &x\in\mathbb{R}^{3},\\ u\in H^{1}(\mathbb{R}^{3}), u>0 &x\in\mathbb{R}^{3}, \end{cases} \end{equation*} where $\varepsilon>0$ is a small parameter, $a, b$ are positive constants, $V, K$ and $Q$ are positive bounded functions and $p\in(4,6]$, $f$ is a continuous superlinear and subcritical nonlinearity. On the one hand, for subcritical case, i.e., $p\in(4,6)$, we prove that there are three families of semiclassical positive solutions for $\varepsilon>0$ small, one is concentrating on the set of minima of $V$, the rest of two families of solutions are concentrating on the sets of maxima of $K$ and $Q$ respectively. On the other hand, we also prove the multiplicity and concentration of positive solutions for critical case($p=6$). The novelty is that we prove some new concentration phenomena for the positive solutions.
Citation: Jun Wang, Lu Xiao. Existence and concentration of solutions for a Kirchhoff type problem with potentials. Discrete & Continuous Dynamical Systems - A, 2016, 36 (12) : 7137-7168. doi: 10.3934/dcds.2016111
References:
[1]

C. O. Alves, F. J. S. A. Corrêa and T. F Ma, Positive solutions for a quasilinear elliptic equation of Kirchhoff type,, Comput. Math. Appl., 49 (2005), 85. doi: 10.1016/j.camwa.2005.01.008. Google Scholar

[2]

C. O. Alves, F. J. S. A. Corrêa and G. M. Figueiredo, On a class of nonlocal elliptic problems with critical growth,, Differential Equations and Applications, 2 (2010), 409. doi: 10.7153/dea-02-25. Google Scholar

[3]

C. O. Alves and G. M. Figueiredo, Nonlinear perturbations of a periodic Kirchhoff equation in $\mathbbR^N$,, Non. Anal., 75 (2012), 2750. doi: 10.1016/j.na.2011.11.017. Google Scholar

[4]

C. O. Alves and M. A. Souto, On existence and concentration behavior of ground state solutions for a class of problems with critical growth,, Comm. Pure Appl. Anal., 1 (2002), 417. doi: 10.3934/cpaa.2002.1.417. Google Scholar

[5]

A. Arosio and S. Panizzi, On the well-posedness of the Kirchhoff string,, Trans. Amer. Math. Soc., 348 (1996), 305. doi: 10.1090/S0002-9947-96-01532-2. Google Scholar

[6]

G. Autuori, F. Colasuonno and P. Pucci, On the existence of stationary solutions for higher order p-Kirchhoff problems,, Commun. Contemp. Math., 16 (2014). doi: 10.1142/S0219199714500023. Google Scholar

[7]

G. Autuori and P. Pucci, Elliptic problems involving the fractional Laplacian in $\mathbbR^N$,, J. Differential Equations, 255 (2013), 2340. doi: 10.1016/j.jde.2013.06.016. Google Scholar

[8]

G. Autuori, A. Fiscella and P. Pucci, Stationary Kirchhoff problems involving a fractional elliptic operator and a critical nonlinearity,, Nonlinear Anal., 125 (2015), 699. doi: 10.1016/j.na.2015.06.014. Google Scholar

[9]

E. DiBenedetto, $C^{1+\alpha}$ local regularity of weak solutions of degenerate results elliptic equations,, Nonl. Anal., 7 (1983), 827. doi: 10.1016/0362-546X(83)90061-5. Google Scholar

[10]

C.-Y. Chen, Y.-C. Kuo and T.-F. Wu, The Nehari manifold for a Kirchhoff type problem involving sign-changing weight functions,, J. Differential Equations, 250 (2011), 1876. doi: 10.1016/j.jde.2010.11.017. Google Scholar

[11]

M. Chipot and B. Lovat, Some remarks on nonlocal elliptic and parabolic problems,, Nonlinear Anal., 30 (1997), 4619. doi: 10.1016/S0362-546X(97)00169-7. Google Scholar

[12]

S. Cingolani and M. Lazzo, Multiple positive solutions to nonlinear Schrödinger equations with competing potential functions,, J. Differential Equations, 160 (2000), 118. doi: 10.1006/jdeq.1999.3662. Google Scholar

[13]

S. Cingolani and M. Lazzo, Multiplicity of solutions for $p(x)$-polyharmonic elliptic Kirchhoff equations,, Nonlinear Anal., 74 (2011), 5962. doi: 10.1016/j.na.2011.05.073. Google Scholar

[14]

P. D'Ancona and S. Spagnolo, Multiple positive solutions to nonlinear Schrödinger equations with competing potential functions,, Invent. Math., 108 (1992), 247. Google Scholar

[15]

D. G. de Figueiredo and J. Yang, Decay, symmetry and existence of solutions of semilinear elliptic systems,, Nonlinear Anal., 33 (1998), 211. doi: 10.1016/S0362-546X(97)00548-8. Google Scholar

[16]

M. del Pino and P. L. Felmer, Multiple positive solutions to nonlinear Schrödinger equations with competing potential functions,, Calc. Var. Partial Differential Equations, 4 (1996), 121. doi: 10.1007/BF01189950. Google Scholar

[17]

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. doi: 10.1016/S0294-1449(97)89296-7. Google Scholar

[18]

M. del Pino, M. Kowalczyk and J. C. Wei, Concentration on curves for nonlinear Schrödinger equations,, Comm. Pure Appl. Math., 60 (2007), 113. doi: 10.1002/cpa.20135. Google Scholar

[19]

Y. H. Ding and X. Liu, Semi-classical limits of ground states of a nonlinear Dirac equation,, J. Differ. Equ., 252 (2012), 4962. doi: 10.1016/j.jde.2012.01.023. Google Scholar

[20]

Y. H. Ding and X. Liu, Semi-classical limits of ground states of a nonlinear Dirac equation,, Manuscripta Math., 140 (2013), 51. doi: 10.1016/j.jde.2012.01.023. Google Scholar

[21]

G. M. Figueiredo, N. Ikoma and J. R. Santos Juior, Semi-classical limits of ground states of a nonlinear Dirac equation,, Arch. Rational Mech. Anal., 213 (2014), 931. doi: 10.1007/s00205-014-0747-8. Google Scholar

[22]

G. M. Figueiredo and J. R. Santos Juior, Multiplicity of solutions for a Kirchhoff equation with subcritical or critical growth,, Differ. Integr. Equ., 25 (2012), 853. Google Scholar

[23]

G. M. Figueiredo and J. R. Santos Juior, Multiplicity and concentration behavior of positive solutions for a Schrödinger-Kirchhoff type problem via penalization method,, ESAIM Control Optim. Calc Var., 20 (2014), 389. doi: 10.1051/cocv/2013068. Google Scholar

[24]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order,, 2nd ed., (1983). doi: 10.1007/978-3-642-61798-0. Google Scholar

[25]

X. M. He and W. M. Zou, Existence and Concentration Behavior of Positive Solutions for a Kirchhoff Equation in $\mathbbR^{3}$,, J. Differential Equations, 252 (2012), 1813. doi: 10.1016/j.jde.2011.08.035. Google Scholar

[26]

X. M. He and W. M. Zou, Infinitely many positive solutions for Kirchhoff-type problems,, Nonl. Anal., 70 (2009), 1407. doi: 10.1016/j.na.2008.02.021. Google Scholar

[27]

X. M. He and W. M. Zou, Multiplicity of solutions for a class of Kirchhoff type problems,, Acta Math. Appl., 26 (2010), 387. doi: 10.1007/s10255-010-0005-2. Google Scholar

[28]

L. Jeanjean and K. Tanaka, Singularly perturbed elliptic problems with superlinear or asymptotically linear nonlinearities,, Calc. Var. Partial Differential Equations, 21 (2004), 287. doi: 10.1007/s00526-003-0261-6. Google Scholar

[29]

J. H. Jin and X. Wu, Infinitely many radial solutions for Kirchhoff type problems in $\mathbbR^N$,, J. Math. Anal. Appl., 369 (2010), 564. doi: 10.1016/j.jmaa.2010.03.059. Google Scholar

[30]

W. Kryszewski and A. Szulkin, Generalized linking theorem with an application to semilinear Schrödinger equations,, Adv. Differential Equations, 3 (1998), 441. Google Scholar

[31]

G. Kirchhoff, Mechanik,, eubner, (1883). Google Scholar

[32]

G.-B. Li and H.-Y. Ye, Existence of positive solutions for nonlinear Kirchhoff type problems in $\mathbbR^{3}$ with critical Sobolev exponent and sign-changing nonlinearities,, Math. Methods Appl. Sci., 37 (2014), 2570. doi: 10.1002/mma.3000. Google Scholar

[33]

G.-B. Li and H.-Y. Ye, Existence of positive ground state solutions for the nonlinear Kirchhoff type equations in $\mathbbR^{3}$,, J. Differential Equations, 257 (2014), 566. doi: 10.1016/j.jde.2014.04.011. Google Scholar

[34]

Y.-H. Li, F.-Y. Li and J.-P. Shi, Existence of a positive solution to Kirchhoff type problems without compactness conditions,, J. Differ. Equ., 253 (2012), 2285. doi: 10.1016/j.jde.2012.05.017. Google Scholar

[35]

Z.-P. Liang, F.-Y. Li and J.-P. Shi, Positive solutions to Kirchhoff type equations with nonlinearity having prescribed asymptotic behavior,, Ann. Inst. H. Poincaré Anal. Non Linéaire, 31 (2014), 155. doi: 10.1016/j.anihpc.2013.01.006. Google Scholar

[36]

J.-L. Lions, On some questions in boundary value problems of mathematical physics. Contemporary developments in continuum mechanics and partial differential equations (Proc. Internat. Sympos., Inst. Mat., Univ. Fed. Rio de Janeiro, Rio de Janeiro, 1977),, North- Holland Mathematical Studies, 30 (1978), 284. Google Scholar

[37]

P. L. Lions, The concentration compactness principle in the calculus of variations: The locally compact case. Parts 1, 2,, Ann. Inst. H. Poincaré Anal. Non Linéaire, 1 (1984), 109. Google Scholar

[38]

W. Liu and X. M. He, Multiplicity of high energy solutions for superlinear Kirchhoff equations,, J. Appl. Math. Comput., 39 (2012), 473. doi: 10.1007/s12190-012-0536-1. Google Scholar

[39]

Z.-S. Liu and S.-J. Guo, Existence and concentration of positive ground states for a Kirchhoff equation involving critical Sobolev exponent,, Z. Angew. Math. Phys., 66 (2015), 747. doi: 10.1007/s00033-014-0431-8. Google Scholar

[40]

T. F. Ma and J. E. Munoz Rivera, Positive solutions for a nonlinear nonlocal elliptic transmission problem,, Appl. Math. Lett., 16 (2003), 243. doi: 10.1016/S0893-9659(03)80038-1. Google Scholar

[41]

A. Pankov, On decay of solution to nonlinear Schrödinger equations,, Proc. Amer. Math. Soc., 136 (2008), 2565. doi: 10.1090/S0002-9939-08-09484-7. Google Scholar

[42]

P. Pucci, M.-Q. Xiang and B.-L. Zhang, Multiple solutions for nonhomogeneous Schrödinger-Kirchhoff type equations involving the fractional p-Laplacian in $\mathbbR^N$,, Calc. Var. Partial Differential Equations, 54 (2015), 2785. doi: 10.1007/s00526-015-0883-5. Google Scholar

[43]

P. Pucci and Q.-H. Zhang, Existence of entire solutions for a class of variable exponent elliptic equations,, J. Differential Equations, 257 (2014), 1529. doi: 10.1016/j.jde.2014.05.023. Google Scholar

[44]

P. H. Rabinowitz, On a class of nonlinear Schrödinger equations,, Z. Angew. Math. Phys., 43 (1992), 270. doi: 10.1007/BF00946631. Google Scholar

[45]

K. Perera and Z. Zhang, On a class of nonlinear Schrödinger equations,, J. Diff. Eqns., 221 (2006), 246. Google Scholar

[46]

W. Strauss, Existence of solitary waves in higher dimensions,, Commun. Math. Phys., 55 (1977), 149. doi: 10.1007/BF01626517. Google Scholar

[47]

A. Szulkin and T. Weth, The Method of Nehari Manifold,, Handbook of Nonconvex Analysis and Applications, (2010), 597. Google Scholar

[48]

P. Tolksdorf, Regularity for some general class of quasilinear elliptic equations,, J. Differential Equations, 51 (1984), 126. doi: 10.1016/0022-0396(84)90105-0. Google Scholar

[49]

X. F. Wang, On concentration of positive bound states of nonlinear Schrödinger equations,, Commun. Math. Phys., 153 (1993), 229. doi: 10.1007/BF02096642. Google Scholar

[50]

J. Wang, L. X. Tian, J. X. Xu and F. B. Zhang, Multiplicity and concentration of positive solutions for a Kirchhoff type problem with critical growth,, J. Differential Equations, 253 (2012), 2314. doi: 10.1016/j.jde.2012.05.023. Google Scholar

[51]

J. Wang, J. X. Xu and F. B. Zhang, Multiple positive solutions for Schrödinger-Poisson systems with critical growth,, Preprint., (). Google Scholar

[52]

M. Willem, Minimax Theorems,, Progr. Nonlinear Differential Equations Appl., (1996). doi: 10.1007/978-1-4612-4146-1. Google Scholar

[53]

X. Wu, Existence of nontrivial solutions and high energy solutions for Schrödinger Kirchhoff-type equations in $\mathbbR^N$,, Nonlinear Anal. Real World Appl., 12 (2011), 1278. doi: 10.1016/j.nonrwa.2010.09.023. Google Scholar

[54]

Y.-W. Ye and C.-L. Tang, Multiple solutions for Kirchhoff-type equations in $\mathbbR^N$,, J. Math. Phys., 54 (2013). doi: 10.1063/1.4819249. Google Scholar

[55]

Y.-W. Ye, Multiple solutions for Kirchhoff-type equations in $\mathbbR^N$,, Differ. Equ. Appl., 5 (2013), 83. doi: 10.7153/dea-05-06. Google Scholar

show all references

References:
[1]

C. O. Alves, F. J. S. A. Corrêa and T. F Ma, Positive solutions for a quasilinear elliptic equation of Kirchhoff type,, Comput. Math. Appl., 49 (2005), 85. doi: 10.1016/j.camwa.2005.01.008. Google Scholar

[2]

C. O. Alves, F. J. S. A. Corrêa and G. M. Figueiredo, On a class of nonlocal elliptic problems with critical growth,, Differential Equations and Applications, 2 (2010), 409. doi: 10.7153/dea-02-25. Google Scholar

[3]

C. O. Alves and G. M. Figueiredo, Nonlinear perturbations of a periodic Kirchhoff equation in $\mathbbR^N$,, Non. Anal., 75 (2012), 2750. doi: 10.1016/j.na.2011.11.017. Google Scholar

[4]

C. O. Alves and M. A. Souto, On existence and concentration behavior of ground state solutions for a class of problems with critical growth,, Comm. Pure Appl. Anal., 1 (2002), 417. doi: 10.3934/cpaa.2002.1.417. Google Scholar

[5]

A. Arosio and S. Panizzi, On the well-posedness of the Kirchhoff string,, Trans. Amer. Math. Soc., 348 (1996), 305. doi: 10.1090/S0002-9947-96-01532-2. Google Scholar

[6]

G. Autuori, F. Colasuonno and P. Pucci, On the existence of stationary solutions for higher order p-Kirchhoff problems,, Commun. Contemp. Math., 16 (2014). doi: 10.1142/S0219199714500023. Google Scholar

[7]

G. Autuori and P. Pucci, Elliptic problems involving the fractional Laplacian in $\mathbbR^N$,, J. Differential Equations, 255 (2013), 2340. doi: 10.1016/j.jde.2013.06.016. Google Scholar

[8]

G. Autuori, A. Fiscella and P. Pucci, Stationary Kirchhoff problems involving a fractional elliptic operator and a critical nonlinearity,, Nonlinear Anal., 125 (2015), 699. doi: 10.1016/j.na.2015.06.014. Google Scholar

[9]

E. DiBenedetto, $C^{1+\alpha}$ local regularity of weak solutions of degenerate results elliptic equations,, Nonl. Anal., 7 (1983), 827. doi: 10.1016/0362-546X(83)90061-5. Google Scholar

[10]

C.-Y. Chen, Y.-C. Kuo and T.-F. Wu, The Nehari manifold for a Kirchhoff type problem involving sign-changing weight functions,, J. Differential Equations, 250 (2011), 1876. doi: 10.1016/j.jde.2010.11.017. Google Scholar

[11]

M. Chipot and B. Lovat, Some remarks on nonlocal elliptic and parabolic problems,, Nonlinear Anal., 30 (1997), 4619. doi: 10.1016/S0362-546X(97)00169-7. Google Scholar

[12]

S. Cingolani and M. Lazzo, Multiple positive solutions to nonlinear Schrödinger equations with competing potential functions,, J. Differential Equations, 160 (2000), 118. doi: 10.1006/jdeq.1999.3662. Google Scholar

[13]

S. Cingolani and M. Lazzo, Multiplicity of solutions for $p(x)$-polyharmonic elliptic Kirchhoff equations,, Nonlinear Anal., 74 (2011), 5962. doi: 10.1016/j.na.2011.05.073. Google Scholar

[14]

P. D'Ancona and S. Spagnolo, Multiple positive solutions to nonlinear Schrödinger equations with competing potential functions,, Invent. Math., 108 (1992), 247. Google Scholar

[15]

D. G. de Figueiredo and J. Yang, Decay, symmetry and existence of solutions of semilinear elliptic systems,, Nonlinear Anal., 33 (1998), 211. doi: 10.1016/S0362-546X(97)00548-8. Google Scholar

[16]

M. del Pino and P. L. Felmer, Multiple positive solutions to nonlinear Schrödinger equations with competing potential functions,, Calc. Var. Partial Differential Equations, 4 (1996), 121. doi: 10.1007/BF01189950. Google Scholar

[17]

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. doi: 10.1016/S0294-1449(97)89296-7. Google Scholar

[18]

M. del Pino, M. Kowalczyk and J. C. Wei, Concentration on curves for nonlinear Schrödinger equations,, Comm. Pure Appl. Math., 60 (2007), 113. doi: 10.1002/cpa.20135. Google Scholar

[19]

Y. H. Ding and X. Liu, Semi-classical limits of ground states of a nonlinear Dirac equation,, J. Differ. Equ., 252 (2012), 4962. doi: 10.1016/j.jde.2012.01.023. Google Scholar

[20]

Y. H. Ding and X. Liu, Semi-classical limits of ground states of a nonlinear Dirac equation,, Manuscripta Math., 140 (2013), 51. doi: 10.1016/j.jde.2012.01.023. Google Scholar

[21]

G. M. Figueiredo, N. Ikoma and J. R. Santos Juior, Semi-classical limits of ground states of a nonlinear Dirac equation,, Arch. Rational Mech. Anal., 213 (2014), 931. doi: 10.1007/s00205-014-0747-8. Google Scholar

[22]

G. M. Figueiredo and J. R. Santos Juior, Multiplicity of solutions for a Kirchhoff equation with subcritical or critical growth,, Differ. Integr. Equ., 25 (2012), 853. Google Scholar

[23]

G. M. Figueiredo and J. R. Santos Juior, Multiplicity and concentration behavior of positive solutions for a Schrödinger-Kirchhoff type problem via penalization method,, ESAIM Control Optim. Calc Var., 20 (2014), 389. doi: 10.1051/cocv/2013068. Google Scholar

[24]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order,, 2nd ed., (1983). doi: 10.1007/978-3-642-61798-0. Google Scholar

[25]

X. M. He and W. M. Zou, Existence and Concentration Behavior of Positive Solutions for a Kirchhoff Equation in $\mathbbR^{3}$,, J. Differential Equations, 252 (2012), 1813. doi: 10.1016/j.jde.2011.08.035. Google Scholar

[26]

X. M. He and W. M. Zou, Infinitely many positive solutions for Kirchhoff-type problems,, Nonl. Anal., 70 (2009), 1407. doi: 10.1016/j.na.2008.02.021. Google Scholar

[27]

X. M. He and W. M. Zou, Multiplicity of solutions for a class of Kirchhoff type problems,, Acta Math. Appl., 26 (2010), 387. doi: 10.1007/s10255-010-0005-2. Google Scholar

[28]

L. Jeanjean and K. Tanaka, Singularly perturbed elliptic problems with superlinear or asymptotically linear nonlinearities,, Calc. Var. Partial Differential Equations, 21 (2004), 287. doi: 10.1007/s00526-003-0261-6. Google Scholar

[29]

J. H. Jin and X. Wu, Infinitely many radial solutions for Kirchhoff type problems in $\mathbbR^N$,, J. Math. Anal. Appl., 369 (2010), 564. doi: 10.1016/j.jmaa.2010.03.059. Google Scholar

[30]

W. Kryszewski and A. Szulkin, Generalized linking theorem with an application to semilinear Schrödinger equations,, Adv. Differential Equations, 3 (1998), 441. Google Scholar

[31]

G. Kirchhoff, Mechanik,, eubner, (1883). Google Scholar

[32]

G.-B. Li and H.-Y. Ye, Existence of positive solutions for nonlinear Kirchhoff type problems in $\mathbbR^{3}$ with critical Sobolev exponent and sign-changing nonlinearities,, Math. Methods Appl. Sci., 37 (2014), 2570. doi: 10.1002/mma.3000. Google Scholar

[33]

G.-B. Li and H.-Y. Ye, Existence of positive ground state solutions for the nonlinear Kirchhoff type equations in $\mathbbR^{3}$,, J. Differential Equations, 257 (2014), 566. doi: 10.1016/j.jde.2014.04.011. Google Scholar

[34]

Y.-H. Li, F.-Y. Li and J.-P. Shi, Existence of a positive solution to Kirchhoff type problems without compactness conditions,, J. Differ. Equ., 253 (2012), 2285. doi: 10.1016/j.jde.2012.05.017. Google Scholar

[35]

Z.-P. Liang, F.-Y. Li and J.-P. Shi, Positive solutions to Kirchhoff type equations with nonlinearity having prescribed asymptotic behavior,, Ann. Inst. H. Poincaré Anal. Non Linéaire, 31 (2014), 155. doi: 10.1016/j.anihpc.2013.01.006. Google Scholar

[36]

J.-L. Lions, On some questions in boundary value problems of mathematical physics. Contemporary developments in continuum mechanics and partial differential equations (Proc. Internat. Sympos., Inst. Mat., Univ. Fed. Rio de Janeiro, Rio de Janeiro, 1977),, North- Holland Mathematical Studies, 30 (1978), 284. Google Scholar

[37]

P. L. Lions, The concentration compactness principle in the calculus of variations: The locally compact case. Parts 1, 2,, Ann. Inst. H. Poincaré Anal. Non Linéaire, 1 (1984), 109. Google Scholar

[38]

W. Liu and X. M. He, Multiplicity of high energy solutions for superlinear Kirchhoff equations,, J. Appl. Math. Comput., 39 (2012), 473. doi: 10.1007/s12190-012-0536-1. Google Scholar

[39]

Z.-S. Liu and S.-J. Guo, Existence and concentration of positive ground states for a Kirchhoff equation involving critical Sobolev exponent,, Z. Angew. Math. Phys., 66 (2015), 747. doi: 10.1007/s00033-014-0431-8. Google Scholar

[40]

T. F. Ma and J. E. Munoz Rivera, Positive solutions for a nonlinear nonlocal elliptic transmission problem,, Appl. Math. Lett., 16 (2003), 243. doi: 10.1016/S0893-9659(03)80038-1. Google Scholar

[41]

A. Pankov, On decay of solution to nonlinear Schrödinger equations,, Proc. Amer. Math. Soc., 136 (2008), 2565. doi: 10.1090/S0002-9939-08-09484-7. Google Scholar

[42]

P. Pucci, M.-Q. Xiang and B.-L. Zhang, Multiple solutions for nonhomogeneous Schrödinger-Kirchhoff type equations involving the fractional p-Laplacian in $\mathbbR^N$,, Calc. Var. Partial Differential Equations, 54 (2015), 2785. doi: 10.1007/s00526-015-0883-5. Google Scholar

[43]

P. Pucci and Q.-H. Zhang, Existence of entire solutions for a class of variable exponent elliptic equations,, J. Differential Equations, 257 (2014), 1529. doi: 10.1016/j.jde.2014.05.023. Google Scholar

[44]

P. H. Rabinowitz, On a class of nonlinear Schrödinger equations,, Z. Angew. Math. Phys., 43 (1992), 270. doi: 10.1007/BF00946631. Google Scholar

[45]

K. Perera and Z. Zhang, On a class of nonlinear Schrödinger equations,, J. Diff. Eqns., 221 (2006), 246. Google Scholar

[46]

W. Strauss, Existence of solitary waves in higher dimensions,, Commun. Math. Phys., 55 (1977), 149. doi: 10.1007/BF01626517. Google Scholar

[47]

A. Szulkin and T. Weth, The Method of Nehari Manifold,, Handbook of Nonconvex Analysis and Applications, (2010), 597. Google Scholar

[48]

P. Tolksdorf, Regularity for some general class of quasilinear elliptic equations,, J. Differential Equations, 51 (1984), 126. doi: 10.1016/0022-0396(84)90105-0. Google Scholar

[49]

X. F. Wang, On concentration of positive bound states of nonlinear Schrödinger equations,, Commun. Math. Phys., 153 (1993), 229. doi: 10.1007/BF02096642. Google Scholar

[50]

J. Wang, L. X. Tian, J. X. Xu and F. B. Zhang, Multiplicity and concentration of positive solutions for a Kirchhoff type problem with critical growth,, J. Differential Equations, 253 (2012), 2314. doi: 10.1016/j.jde.2012.05.023. Google Scholar

[51]

J. Wang, J. X. Xu and F. B. Zhang, Multiple positive solutions for Schrödinger-Poisson systems with critical growth,, Preprint., (). Google Scholar

[52]

M. Willem, Minimax Theorems,, Progr. Nonlinear Differential Equations Appl., (1996). doi: 10.1007/978-1-4612-4146-1. Google Scholar

[53]

X. Wu, Existence of nontrivial solutions and high energy solutions for Schrödinger Kirchhoff-type equations in $\mathbbR^N$,, Nonlinear Anal. Real World Appl., 12 (2011), 1278. doi: 10.1016/j.nonrwa.2010.09.023. Google Scholar

[54]

Y.-W. Ye and C.-L. Tang, Multiple solutions for Kirchhoff-type equations in $\mathbbR^N$,, J. Math. Phys., 54 (2013). doi: 10.1063/1.4819249. Google Scholar

[55]

Y.-W. Ye, Multiple solutions for Kirchhoff-type equations in $\mathbbR^N$,, Differ. Equ. Appl., 5 (2013), 83. doi: 10.7153/dea-05-06. Google Scholar

[1]

Rui-Qi Liu, Chun-Lei Tang, Jia-Feng Liao, Xing-Ping Wu. Positive solutions of Kirchhoff type problem with singular and critical nonlinearities in dimension four. Communications on Pure & Applied Analysis, 2016, 15 (5) : 1841-1856. doi: 10.3934/cpaa.2016006

[2]

Xing Liu, Yijing Sun. Multiple positive solutions for Kirchhoff type problems with singularity. Communications on Pure & Applied Analysis, 2013, 12 (2) : 721-733. doi: 10.3934/cpaa.2013.12.721

[3]

Wenjing Chen. Multiplicity of solutions for a fractional Kirchhoff type problem. Communications on Pure & Applied Analysis, 2015, 14 (5) : 2009-2020. doi: 10.3934/cpaa.2015.14.2009

[4]

To Fu Ma. Positive solutions for a nonlocal fourth order equation of Kirchhoff type. Conference Publications, 2007, 2007 (Special) : 694-703. doi: 10.3934/proc.2007.2007.694

[5]

Ling Ding, Shu-Ming Sun. Existence of positive solutions for a class of Kirchhoff type equations in $\mathbb{R}^3$. Discrete & Continuous Dynamical Systems - S, 2016, 9 (6) : 1663-1685. doi: 10.3934/dcdss.2016069

[6]

Jiafeng Liao, Peng Zhang, Jiu Liu, Chunlei Tang. Existence and multiplicity of positive solutions for a class of Kirchhoff type problems at resonance. Discrete & Continuous Dynamical Systems - S, 2016, 9 (6) : 1959-1974. doi: 10.3934/dcdss.2016080

[7]

Qi-Lin Xie, Xing-Ping Wu, Chun-Lei Tang. Existence and multiplicity of solutions for Kirchhoff type problem with critical exponent. Communications on Pure & Applied Analysis, 2013, 12 (6) : 2773-2786. doi: 10.3934/cpaa.2013.12.2773

[8]

Xiao-Jing Zhong, Chun-Lei Tang. The existence and nonexistence results of ground state nodal solutions for a Kirchhoff type problem. Communications on Pure & Applied Analysis, 2017, 16 (2) : 611-628. doi: 10.3934/cpaa.2017030

[9]

Jia-Feng Liao, Yang Pu, Xiao-Feng Ke, Chun-Lei Tang. Multiple positive solutions for Kirchhoff type problems involving concave-convex nonlinearities. Communications on Pure & Applied Analysis, 2017, 16 (6) : 2157-2175. doi: 10.3934/cpaa.2017107

[10]

Yuanxiao Li, Ming Mei, Kaijun Zhang. Existence of multiple nontrivial solutions for a $p$-Kirchhoff type elliptic problem involving sign-changing weight functions. Discrete & Continuous Dynamical Systems - B, 2016, 21 (3) : 883-908. doi: 10.3934/dcdsb.2016.21.883

[11]

Guozhen Lu and Juncheng Wei. On positive entire solutions to the Yamabe-type problem on the Heisenberg and stratified groups. Electronic Research Announcements, 1997, 3: 83-89.

[12]

Jiu Liu, Jia-Feng Liao, Chun-Lei Tang. Positive solution for the Kirchhoff-type equations involving general subcritical growth. Communications on Pure & Applied Analysis, 2016, 15 (2) : 445-455. doi: 10.3934/cpaa.2016.15.445

[13]

Nemat Nyamoradi, Kaimin Teng. Existence of solutions for a Kirchhoff-type-nonlocal operators of elliptic type. Communications on Pure & Applied Analysis, 2015, 14 (2) : 361-371. doi: 10.3934/cpaa.2015.14.361

[14]

Raffaella Servadei, Enrico Valdinoci. Variational methods for non-local operators of elliptic type. Discrete & Continuous Dynamical Systems - A, 2013, 33 (5) : 2105-2137. doi: 10.3934/dcds.2013.33.2105

[15]

Patrizia Pucci, Mingqi Xiang, Binlin Zhang. A diffusion problem of Kirchhoff type involving the nonlocal fractional p-Laplacian. Discrete & Continuous Dynamical Systems - A, 2017, 37 (7) : 4035-4051. doi: 10.3934/dcds.2017171

[16]

Mingqi Xiang, Binlin Zhang. A critical fractional p-Kirchhoff type problem involving discontinuous nonlinearity. Discrete & Continuous Dynamical Systems - S, 2019, 12 (2) : 413-433. doi: 10.3934/dcdss.2019027

[17]

Peter Monk, Virginia Selgas. Sampling type methods for an inverse waveguide problem. Inverse Problems & Imaging, 2012, 6 (4) : 709-747. doi: 10.3934/ipi.2012.6.709

[18]

Shu-Zhi Song, Shang-Jie Chen, Chun-Lei Tang. Existence of solutions for Kirchhoff type problems with resonance at higher eigenvalues. Discrete & Continuous Dynamical Systems - A, 2016, 36 (11) : 6453-6473. doi: 10.3934/dcds.2016078

[19]

Norihisa Ikoma. Existence of ground state solutions to the nonlinear Kirchhoff type equations with potentials. Discrete & Continuous Dynamical Systems - A, 2015, 35 (3) : 943-966. doi: 10.3934/dcds.2015.35.943

[20]

Helin Guo, Yimin Zhang, Huansong Zhou. Blow-up solutions for a Kirchhoff type elliptic equation with trapping potential. Communications on Pure & Applied Analysis, 2018, 17 (5) : 1875-1897. doi: 10.3934/cpaa.2018089

2018 Impact Factor: 1.143

Metrics

  • PDF downloads (22)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]