Existence of positive ground state solutions for Kirchhoff type equation with general critical growth
Keywords
Kirchhoff type problem, ground state solution, variational methodAbstract
We study the existence of positive ground state solutions for the nonlinear Kirchhoff type equation $$ \begin{cases} \displaystyle -\bigg(a+b\int_{\mathbb R^3}|\nabla u|^2\bigg)\Delta {u}+V(x)u =f(u) & \mbox{in }\mathbb R^3, \\ \noalign{\medskip} u\in H^1(\mathbb R^3), \quad u> 0 & \mbox{in } \mathbb R^3, \end{cases} $$% where $a,b> 0$ are constants, $f\in C(\mathbb R,\mathbb R)$ has general critical growth. We generalize a Berestycki-Lions theorem about the critical case of Schrödinger equation to Kirchhoff type equation via variational methods. Moreover, some subcritical works on Kirchhoff type equation are extended to the current critical case.References
C. Alves and F. Corrêa, On existence of solutions for a class of problem involving a nonlinear operator, Appl. Nonlinear Anal. 8 (2001), 43–56.
C. Alves, F. Corrêa and T. Ma, Positive solutions for a quasilinear elliptic equation of Kirchhoff type, Comput. Math. Appl. 49 (2005), 85–93.
C. Alves and G. Figueiredo, Nonliear perturbations of peiodic Krichhoff equation in R^N , Nonlinear Anal. 75 (2012), 2750–2759.
A. Azzollini, The elliptic Kirchhoff equation in RN perturbed by a local nonlinearity, Differ. Integral Equ. 25 (2012), 543–554.
P. D’Ancona and S. Spagnolo, Global solvability for the degenerate Kirchhoff equation with real analytic data, Invent. Math. 108 (1992), 247–262.
H. Berestycki and P. Lions, Nonlinear scalar field equations. I. Existence of a ground state, Arch. Ration. Mech. Anal. 82 (1983), 313–345.
H. Berestycki, T. Gallouët and O. Kavian, Equations de champs scalaires euclidiens non linéaire dans le plan, C.R. Mech. Acad. Sci. Paris Sér. I Math. 297 (1983), 307–310.
H. Brezis and E. Lieb, A relation between pointwise convergence of functions and convergence of functionals, Proc. Amer. Math. Soc. 8 (1983), 486–490.
C. Chen, Y. Kuo and T. Wu, 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.
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, Concentration bound states for Kirchhoff type problems in R^3 involving critical Sobolev exponents, Adv. Nonlinear Stud. 14 (2) (2014), 483–510.
L. Jeanjean, On the existence of bounded Palais–Smale sequence and application to a Landesman–Lazer type problem set on RN , Proc. Roy. Soc. Edinburgh Sect. A 129 (1999), 787–809.
J. Jin and X. Wu, Infinitely many radial solutions for Kirchhoff-type problems in R^N, J. Math. Anal. Appl. 369 (2010), 564–574.
G. Kirchhoff, LATEX – Mechanik, Teubner, Leipzig, 1883.
G. Li and Y. He 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, in: Contemporary Developments in Continuum Mechanics and Partial Differential Equations. Proc. Internat. Sympos. Inst. Mat. Univ. Fed. Rio de Janeiro, (1997), in: North-Holl and Math. Stud. 30 (1978), 284–346.
P. 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; Ann. Inst. H. Poincaré Anal. Non Linéaire 2 (1984), 223–283.
Z. Liu and S. Guo, Positive solutions for asymptotically linear Schrödinger–Kirchhofftype equations, Math. Meth. Appl. Sci. 37 (2014), 571–580.
Z. Liu and S. Guo, Existence and concentration of positive ground states for a Kirchhoff equation involving critical Sobolev exponent, Z. Angew. Math. Phys. 66 (2015), 747–769.
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 S. Guo, Existence of positive ground state solutions for Kirchhoff type problems, Nonlinear Anal. 120 (2015), 1–13.
T. Ma and J. Rivera, Positive solutions for a nonlinear nonlocal elliptic transmission problem, Appl. Math. Lett. 16 (2003), 243–248.
A. Mao and Z. Zhang, Sign-changing and multiple solutions of Kirchhoff type problems without the P.S. condition, Nonlinear Anal. 70 (2009), 1275–1287.
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.
K. Perera and Z. Zhang, Nontrivial solutions of Kirchhoff-type problems via the Yang index, J. Differential Equations 221 (2006), 246–255.
W. Strauss, Existence of solitary waves in higher dimensions, Comm. Math. Phys. 55 (1977), 149–162.
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. 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, LATEX – Minimax Theorems, Birkhäuser, Boston, 1996.
X. Wu, Existence of nontrivial solutions and high energy solutions for Schrödinger–Kirchhoff-type equations in RN , Nonlinear Anal. 12 (2011), 1278–1287.
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.
J. Zhang and W. Zou, The critical case for a Berestycki–Lions theorem, Sci. China Math. 14 (2014), 541–554.
J.J. Zhang and W. Zou, A Berestycki–Lions theorem revisted, Commun. Contemp. Math. 14 (2012), 14 pp.
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