Zhang, Zhitao; Perera, Kanishka Sign changing solutions of Kirchhoff type problems via invariant sets of descent flow. (English) Zbl 1100.35008 J. Math. Anal. Appl. 317, No. 2, 456-463 (2006). 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. Reviewer: Messoud A. Efendiev (Berlin) Cited in 1 ReviewCited in 302 Documents 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 PDFBibTeX XMLCite \textit{Z. Zhang} and \textit{K. Perera}, J. Math. Anal. Appl. 317, No. 2, 456--463 (2006; Zbl 1100.35008) Full Text: DOI 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: Teubner Leipzig · JFM 08.0542.01 [12] Lions, J.-L., On some questions in boundary value problems of mathematical physics, (Contemporary Developments in Continuum Mechanics and Partial Differential Equations. Contemporary Developments in Continuum Mechanics and Partial Differential Equations, Proc. Internat. Sympos., Inst. Mat., Univ. Fed. Rio de Janeiro, Rio de Janeiro, 1977. Contemporary Developments in Continuum Mechanics and Partial Differential Equations. Contemporary Developments in Continuum Mechanics and Partial Differential Equations, Proc. Internat. Sympos., Inst. Mat., Univ. Fed. Rio de Janeiro, Rio de Janeiro, 1977, North-Holland Math. Stud., vol. 30 (1978), North-Holland: North-Holland Amsterdam), 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; 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 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.