# American Institute of Mathematical Sciences

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:
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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

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##### 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. 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