Article Contents
Article Contents

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

• 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.
Mathematics Subject Classification: Primary: 35J61, 35J20, 35Q55; Secondary: 49J40.

 Citation:

•  [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), 85-93.doi: 10.1016/j.camwa.2005.01.008. [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), 409-417.doi: 10.7153/dea-02-25. [3] C. O. Alves and G. M. Figueiredo, Nonlinear perturbations of a periodic Kirchhoff equation in $\mathbbR^N$, Non. Anal., 75 (2012), 2750-2759.doi: 10.1016/j.na.2011.11.017. [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), 417-431.doi: 10.3934/cpaa.2002.1.417. [5] A. Arosio and S. Panizzi, On the well-posedness of the Kirchhoff string, Trans. Amer. Math. Soc., 348 (1996), 305-330.doi: 10.1090/S0002-9947-96-01532-2. [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), 1450002, 43 pp.doi: 10.1142/S0219199714500023. [7] G. Autuori and P. Pucci, Elliptic problems involving the fractional Laplacian in $\mathbbR^N$, J. Differential Equations, 255 (2013), 2340-2362.doi: 10.1016/j.jde.2013.06.016. [8] G. Autuori, A. Fiscella and P. Pucci, Stationary Kirchhoff problems involving a fractional elliptic operator and a critical nonlinearity, Nonlinear Anal., 125 (2015), 699-714.doi: 10.1016/j.na.2015.06.014. [9] E. DiBenedetto, $C^{1+\alpha}$ local regularity of weak solutions of degenerate results elliptic equations, Nonl. Anal., 7 (1983), 827-850.doi: 10.1016/0362-546X(83)90061-5. [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), 1876-1908.doi: 10.1016/j.jde.2010.11.017. [11] M. Chipot and B. Lovat, Some remarks on nonlocal elliptic and parabolic problems, Nonlinear Anal., 30 (1997), 4619-4627.doi: 10.1016/S0362-546X(97)00169-7. [12] S. Cingolani and M. Lazzo, Multiple positive solutions to nonlinear Schrödinger equations with competing potential functions, J. Differential Equations, 160 (2000), 118-138.doi: 10.1006/jdeq.1999.3662. [13] S. Cingolani and M. Lazzo, Multiplicity of solutions for $p(x)$-polyharmonic elliptic Kirchhoff equations, Nonlinear Anal., 74 (2011), 5962-5974.doi: 10.1016/j.na.2011.05.073. [14] P. D'Ancona and S. Spagnolo, Multiple positive solutions to nonlinear Schrödinger equations with competing potential functions, Invent. Math., 108 (1992), 247-262. [15] D. G. de Figueiredo and J. Yang, Decay, symmetry and existence of solutions of semilinear elliptic systems, Nonlinear Anal., 33 (1998), 211-234.doi: 10.1016/S0362-546X(97)00548-8. [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), 121-137.doi: 10.1007/BF01189950. [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), 127-149.doi: 10.1016/S0294-1449(97)89296-7. [18] M. del Pino, M. Kowalczyk and J. C. Wei, Concentration on curves for nonlinear Schrödinger equations, Comm. Pure Appl. Math., 60 (2007), 113-146.doi: 10.1002/cpa.20135. [19] Y. H. Ding and X. Liu, Semi-classical limits of ground states of a nonlinear Dirac equation, J. Differ. Equ., 252 (2012), 4962-4987.doi: 10.1016/j.jde.2012.01.023. [20] Y. H. Ding and X. Liu, Semi-classical limits of ground states of a nonlinear Dirac equation, Manuscripta Math., 140 (2013), 51-82.doi: 10.1016/j.jde.2012.01.023. [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), 931-979.doi: 10.1007/s00205-014-0747-8. [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), 853-868. [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), 389-415.doi: 10.1051/cocv/2013068. [24] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, 2nd ed., Grundlehren der Mathematischen Wissenschaften 224, Springer, Berlin, 1983.doi: 10.1007/978-3-642-61798-0. [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), 1813-1834.doi: 10.1016/j.jde.2011.08.035. [26] X. M. He and W. M. Zou, Infinitely many positive solutions for Kirchhoff-type problems, Nonl. Anal., 70 (2009), 1407-1414.doi: 10.1016/j.na.2008.02.021. [27] X. M. He and W. M. Zou, Multiplicity of solutions for a class of Kirchhoff type problems, Acta Math. Appl., Sin. (Engl. Ser.), 26 (2010), 387-394.doi: 10.1007/s10255-010-0005-2. [28] L. Jeanjean and K. Tanaka, Singularly perturbed elliptic problems with superlinear or asymptotically linear nonlinearities, Calc. Var. Partial Differential Equations, 21 (2004), 287-318.doi: 10.1007/s00526-003-0261-6. [29] J. H. Jin and X. Wu, Infinitely many radial solutions for Kirchhoff type problems in $\mathbbR^N$, J. Math. Anal. Appl., 369 (2010), 564-574.doi: 10.1016/j.jmaa.2010.03.059. [30] W. Kryszewski and A. Szulkin, Generalized linking theorem with an application to semilinear Schrödinger equations, Adv. Differential Equations, 3 (1998), 441-472. [31] G. Kirchhoff, Mechanik, eubner, Leipzig, 1883. [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), 2570-2584.doi: 10.1002/mma.3000. [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), 566-600.doi: 10.1016/j.jde.2014.04.011. [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), 2285-2294.doi: 10.1016/j.jde.2012.05.017. [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), 155-167.doi: 10.1016/j.anihpc.2013.01.006. [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, North-Holland, Amsterdam, 30 (1978), 284-346. [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), 109-145 and 223-283. [38] W. Liu and X. M. He, Multiplicity of high energy solutions for superlinear Kirchhoff equations, J. Appl. Math. Comput., 39 (2012), 473-487.doi: 10.1007/s12190-012-0536-1. [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), 747-769.doi: 10.1007/s00033-014-0431-8. [40] T. F. Ma and J. E. Munoz Rivera, Positive solutions for a nonlinear nonlocal elliptic transmission problem, Appl. Math. Lett., 16 (2003), 243-248.doi: 10.1016/S0893-9659(03)80038-1. [41] A. Pankov, On decay of solution to nonlinear Schrödinger equations, Proc. Amer. Math. Soc., 136 (2008), 2565-2570.doi: 10.1090/S0002-9939-08-09484-7. [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), 2785-2806.doi: 10.1007/s00526-015-0883-5. [43] P. Pucci and Q.-H. Zhang, Existence of entire solutions for a class of variable exponent elliptic equations, J. Differential Equations, 257 (2014), 1529-1566.doi: 10.1016/j.jde.2014.05.023. [44] P. H. Rabinowitz, On a class of nonlinear Schrödinger equations, Z. Angew. Math. Phys., 43 (1992), 270-291.doi: 10.1007/BF00946631. [45] K. Perera and Z. Zhang, On a class of nonlinear Schrödinger equations, J. Diff. Eqns., 221 (2006), 246-255. [46] W. Strauss, Existence of solitary waves in higher dimensions, Commun. Math. Phys., 55 (1977), 149-162.doi: 10.1007/BF01626517. [47] A. Szulkin and T. Weth, The Method of Nehari Manifold, Handbook of Nonconvex Analysis and Applications, D.Y. Gao and D. Motreanu eds., International Press, Boston, 2010, 597-632. [48] P. Tolksdorf, Regularity for some general class of quasilinear elliptic equations, J. Differential Equations, 51 (1984), 126-150.doi: 10.1016/0022-0396(84)90105-0. [49] X. F. Wang, On concentration of positive bound states of nonlinear Schrödinger equations, Commun. Math. Phys., 153 (1993), 229-244.doi: 10.1007/BF02096642. [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), 2314-2351.doi: 10.1016/j.jde.2012.05.023. [51] J. Wang, J. X. Xu and F. B. Zhang, Multiple positive solutions for Schrödinger-Poisson systems with critical growth, Preprint. [52] M. Willem, Minimax Theorems, Progr. Nonlinear Differential Equations Appl., vol. 24, Birkhäuser, Basel, 1996.doi: 10.1007/978-1-4612-4146-1. [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), 1278-1287.doi: 10.1016/j.nonrwa.2010.09.023. [54] Y.-W. Ye and C.-L. Tang, Multiple solutions for Kirchhoff-type equations in $\mathbbR^N$, J. Math. Phys., 54 (2013), 081508, 16 pp.doi: 10.1063/1.4819249. [55] Y.-W. Ye, Multiple solutions for Kirchhoff-type equations in $\mathbbR^N$, Differ. Equ. Appl., 5 (2013), 83-92.doi: 10.7153/dea-05-06.