Positive solutions of Kirchhoff-Hénon type elliptic equations with critical Sobolev growth

Kazune Takahashi

Abstract


We investigate the following Kirchhoff-Hénon type equation involving the critical Sobolev exponent with Dirichlet boundary condition: \[ - \bigg( a + b \bigg( \int_\Omega \lvert Du \rvert^2 dx \bigg)^{(p-2)/2} \bigg) \Delta u = \Psi u^{q-1} + \lvert x \rvert^\alpha u^{2^* - 1} \] in $\Omega$ included in a unit ball under several conditions. Here, $a, b \geq 0$, $a + b > 0$, $2 < p < q < 2^*$ and $\Psi \in L^\infty(\Omega) \setminus \{ 0 \}$ is a given non-negative function with several conditions. We show that, if either $N = 3$ with $4 < q < 2^* = 6$ or $N \geq 4$, there exists a positive solution for small $\alpha \geq 0$. Our methods includes the mountain pass theorem and the Talenti function.

Keywords


Critical Sobolev exponent; Kirchhoff equation; Henon equation; mountain pass theorem; Talenti function

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References


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), no. 1, 85–93.

A. Ambrosetti and P. H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Funct. Anal. 14 (1973), 349–381.

A. Azzollini, P. d’Avenia and A. Pomponio, Multiple critical points for a class of nonlinear functionals, Ann. Mat. Pura Appl. (4) 190 (2011), no. 3, 507–523.

H. Brézis, J.-M. Coron and L. Nirenberg, Free vibrations for a nonlinear wave equation and a theorem of P. Rabinowitz, Comm. Pure Appl. Math. 33 (1980), no. 5, 667–684.

H. Brézis and L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Comm. Pure Appl. Math. 36 (1983), no. 4, 437–477.

P. Chen and X. Liu, Multiplicity of solutions to Kirchhoff type equations with critical Sobolev exponent, Commun. Pure Appl. Anal. 17 (2018), no. 1, 113–125.

Y. Duan, X. Sun and J.-F. Liao, Multiplicity of positive solutions for a class of critical Sobolev exponent problems involving Kirchhoff-type nonlocal term, Comput. Math. Appl. 75 (2018), no. 12, 4427–4437.

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), no. 2, 706–713.

Y. Li, F. Li i and J. Shi, Existence of a positive solution to Kirchhoff type problems without compactness conditions, J. Differential Equations 253 (2012), no. 7, 2285–2294.

P.-L. Lions, The concentration-compactness principle in the calculus of variations. The limit case. I, Rev. Mat. Iberoamericana 1 (1985), no. 1, 145–201.

Z. Liu and Ch. Luo, Existence of positive ground state solutions for Kirchhoff type equation with general critical growth, Topol. Methods Nonlinear Anal. 49 (2017), no. 1, 165–182.

W. Long and J. Yang, Existence for critical Hénon-type equations, Differential Integral Equations 25 (2012), no. 5–6, 567–578.

T.F. Ma and J.E. Muñoz Rivera, Positive solutions for a nonlinear nonlocal elliptic transmission problem, Appl. Math. Lett. 16 (2003), no. 2, 243–248.

D. Naimen, The critical problem of Kirchhoff type elliptic equations in dimension four, J. Differential Equations 257 (2014), no. 4, 1168–1193.

D. Naimen, Positive solutions of Kirchhoff type elliptic equations involving a critical Sobolev exponent, NoDEA Nonlinear Differential Equations Appl. 21 (2014), no. 6, 885–914.

R.S. Palais and S. Smale, A generalized Morse theory, Bull. Amer. Math. Soc. 70 (1964), 165–172.

S. Secchi, The Brezis–Nirenberg problem for the Hénon equation: ground state solutions, Adv. Nonlinear Stud. 12 (2012), no. 2, 383–394.

K. Takahashi, Positive solution for Hénon type equations with critical Sobolev growth, Electron. J. Differential Equations 2018 (2018), no. 194, 1–17.

G. Talenti, Best constant in Sobolev inequality, Ann. Mat. Pura Appl. (4) 110 (1976), 353–372.

X. H. Tang and S. Chen, Ground state solutions of Nehari–Pohozaev type for Kirchhofftype problems with general potentials, Calc. Var. Partial Differential Equations 56 (2017), no. 4, Art. 110, pp. 25.

L. Zeng and Ch.L. Tang, Existence of a positive ground state solution for a Kirchhoff type problem involving a critical exponent, Ann. Polon. Math. 117 (2016), no. 2, 163–180.

Ch. Zhang and Z. Liu, Multiplicity of nontrivial solutions for a critical degenerate Kirchhoff type problem, Appl. Math. Lett. 69 (2017), 87–93.

J. Zhang and Y. Ji, The existence of nontrivial solutions for the critical Kirchhoff type problem in RN , Comput. Math. Appl. 74, (2017), no. 12, 3080–3094.

X.-J. Zhong and Ch.-L. Tang, Multiple positive solutions to a Kirchhoff type problem involving a critical nonlinearity, Comput. Math. Appl. 72 (2016), no. 12, 2865–2877.


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