Existence of solution for a Kirchhoff type system with weight and nonlinearity involving a $(p,q)$-superlinear term and critical Caffarelli-Kohn-Nirenberg growth

Rodrigo da Silva Rodrigues, Mateus Balbino Guimarães


We study a $(p,q)$-Laplacian system of Kirchhoff type equations with weight and nonlinearity involving a $(p,q)$-superlinear term, in which $p$ may be different from $q$, and with critical Caffarelli-Kohn-Nirenberg exponent. Using the Mountain Pass Theorem, we obtain a nontrivial solution to the problem.


Nonlocal problems; variational methods; critical exponents; Kirchhoff type equations; nonlinear elliptic systems; mountain pass theorem

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C. Alves, F. Corrêa and T.F. Ma, Positive solutions for a quasilinear elliptic equation of Kirchhoff type, Comput. Math. Appl. 49 (2005), 85–93.

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

H. Brézis and E. Lieb, A relation between pointwise convergence of functions and convergence of functionals, Proc. Amer. Math. Soc. 88 (1983), 486–490.

L. Caffarelli, R. Kohn and L. Nirenberg, First order interpolation inequalities with weights, Compos. Math. 53 (1984), 259–275.

B. Cheng, X. Wu and J. Liu, Multiplicity of nontrivial solutions for Kirchhoff type problems, Bound. Value Probl. (2010), Article ID 268946, 13 p.

N.T. Chung, An existence result for a class of Kirchhoff type systems via sub and supersolutions method, Applied Mathematics Letters 35 (2014), 95–101.

N.T Chung and H.Q Toan, Existence and multiplicity of solutions for a degenerate nonlocal elliptic differential equation, Electron. J. Differential Equations 148 (2013), 1–13.

F.J.S.A. Corrêa and R.G. Nascimento, On a nonlocal elliptic system of p-Kirchhoff type under Neumann boundary condition, Math. Comput. Modelling 49 (2009), 598–604.

F.J.S.A. Corrêa and R.G. Nascimento, On the existence of solutions of a nonlocal elliptic equation with a p-Kirchhoff type term, Int. J. Math. Math. Sci. (2008), Article ID 364085, 25 p.

N. Ghoussoub and C. Yuan, Multiple solutions for quasi-linear PDEs involving the critical Sobolev and Hardy exponents, Trans. Amer. Math. Soc. 352 (1998), 5703–5743.

G. Figueiredo and J. dos Santos Junior, Multiplicity of solutions for a Kirchhoff equation with subcritical or critical growth, Differential Integral Equations 25 (2012), 853–868.

G. Figueiredo and J. dos Santos Junior, On a p-Kirchhoff equation via Krasnosel’skiı̆’s genus, Appl. Math. Lett. 22 (2009), 819–822.

T. Horiuchi, Best constant in weighted Sobolev inequality with weights being powers of distance from origin, J. Inequal. Appl. 1 (1997), 275–292.

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

P. Lions, The concentration-compactness principle in the calculus of variations. The limit case, Rev. Mat. Iberoam. 1 (1985), 145–201.

E. Silva and M. Xavier, Quasilinear elliptic system with coupling on nonhomogeneous critical term, Nonlinear Anal. 69 (2008), 1164–1178.

B. Xuan, The solvability of quasilinear Brezis–Nirenberg-type problems with singular weights, Nonlinear Anal. 62 (2005), 703–725.

M. Willem, Minimax theorems, Birkhäuser, 1996.


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