Previous |  Up |  Next

Article

Keywords:
genus theory; variational method; half-line; Kirchhoff equation; Palais-Smale condition
Summary:
We study the existence and multiplicity of infinitely distinct pairs of nontrivial solutions for a boundary value problem of Kirchhoff type posed on the half-line via the genus theory, which is a technique in critical point theory.
References:
[1] 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 (2005), 85-93. DOI 10.1016/j.camwa.2005.01.008 | MR 2123187 | Zbl 1130.35045
[2] Ambrosetti, A., Malchiodi, A.: Nonlinear Analysis and Semilinear Elliptic Problems. Cambridge Studies in Advanced Mathematics 104. Cambridge University Press, Cambridge (2007). DOI 10.1017/CBO9780511618260 | MR 2292344 | Zbl 1125.47052
[3] Arosio, A., Panizzi, S.: On the well-posedness of the Kirchhoff string. Trans. Am. Math. Soc. 348 (1996), 305-330. DOI 10.1090/S0002-9947-96-01532-2 | MR 1333386 | Zbl 0858.35083
[4] Badiale, M., Serra, E.: Semilinear Elliptic Equations for Beginners: Existence Results via the Variational Approach. Universitext. Springer, London (2011). DOI 10.1007/978-0-85729-227-8 | MR 2722059 | Zbl 1214.35025
[5] Benhamida, G., Moussaoui, T.: Existence and multiplicity of solutions for a fractional $p$-Laplacian problem of Kirchhoff type via Krasnoselskii's genus. Math. Bohem. 143 (2018), 189-200. DOI 10.21136/MB.2017.0010-17 | MR 3831486 | Zbl 1463.35493
[6] Brezis, H.: Functional Analysis, Sobolev Spaces and Partial Differential Equations. Universitext. Springer, New York (2011). DOI 10.1007/978-0-387-70914-7 | MR 2759829 | Zbl 1220.46002
[7] Castro, A.: Metodos Variacionales y Analisis Funcional no Linear. X. Coloquio Colombiano de Matematicas (1980), Spanish.
[8] Clark, D. C.: A variant of the Lusternik-Schnirelman theory. Indiana Univ. Math. J. 22 (1972), 65-74. DOI 10.1512/iumj.1972.22.22008 | MR 0296777 | Zbl 0228.58006
[9] Corrêa, F. J. S. A., Figueiredo, G. M.: On a $p$-Kirchhoff equation via Krasnoselskii's genus. Appl. Math. Lett. 22 (2009), 819-822. DOI 10.1016/j.aml.2008.06.042 | MR 2523587 | Zbl 1171.35371
[10] D'Ancona, P., Spagnolo, S.: Global solvability for the degenerate Kirchhoff equation with real analytic data. Invent. Math. 108 (1992), 247-262. DOI 10.1007/BF02100605 | MR 1161092 | Zbl 0785.35067
[11] Frites, O., Moussaoui, T., O'Regan, D.: Existence of solutions for a variational inequality on the half-line. Bull. Iran. Math. Soc. 43 (2017), 223-237. MR 3622368 | Zbl 1373.47055
[12] He, X., Zou, W.: Infinitely many positive solutions for Kirchhoff-type problems. Nonlinear Anal., Theory Methods Appl., Ser. A 70 (2009), 1407-1414. DOI 10.1016/j.na.2008.02.021 | MR 2474927 | Zbl 1157.35382
[13] Jin, J., Wu, X.: Infinitely many radial solutions for Kirchhoff-type problems in $\Bbb R^{N}$. J. Math. Anal. Appl. 369 (2010), 564-574. DOI 10.1016/j.jmaa.2010.03.059 | MR 2651702 | Zbl 1196.35221
[14] Kavian, O.: Introduction à la théorie des points critiques et applications aux problèmes elliptiques. Mathématiques & Applications (Berlin) 13. Springer, Paris (1993), French. MR 1276944 | Zbl 0797.58005
[15] Krasnosel'skii, M. A.: Topological Methods in the Theory of Nonlinear Integral Equations. International Series of Monographs on Pure and Applied Mathematics 45. MacMillan, New York (1964). MR 0159197 | Zbl 0111.30303
[16] Mokhtari, A., Moussaoui, T., O'Regan, D.: Multiplicity results for an impulsive boundary value problem of $p(t)$-Kirchhoff type via critical point theory. Opsc. Math. 36 (2016), 631-649. DOI 10.7494/OpMath.2016.36.5.631 | MR 3520804 | Zbl 1358.34038
[17] Peral, I.: Multiplicity of solutions for the $p$-Laplacian. Second School of Nonlinear Functional Analysis and Applications to Differential Equations 1997. Available at {\def\let \relax \brokenlink{ https://matematicas.uam.es}{/ ireneo.peral/ICTP.pdf}}.
[18] Rabinowitz, P. H.: Minimax Methods in Critical Point Theory with Applications to Differential Equations. Regional Conference Series in Mathematics 65. AMS, Providence (1986). DOI 10.1090/cbms/065 | MR 0845785 | Zbl 0609.58002
[19] Triebel, H.: Interpolation Theory, Function Spaces, Differential Operators. North-Holland Mathematical Library 18. North-Holland, Amsterdam (1978). MR 0503903 | Zbl 0387.46032
Partner of
EuDML logo