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Multiplicity results for a fourth-order semilinear elliptic problem. (English) Zbl 0898.35032
The paper studies the fourth order elliptic problem \[ \Delta^2 u+c\Delta u= bg (x,u)\quad\text{in } \Omega, \quad\Delta u=0,\;u=0\quad \text{on } \partial \Omega \] by studying the critical points of the related functional \[ f_b(u)= {1\over 2} \left(\int_\Omega (\Delta u)^2-c \int_\Omega |\nabla u|^2\right) -b\int_\Omega G(x,u), \quad u\in H^2 (\Omega) \cap H^1_0 (\Omega) \] where \(G\) is an antiderivative of \(g\) with respect to \(u\). The existence of multiple critical points of \(f_b\) is established by applying the “linking” theorems of M. Schechter and K. Tintarev [Bull. Soc. Math. Belg., Ser. B 44, 249-261 (1992; Zbl 0785.58017)] and A. Marino, A. M. Micheletti and A. Pistoia [Topol. Methods Nonlinear Anal. 4, 289-339 (1994; Zbl 0844.35035)]. The paper finishes with an application of Leray-Schauder degree to prove (in the nondegenerate case) the existence of at least three nontrivial solutions of the equation in the special case \(g=\max \{(u+1)^+ -1,0\}\).

MSC:
35J35 Variational methods for higher-order elliptic equations
35J65 Nonlinear boundary value problems for linear elliptic equations
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[1] Lazer, A.C.; McKenna, P.J., Large-amplitude periodic oscillations in suspension bridges: some new connections with nonlinear analysis, SIAM review, 32, 537-578, (1990) · Zbl 0725.73057
[2] McKenna, P.J.; Walter, W., Nonlinear oscillations in a suspension bridge, Arch. ration. mech. anal., 98, 167-177, (1987) · Zbl 0676.35003
[3] Tarantello, G., A note on a semilinear elliptic problem, Diff. integ. equat., 5, 3, 561-565, (1992) · Zbl 0786.35060
[4] Lazer, A.C.; McKenna, P.J., Global bifurcation and a theorem of tarantello, Journ. math. anal. appl., 181, 648-655, (1994) · Zbl 0797.34021
[5] Marino, A.; Micheletti, A.M.; Pistoia, A., Some variational results on semilinear problems with asympotically nonsymmetric behaviour, (), 243-256 · Zbl 0849.35035
[6] Schechter, M.; Tintarev, K., Pairs of critical points produced by linking subsets with application to semilinear elliptic problems, Bull. soc. math. belg., 44, 3, 249-261, (1992), ser. B · Zbl 0785.58017
[7] Rabinowitz, P., Minimax methods in critical point theory with applications to differential equations, ()
[8] Marino, A.; Micheletti, A.M.; Pistoia, A., A nonsymmetric asymptotically linear elliptic problem, Topol. meth. nonlin. anal., 4, 289-339, (1994) · Zbl 0844.35035
[9] Gilbarg, D.; Trudinger, N.S., Elliptic partial differential eqiuations of second order, (1983), Springer-Verlag New York · Zbl 0691.35001
[10] Dancer, E.N., On the Dirichlet problem for weakly nonlinear elliptic partial differential equations, (), 283-300 · Zbl 0351.35037
[11] Ramos, M., Teoremas de enlaces na teoria DoS pontos críticos, (1993), Faculdade de Ciéncias da Universidade de Lisboa, Departmento del Matemática · Zbl 0964.58011
[12] Dancer, E.N., Generic domain dependence for nonsmooth equations and the open set problem for jumping nonlinearities, Top. met. nonlin. an., 1, 139-150, (1993) · Zbl 0817.35026
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