### Existence and concentration of ground state sign-changing solutions for Kirchhoff type equations with steep potential well and nonlinearity

DOI: http://dx.doi.org/10.12775/TMNA.2017.062

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T. Bartsch, Z. Liu and T. Weth, Sign-changing solutions of superlinear Schrödinger equations, Comm. Partial Differential Equations 29 (2004), 25–42.

T. Bartsch and Z.Q. Wang, Existence and multiplicity results for some superlinear elliptic problems on RN , Comm. Partial Differential Equations 20 (1995), 1725–1741.

T. Bartsch and T. Weth, Three nodal solutions of singularly perturbed elliptic equations on domains without topology, Ann. Inst. H. Poincaré Anal. Non Linéaire 22 (2005), 259–281.

S.J. Chen and L. Li, Multiple solutions for the nonhomogeneous Kirchhoff equation on RN , Nonlinear Anal. 14 (2013), 1477–1486.

S.T. Chen and X.H. Tang, Ground state sign-changing solutions for a class of Schrödinger–Poisson type problems in R3 , Z. Angew. Math. Phys. (2016), 67–102.

B. Cheng, New existence and multiplicity of nontrivial solutions for nonlocal elliptic Kirchhoff type problems, J. Math. Anal. Appl. 394 (2012), 488–495.

B. Cheng and X. Wu, Existence results of positive solutions of Kirchhoff type problems, Nonlinear Anal. 71 (2009), 4883–4892.

B. Cheng, X. Wu and J. Liu, Multiple solutions for a class of Kirchhoff type problems with concave nonlinearity, Nonlinear Differential Equ. Appl. 19 (2012), 521–537.

M. Du, L. Tian, J. Wang and F. Zhang, Existence of ground state solutions for a superbiquadratic Kirchhoff-type equation with steep potential well, Appl. Anal. 95 (2016), 627–645.

G.M. Figueiredo, Existence of a positive solution for a Kirchhoff problem type with critical growth via truncation argument, J. Math. Anal. Appl. 401 (2013), 706–713.

Z. Gao, X.H. Tang and W. Zhang, Multiplicity and concentration of solutions for fractional Schrödinger equations, Taiwanese J. Math. 21 (2017), 187–210.

Y. Guo and J. Nie, Existence and multiplicity of nontrivial solutions for p-Laplacian Schrödinger–Kirchhoff-type equations, J. Math. Anal. Appl. 428 (2015), 1054–1069.

X. He and W. Zou, Existence and concentration behavior of positive solutions for a Kirchhoff equation in R3 , J. Differential Equations 252 (2012), 1813–1834.

Y. He, G. Li and S. Peng, Concentrating bound states for Kirchhoff type problems in R3 involving critical Sobolev exponents, Adv. Nonlinear Stud. 14 (2014), 483–510.

Y. Jiang and H.S. Zhou, Schrödinger–Poisson system with steep potential well, J. Differential Equations 251 (2011), 582–608.

J. Jin and X. Wu, Infinitely many radial solutions for Kirchhoff-type problems in RN , J. Math. Anal. Appl. 369 (2010), 564–574.

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

F. Li, C. Gao and X. Zhu, Existence and concentration of sign-changing solutions to Kirchhoff-type system with Hartree-type nonlinearity, J. Math. Anal. Appl. 448 (2017), 60–80.

Y. Li, F. Li and J. Shi, Existence of a positive solution to Kirchhoff type problems without compactness conditions, J. Differential Equations 253 (2012), 2285–2294.

Y. Li, F. Li and J. Shi, Existence of positive solutions to Kirchhoff type problems with zero mass, J. Math. Anal. Appl. 410 (2014), 361–374.

S. Liang and S. Shi, Soliton solutions to Kirchhoff type problems involving the critical growth in RN , Nonlinear Anal. 81 (2013), 31–41.

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.

C. Miranda, Un’osservazione su un teorema di Brouwer, Boll. Unione Mat. Ital. 3 (1940), 5–7.

D. Naimen, The critical problem of Kirchhoff type elliptic equations in dimension four, J. Differential Equations 257 (2014), 1168–1193.

J. Nie, Exitence and multiplicity of nontrivial solutions for a class of Schrödinger–Kirchhoff-type equations, J. Math. Anal. Appl. 417 (2014), 65–79.

J. Nie and X. Wu, Existence and multiplicity of non-trivial solutions for Schrödinger–Kirchhoff-type equations with radial potential, Nonlinear Anal. 75 (2012), 3470–3479.

E.S. Noussair and J. Wei, On the effect of the domain geometry on the existence and profile of nodal solution of some singularly perturbed semilinear Dirichlet problem, Indiana Univ. Math. J. 46 (1997), 1321–1332.

D. Qin, Y. He and X.H. Tang, Ground state solutions for Kirchhoff type equations with asymptotically 4-linear nonlinearity, Comput. Math. Appl. 71 (2016), 1524–1536.

P.H. Rabinowitz, Minimax methods in critical point theory with applications to differential equations, CBMS Reg. Conf. Ser. Math., Vol. 65, American Mathematical Society, Providence, RI, 1986.

W. Shuai, Sign-changing solutions for a class of Kirchhoff-type problem in bounded domains, J. Differential Equations 259 (2015), 1256–1274.

W. Shuai and Q.F. Wang, Existence and asymptotic behavior of sign-changing solutions for the nonlinear Schrödinger–Poisson system in R3 , Z. Angew. Math. Phys. 66 (2015), 3267–3282.

J. Sun and T. Wu, Ground state solutions for an indefinite Kirchhoff type problem with steep potential well, J. Differential Equations 256 (2014), 1771–1792.

J.J. Sun and C.L. Tang, Existence and multiplicity of solutions for Kirchhoff type equations, Nonlinear Anal. 74 (2011), 1212–1222.

Y. Sun and X. Liu, Existence of positive solutions for Kirchhoff type problems with critical exponent, J. Partial Differential Equations 25 (2012), 187–198.

X.H. Tang, Non-Nehari manifold method for superlinear Schrödinger equation, Taiwanese J. Math. 18 (2014), 1957–1979.

X.H. Tang, New Super-quadratic conditions on ground state solutions for superlinear Schrödinger equation, Adv. Nonlinear Stud. 14 (2014), 349–361.

X.H. Tang, Non-Nehari-manifold method for asymptotically Schrödinger equation, J. Aust. Math. Soc. 98 (2015), 104–116.

X.H. Tang and B. Cheng, Ground state sign-changing solutions for Kirchhoff type problems in bounded domains, J. Differential Equations 261 (2016), 2384–2402.

J. Wang, L. Tian, J. Xu and F. Zhang, Multiplicity and concentration of positive solutions for a Kirchhoff type problem with critical growth, J. Differential Equations 253 (2012), 2314–2351.

M. Willem, Minimax Theorems, Birkhäuser, Boston, 1996.

K. Wu and X. Wu, Infinitely many small energy solutions for a modified Kirchhoff-type equation in R3 , Comput. Math. Appl. 70 (2015), 592–602.

X. Wu, Existence of nontrivial solutions and high energy solutions for Schrödinger–Kirchhoff-type equations in R3 , Nonlinear Anal. 12 (2011), 1278–1287.

Q. Xie and S. Ma, Existence and concentration of positive solutions for Kirchhoff-type problems with a steep well potential, J. Math. Anal. Appl. 431 (2015), 1210–1223.

L. Yang and Z. Liu, Multiplicity and concentration of solutions for fractional Schrödinger equation with sublinear perturbation and steep potential well, Comput. Math. Appl. 72 (2016), 1629–1640.

H. Ye, The existence of least energy nodal solutions for some class of Kirchhoff equations and Choquard equations in RN , J. Math. Anal. Appl. 431 (2015), 935–954.

H. Zhang and F. Zhang, Ground states for the nonlinear Kirchhoff type problems, J. Math. Anal. Appl. 423 (2015), 1671–1692.

J. Zhang, X.H. Tang and W. Zhang, Existence of multiple solutions of Kirchhoff type equation with sign-changing potential, Appl. Math. Comput. 242 (2014), 491–499.

J. Zhang, X.H. Tang and W. Zhang, Ground states for diffusion system with periodic and asymptotically periodic nonlinearity, Comput. Math. Appl. 71 (2016), 633–641.

W. Zhang, X.H. Tang, B. Cheng and J. Zhang, Sign-chang solutions for fourth order elliptic equations with Kirchoff-type, Commun. Pure Appl. Anal. 15 (2016), 2161–2177.

W. Zhang, X.H. Tang and J. Zhang, Existence and concentration of solutions for Schrödinger–Poisson system with steep potential well, Math. Meth. Appl. Sci. 39 (2016), 2549–2557.

L. Zhao, H. Liu and F. Zhao, Existence and concentration of solutions for the Schrödinger–Poisson equations with steep well potential, J. Differential Equations 255 (2013), 1–23.

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