×

Sign-changing and multiple solutions of Kirchhoff type problems without the P.S. condition. (English) Zbl 1160.35421

From the summary: We use minimax methods and invariant sets of descent flow to prove two existence theorems for the 4-superlinear Kirchhoff type problems
\[ \begin{aligned} -\bigg(a+b \int_\Omega |\nabla u|^2\bigg)\Delta u= f(x,u) &\quad\text{in }\Omega,\\ u=0 &\quad\text{on }\partial\Omega, \end{aligned} \]
without the P.S. condition, one concerning the existence of a nontrivial solution and the other one concerning the existence of sign-changing solutions and multiple solutions.

MSC:

35J65 Nonlinear boundary value problems for linear elliptic equations
35J20 Variational methods for second-order elliptic equations
47J10 Nonlinear spectral theory, nonlinear eigenvalue problems
PDFBibTeX XMLCite
Full Text: DOI

References:

[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: Teubner Leipzig
[3] Bernstein, S., Sur une classe d’équations fonctionnelles aux dérivées partielles, Bull. Acad. Sci. URSS. Sér., 4, 17-26 (1940), (Izvestia Akad. Nauk SSSR) · Zbl 0026.01901
[4] Pohožaev, S. I., A certain class of quasilinear hyperbolic equations, Mat. Sb. (NS), 96, 138 (1975), 152-166, 168 (in Russian) · Zbl 0309.35051
[5] Lions, J. L., On some questions in boundary value problems of mathematical physics, (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 (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, New York), 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.; 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, 6, 701-730 (2001) · Zbl 1007.35049
[9] D’Ancona, 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] Ma, T. F.; Munoz Rivera, J. E., Positive solutions for a nonlinear elliptic transmission problem, Appl. Math. Lett., 16, 2, 243-248 (2003) · Zbl 1135.35330
[11] Bartsch, T.; Ding, Y. H., On a nonlinear Schrödinger equation with periodic potential, Math. Ann., 313, 15-37 (1999) · Zbl 0927.35103
[12] Alama, S.; Li, Y. Y., Existence of solutions for semilinear elliptic equations with indefinite linear part, J. Differential Equations, 96, 89-115 (1992) · Zbl 0766.35009
[13] Buffoni, B.; Jeanjean, L.; Stuart, C. A., Existence of nontrivial solutions to a strongly indefinite semilinear equation, Proc. Amer. Math. Soc., 119, 179-186 (1993) · Zbl 0789.35052
[14] 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
[15] Jeanjean, L., Solutions in spectral gaps for a nonlinear equation of Schrödinger type, J. Differential Equations, 112, 53-80 (1994) · Zbl 0804.35033
[16] 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
[17] Troestler, C.; Willem, M., Nontrivial solution of a semilinear Schrödinger equation, Comm. Partial Differential Equations, 21, 1431-1449 (1996) · Zbl 0864.35036
[18] Ambrosetti, A.; Rabinowitz, P., Dual variational methods in critical point theory and applications, J. Funct. Anal., 41, 349-381 (1973) · Zbl 0273.49063
[19] Perera, K.; Schechter, M., Double resonance problems with respect to the Fucik spectrum, Indiana Univ. Math. J., 52, 1, 1C18 (2003) · Zbl 1030.35079
[20] Schechter, M., A variation of the mountain pass lemma and applications, J. London Math. Soc., 2, 44, 491-502 (1991) · Zbl 0756.35032
[21] Mao, A. M.; Luan, S. X.; Ding, Y. H., Periodic solutions for a class of first order super-quadratic Hamiltonian system, J. Math. Anal. Appl., 330, 1, 584-596 (2007) · Zbl 1122.37048
[22] Dancer, E. N.; Zhang, Z. T., Fucik spectrum, sign-changing, and multiple solutions for semilinear elliptic boundary value problems with resonance at infinity, J. Math. Anal. Appl., 250, 2, 449-464 (2000) · Zbl 0974.35028
[23] Kryszewski, W.; Szulkin, A., Generalized linking theorem with an application to semilinear Schrödinger equations, Adv. Differential Equations, 3, 441-472 (1998) · Zbl 0947.35061
[24] Ding, Y. H., Multiple homoclinics in a Hamiltonian system with asymptotically or super linear terms, Commun. Contemp. Math., 8, 4, 453-480 (2006) · Zbl 1104.70013
[25] Cerami, G., Un criterio di esistenza per i punti critici su varietá illimitate, Istit. Lombardo Accad. Sci. Lett. Rend. A, 112, 332-336 (1978) · Zbl 0436.58006
[26] J.X. Sun, On some problems about nonlinear operators, Ph.D. Thesis, Shandong University, Jinan, 1984; J.X. Sun, On some problems about nonlinear operators, Ph.D. Thesis, Shandong University, Jinan, 1984
[27] Sun, J. X.; Liu, Z. L., Calculus of variations and super- and sub-solutions in reverse order, Acta. Math. Sinica, 37, 4, 512-514 (1994), (in Chinese) · Zbl 0810.47059
[28] Liu, Z. L.; Sun, J. X., Invariant sets of descending flow in critical point theory with applications to nonlinear differential equations, J. Differential Equations, 172, 2, 257-299 (2001) · Zbl 0995.58006
[29] Liu, Z. L.; Sun, J. X., Four versus two solutions of semilinear elliptic boundary value problems, Calc. Var. Partial Differential Equations, 14, 3, 319-327 (2002) · Zbl 0996.35017
[30] Liu, Z. L.; Van Heerden, F. A.; Francois, A.; Wang, Z. Q., Nodal type bound states of Schrödinger equations via invariant set and minimax methods, J. Differential Equations, 214, 2, 358-390 (2005) · Zbl 1210.35044
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.