# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
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 $\bbfR^N$, $a,b>0$ 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. (PDE) 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) · Zbl 0026.01901 [6] 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, 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) · Zbl 08.0542.01 [12] Lions, J. -L.: On some questions in boundary value problems of mathematical physics. North-holland math. Stud. 30, 284-346 (1978) [13] Ma, T. F.; Rivera, J. E. Muñoz: 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) [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