## Sign changing solutions of Kirchhoff type problems via invariant sets of descent flow.(English)Zbl 1100.35008

In the present study the authors obtain sing-changing solutions of the problem: $-\left(a+b\int_\Omega|\nabla u|^2\,dx\right)\Delta u=f (x,u),\quad \text{in }\Omega, \qquad u=0,\quad\text{on }\partial\Omega,\tag{1}$ where $$\Omega$$ is a smooth bounded domain in $$\mathbb{R}^N$$, $$a,b>0$$ and $$f$$ is a given function. Under suitable assumptions on the data, the authors obtain sign-changing solutions for (1). To this end, they use variational methods and invariant sets of descent flow.

### MSC:

 35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs 35J65 Nonlinear boundary value problems for linear elliptic equations 45K05 Integro-partial differential equations

### Keywords:

nonlocal problems; variational methods
Full Text:

### References:

 [1] Alves, C.O.; Correa, F.J.S.A., On existence of solutions for a class of problem involving a nonlinear operator, Comm. appl. nonlinear anal., 8, 43-56, (2001) · Zbl 1011.35058 [2] Alves, C.O.; Corrêa, F.J.S.A.; Ma, T.F., Positive solutions for a quasilinear elliptic equation of Kirchhoff type, Comput. math. appl., 49, 85-93, (2005) · Zbl 1130.35045 [3] Andrade, D.; Ma, T.F., An operator equation suggested by a class of nonlinear stationary problems, Comm. appl. nonlinear anal., 4, 65-71, (1997) · Zbl 0911.47062 [4] Arosio, A.; Panizzi, S., On the well-posedness of the Kirchhoff string, Trans. amer. math. soc., 348, 305-330, (1996) · Zbl 0858.35083 [5] Bernstein, S., Sur une classe d’équations fonctionnelles aux dérivées partielles, Bull. acad. sci. URSS. Sér. math. [izv. akad. nauk SSSR], 4, 17-26, (1940) · JFM 66.0471.01 [6] Cavalcanti, M.M.; Domingos Cavalcanti, V.N.; Soriano, J.A., Global existence and uniform decay rates for the kirchhoff – carrier equation with nonlinear dissipation, Adv. differential equations, 6, 701-730, (2001) · Zbl 1007.35049 [7] Chipot, M.; Lovat, B., Some remarks on nonlocal elliptic and parabolic problems, Nonlinear anal., 30, 4619-4627, (1997) · Zbl 0894.35119 [8] Chipot, M.; Rodrigues, J.-F., On a class of nonlocal nonlinear elliptic problems, RAIRO modél. math. anal. numér., 26, 447-467, (1992) · Zbl 0765.35021 [9] Dancer, E.N.; Zhang, Z., Fucik spectrum, sign-changing, and multiple solutions for semilinear elliptic boundary value problems with resonance at infinity, J. math. anal. appl., 250, 449-464, (2000) · Zbl 0974.35028 [10] D’Ancona, P.; Spagnolo, S., Global solvability for the degenerate Kirchhoff equation with real analytic data, Invent. math., 108, 247-262, (1992) · Zbl 0785.35067 [11] Kirchhoff, G., Mechanik, (1883), Teubner Leipzig · JFM 08.0542.01 [12] Lions, J.-L., On some questions in boundary value problems of mathematical physics, (), 284-346 [13] Ma, T.F.; Muñoz Rivera, J.E., Positive solutions for a nonlinear nonlocal elliptic transmission problem, Appl. math. lett., 16, 243-248, (2003) · Zbl 1135.35330 [14] Pohožaev, S.I., A certain class of quasilinear hyperbolic equations, Mat. sb. (N.S.), 96, 152-166, (1975), 168 [15] J.X. Sun, On Some Problems about Nonlinear Operators, PhD thesis, Shandong University, Jinan, 1984 [16] Sun, J.X., The Schauder condition in the critical point theory, Kexue tongbao (English ed.), 31, 1157-1162, (1986) · Zbl 0603.47045 [17] Vasconcellos, C.F., On a nonlinear stationary problem in unbounded domains, Rev. mat. univ. complut. Madrid, 5, 309-318, (1992) · Zbl 0780.35035
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