Sign-changing solutions of a class of nonlocal quasilinear elliptic boundary value problems. (English) Zbl 1222.35092

Summary: This paper is concerned with nonlocal Kirchhoff’s equation
\[ \begin{cases} -\left(a+b\int_\Omega |\nabla u|^2\right)\Delta u=f(x,u) &\text{in }\Omega,\\ u=0 &\text{on }\partial\Omega, \end{cases} \]
via variational methods and invariant sets of descent flow. We obtain existence of signed and sign-changing solutions with asymptotically 3-linear bounded nonlinearity.


35J65 Nonlinear boundary value problems for linear elliptic equations
35J62 Quasilinear elliptic equations
35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs
Full Text: DOI


[1] Chipot, M.; Lovat, B., Some remarks on nonlocal elliptic and parabolic problems, Nonlinear anal., 30, 7, 4619-4627, (1997) · Zbl 0894.35119
[2] Kirchhoff, G., Mechanik, (1883), Teubner Leipzig · JFM 08.0542.01
[3] Bernstein, S., Sur une classe dʼéquations fonctinnelles aux dérivées partielles, Bull. acad. sci. URSS Sér. [izv. akad. nauk SSSR], 4, 17-26, (1940) · JFM 66.0471.01
[4] Pohožaev, S.I., A certain class of quasilinear hyperbolic equations, Mat. sb. (N.S.), 96, 138, 152-166, (1975), 168 (in Russian)
[5] Lions, J.L., On some questions in boundary value problems of mathematical physics, (), 284-346 · Zbl 0404.35002
[6] Arosio, A.; Panizzi, S., On the well-posedness of the Kirchhoff string, Trans. amer. math. soc., 348, 1, 305-330, (1996) · Zbl 0858.35083
[7] Alves, C.O.; Corra, F.J.S.A.; Ma, T.F., Positive solutions for a quasilinear elliptic equation of Kirchhoff type, Comput. math. appl., 49, 1, 85-93, (2005) · Zbl 1130.35045
[8] Cavalcanti, M.M.; Cavalcanti, V.N.Domingos; Soriano, J.A., Global existence and uniform decay rates for the Kirchhoff-carrier equation with nonlinear dissipation, Adv. differential equations, 6, 6, 701-730, (2001) · Zbl 1007.35049
[9] DʼAncona, P.; Spagnolo, S., Global solvability for the degenerate Kirchhoff equation with real analytic data, Invent. math., 108, 2, 247-262, (1992) · Zbl 0785.35067
[10] Perera, K.; Zhang, Z.T., Nontrivial solutions of Kirchhoff-type problems via the Yang index, J. differential equations, 221, 1, 246-255, (2006) · Zbl 1357.35131
[11] Zhang, Z.T.; Perera, K., Sign changing solutions of Kirchhoff type problems via invariant sets of descent flow, J. math. anal. appl., 317, 2, 456-463, (2006) · Zbl 1100.35008
[12] Mao, A.M.; Zhang, Z.T., Sign-changing and multiple solutions for a class of Kirchhoff type problems without P.S. condition, Nonlinear anal., 70, 1275-1287, (2009) · Zbl 1160.35421
[13] He, X.M.; Zou, W.M., Imfinitely many positive solutions for Kirchhoff-type problems, Nonlinear anal., 70, 1407-1414, (2009) · Zbl 1157.35382
[14] Liu, Z.L.; Su, J.B.; Weth, T., Compactness results for Schrödinger equations with asymptotically linear terms, J. differential equations, 231, 501-512, (2006) · Zbl 1387.35246
[15] J.X. Sun, On some problems about nonlinear operators, PhD thesis, Shandong University, Jinan, 1984.
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