### On ground state solutions of Nehari-Pohozaev type for the nonlinear Kirchhoff type problems with a general critical nonlinearity

DOI: http://dx.doi.org/10.12775/TMNA.2019.010

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C.O. Alves and F. Corrêa, On existence of solutions for a class of problem involving a nonlinear operator, Comm. Appl. Nonlinear Anal. 8 (2001), 43–56.

A. Arosio and S. Panizzi, On the well-posedness of the Kirchhoff string, Trans. Amer. Math. Soc. 348 (1996), 305–330.

H. Berestycki and P. Lions, Nonlinear scalar field equations. I. Existence of a ground state, Arch. Ration. Mech. Anal. 82 (1983), 313–345.

H. Brezis and E.H. Lieb, A relation between pointwise convergence of functions and convergence of functionals, Proc. Amer. Math. Soc. 8 (1983), 486-490.

H. Brezis and L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Comm. Pure Appl. Math. 36 (1983), 437-477.

G. Che and H. Chen, Existence and multiplicity of systems of Kirchhoff-type equations with general potentials, Math. Methods Appl. Sci. 40 (2017), no. 3, 775–785.

C. Chen, Y. Kuo and T.F. Wu, ıt The Nehari manifold for a Kirchhoff type problem involving sign-changing weight functions, J. Differential Equations 250 (2011), 1876–1908.

M. Chipot and B. Lovat, Some remarks on nonlocal elliptic and parabolic problems, Nonlinear Anal. 30 (1997), 4619–4627.

Z. Guo, Ground states for Kirchhoff equations without compact condition, J. Differential Equations 259 (2015), 2884–2902.

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.

L. Jeanjean, On the existence of bounded Palais–Smale sequences and application to a Landesman–Lazer-type problem set on RN , Proc. Roy. Soc. Edinburgh Sect. A 129 (1999), 787–809.

Y. He and G. Li, Standing waves for a class of Kirchhoff type problems in R3 involving critical Sobolev exponents, Calc. Var. 54 (2015), 3067–3106.

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

G. Li and H. Ye, Existence of positive ground state solutions for the nonlinear Kirchhoff type equations in R3 , J. Differential Equations 257 (2014), 566–600.

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.

J. 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 (1997); North-Holland Math. Stud. 30 (1978), 284–346.

H. Liu and H. Chen, Ground-state solution for a class of biharmonic equations including critical exponent, Z. Angew. Math. Phys. 66 (2015), 3333–3343.

Z. Liu and S. Guo, Existence of positive ground state solutions for Kirchhoff type problems, Nonlinear Anal. 120 (2015), 1–13.

Z. Liu and S. Guo, On ground states for the Kirchhoff-type problem with a general critical nonlinearity, J. Math. Anal. Appl. 426 (2015), 267–287.

Z. Liu and C. Luo, Existence of positive ground state solutions for Kirchhoff type equation with general critical growth, Topol. Methods Nonlinear Anal. 49 (2017), 165–182.

Z. Liu, M. Squassina and J. Zhang, Ground states for fractional Kirchhoff equations with critical nonlinearity in low dimension, Nonlinear Differ. Equ. Appl. 24 (2017), 50.

P. Rabinowitz, On a class of nonlinear Schrödinger equations, Z. Angew. Math. Phys. 43 (1992), 270–291.

D. Ruiz, The Schrödinger–Poisson equation under the effect of a nonlinear local term, J. Funct. Anal. 237 (2006), 655–674.

M. Schechter, Linking Methods in Critical Point Theory, Birkhäuser, Boston, 1999.

H. Shi and H.B. Chen, Ground state solutions for asymptotically periodic coupled Kirchhoff-type systems with critical growth, Math. Methods Appl. Sci. 39 (2016), 2193–2201.

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

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

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

X. Tang and S. Chen, Ground state solutions of Nehari–Pohozaev type for Schrödinger–Poisson type problems with general potentials, Discrete Contin. Dyn. Syst. 37 (2017), no. 9, 4973–5002.

X. Tang and S. Chen, Ground state solutions of Nehari–Pohozaev type for Kirchhoff type problems with general potentials, Calc. Var. 56 (2017), 110.

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, Berlin, 1996.

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

W. Xie and H. Chen, Existence and multiplicity of normalized solutions for the nonlinear Kirchhoff type problems, Comput. Math. Appl. 76 (2018), 579–591.

L. Xu and H. Chen, Nontrivial solutions for Kirchhoff-type problems with a parameter, J. Math. Anal. Appl.433 (2016), 455-472.

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

Z. 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.

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