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A three critical points theorem revisited. (English) Zbl 1214.47079
Let $X$ be a reflexive Banach space, $I \subset\Bbb R$ an interval; $\Phi: X \to\Bbb R$ a sequentially weakly lower semicontinuous $C^1$ functional, bounded on each bounded subset of $X$, whose derivative admits a continuous inverse on $X^*$; $J: X \to\Bbb R$ a $C^1$ functional with compact derivative. Assume that $\lim_{\Vert x\Vert \to \infty}(\Phi(x)+\lambda J(x))=+\infty$ for all $\lambda \in I$, and there exists $\rho \in\Bbb R$ such that $\sup_{\lambda \in I}\inf_{x \in X}(\Phi(x)+\lambda(J(x)+\rho))<\inf_{x \in X}\sup_{\lambda \in I}(\Phi(x)+\lambda(J(x)+\rho))$. Then there exists a subset $A \subset I$, $A \neq \emptyset$, and $r>0$ with the following property: for every $\lambda \in A$ and every $C^1$ functional $\Psi: X \to\Bbb R$ with compact derivative, there exists $\delta>0$ such that, for each $\mu \in [0,\delta]$, the equation $\Phi'(x)+\lambda J'(x)+\mu \Psi'(x)=0$ has at least three solutions in $X$ whose norms are less than $r$.

MSC:
47J30Variational methods (nonlinear operator equations)
58E05Abstract critical point theory
49J35Minimax problems (existence)
35J60Nonlinear elliptic equations
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References:
[1] Afrouzi, G. A.; Heidarkhani, S.: Three solutions for a Dirichlet boundary value problem involving the p-Laplacian. Nonlinear anal. 66, 2281-2288 (2007) · Zbl 05144893
[2] G.A. Afrouzi, S. Heidarkhani, Three solutions for a quasilinear boundary value problem, Nonlinear Anal., in press (doi:10.1016/j.na.2007.09.022) · Zbl 1158.34311
[3] G.A. Afrouzi, S. Heidarkhani, Existence of three solutions for a class of Dirichlet quasilinear elliptic systems involving the (p1,...,pn)-Laplacian, Nonlinear Anal., in press (doi:10.1016/j.na.2007.11.038) · Zbl 1161.35371
[4] Bai, Z. B.; Ge, W. G.: Existence of three positive solutions for some second-order boundary value problems. Comput. math. Appl. 48, 699-707 (2004) · Zbl 1066.34019
[5] Barletta, G.; Livrea, R.: Existence of three periodic solutions for a non autonomous system. Matematiche 57, 205-215 (2002) · Zbl 1195.34062
[6] Bonanno, G.: Existence of three solutions for a two point boundary value problem. Appl. math. Lett. 13, 53-57 (2000)
[7] Bonanno, G.: A minimax inequality and its applications to ordinary differential equations. J. math. Anal. appl. 270, 210-229 (2002) · Zbl 1009.49004
[8] Bonanno, G.: Multiple solutions for a Neumann boundary value problem. J. nonlinear convex anal. 4, 287-290 (2003) · Zbl 1042.34034
[9] Bonanno, G.: Some remarks on a three critical points theorem. Nonlinear anal. 54, 651-665 (2003) · Zbl 1031.49006
[10] Bonanno, G.; Candito, P.: Three solutions to a Neumann problem for elliptic equations involving the p-Laplacian. Arch. math. (Basel) 80, 424-429 (2003) · Zbl 1161.35382
[11] Bonanno, G.; Livrea, R.: Multiplicity theorems for the Dirichlet problem involving the p-Laplacian. Nonlinear anal. 54, 1-7 (2003) · Zbl 1163.35367
[12] Bonanno, G.; Candito, P.: On a class of nonlinear variational--hemivariational inequalities. Appl. anal. 83, 1229-1244 (2004) · Zbl 1149.35354
[13] Cammaroto, F.; Chinnì, A.; Di Bella, B.: Multiple solutions for a two point boundary value problem. J. math. Anal. appl. 323, 530-534 (2006) · Zbl 1112.34307
[14] Cammaroto, F.; Chinnì, A.; Di Bella, B.: Multiple solutions for a quasilinear elliptic variational system on strip-like domains. Proc. edinb. Math. soc. 50, 597-603 (2007) · Zbl 1136.35062
[15] Cammaroto, F.; Chinnì, A.; Di Bella, B.: Multiplicity results for a perturbed nonlinear Schrödinger equation. Glasg. math. J. 49, 423-429 (2007) · Zbl 1132.35474
[16] Cammaroto, F.; Chinnì, A.; Di Bella, B.: Multiple solutions for a Dirichlet problem involving the p-Laplacian. Dynam. systems appl. 16, 673-679 (2007) · Zbl 1146.35364
[17] Cordaro, G.: On a minimax problem of ricceri. J. inequal. Appl. 6, 261-285 (2001) · Zbl 0986.49003
[18] Cordaro, G.: Three periodic solutions to an eigenvalue problem for a class of second order Hamiltonian systems. Abstr. appl. Anal. 18, 1037-1045 (2003) · Zbl 1093.34019
[19] Cordaro, G.: Further results related to a minimax problem of ricceri. J. inequal. Appl., 523-533 (2005) · Zbl 1105.49010
[20] Faraci, F.; Iannizzotto, A.; Kupán, P.; Varga, Cs.: Existence and multiplicity results for hemivariational inequalities two parameters. Nonlinear anal. 67, 2654-2669 (2007) · Zbl 1130.47042
[21] Jiang, L.; Zhou, Z.: Three solutions to Dirichlet boundary value problems for p-Laplacian difference equations. Adv. difference equ., 10pp (2008) · Zbl 1146.39028
[22] Kristály, A.: Multiplicity results for an eigenvalue problem for hemivariational inequalities in strip-like domains. Set-valued anal. 13, 85-103 (2005) · Zbl 1080.35058
[23] Kristály, A.: Existence of two nontrivial solutions for a class of quasilinear elliptic variational systems on strip-like domains. Proc. edinb. Math. soc. 48, 465-477 (2005) · Zbl 1146.35349
[24] Kristály, A.; Varga, Cs.: On a class of nonlinear eigenvalue problems in RN. Math. nachr. 278, 1756-1765 (2005) · Zbl 1161.35456
[25] Kristály, A.: A double eigenvalue problem for Schrödinger equations involving sublinear nonlinearities at infinity. Electron. J. Differential equations, No. 42, 11 (2007) · Zbl 1189.35096
[26] Kristály, A.; Varga, Cs.: Multiple solutions for elliptic problems with singular and sublinear potentials. Proc. amer. Math. soc. 135, 2121-2126 (2007) · Zbl 1123.35020
[27] Kristály, A.: Multiple solutions for a sublinear Schrödinger equation. Nodea nonlinear differential equations appl. 14, 291-301 (2007) · Zbl 1136.35090
[28] Kristály, A.; Lisei, H.; Varga, Cs.: Multiple solutions for p-Laplacian type equations. Nonlinear anal. 68, 1375-1381 (2008) · Zbl 1136.35034
[29] A. Kristály, V. Rădulescu, Sublinear eigenvalue problems on compact Riemannian manifolds with applications in Emden--Fowler equations (preprint) · Zbl 1213.58022
[30] A. Kristály, M. Mihăilescu, V. Rădulescu, Two nontrivial solutions for a non-homogeneous Neumann problem: An Orlicz--Sobolev setting, Proc. Roy. Soc. Edinburgh Sect. A (in press)
[31] C. Li, C.-L. Tang, Three solutions for a class of quasilinear elliptic systems involving the (p,q)-Laplacian, Nonlinear Anal., in press (doi:10.1016/j.na.2007.09.021)
[32] Lisei, H.; Varga, Cs.; Horváth, A.: Multiplicity results for a class of quasilinear problems on unbounded domains. Arch. math. (Basel) 90, 256-266 (2008) · Zbl 1142.35062
[33] H. Lisei, G. Moroşanu, Cs. Varga, Multiplicity results for double eigenvalue problems involving the p-Laplacian, Taiwanese J. Math. (in press)
[34] Liu, X. -L.; Li, W. -T.: Existence and multiplicity of solutions for fourth-order boundary value problems with parameters. J. math. Anal. appl. 327, 362-375 (2007) · Zbl 1109.34015
[35] Liu, Q.: Existence of three solutions for $p(x)$-Laplacian equations. Nonlinear anal. 68, 2119-2127 (2008) · Zbl 1135.35329
[36] Livrea, R.: Existence of three solutions for a quasilinear two point boundary value problem. Arch. math. (Basel) 79, 288-298 (2002) · Zbl 1015.34012
[37] Marano, S. A.; Motreanu, D.: On a three critical points theorem for non-differentiable functions and applications to nonlinear boundary value problems. Nonlinear anal. 48, 37-52 (2002) · Zbl 1014.49004
[38] Mihăilescu, M.: Existence and multiplicity of solutions for a Neumann problem involving the $p(x)$-Laplace operator. Nonlinear anal. 67, 1419-1425 (2007) · Zbl 1163.35381
[39] Pucci, P.; Serrin, J.: A mountain pass theorem. J. differential equations 60, 142-149 (1985) · Zbl 0585.58006
[40] Ricceri, B.: On a three critical points theorem. Arch. math. (Basel) 75, 220-226 (2000) · Zbl 0979.35040
[41] Ricceri, B.: Minimax theorems for limits of parametrized functions having at most one local minimum lying in a certain set. Topology appl. 153, 3308-3312 (2006) · Zbl 1101.49008
[42] Ricceri, B.: Recent advances in minimax theory and applications. Pareto optimality, game theory and equilibria, 23-52 (2008) · Zbl 1194.90115
[43] Salvati, R.: Multiple solutions for a mixed boundary value problem. Math. sci. Res. J. 7, 275-283 (2003) · Zbl 1055.34037
[44] Shilgba, L. K.: Multiplicity of periodic solutions for a boundary eigenvalue problem. Dyn. syst. 20, 223-232 (2005) · Zbl 1098.34034
[45] Wu, X.: Saddle-point characterization and multiplicity of periodic solutions of non-autonomous second-order systems. Nonlinear anal. 58, 899-907 (2004) · Zbl 1058.34053
[46] Wu, X.; Chen, S.; Teng, K.: On variational methods for a class of damped vibration problems. Nonlinear anal. 68, 1432-1441 (2008) · Zbl 1141.34011
[47] Zeidler, E.: Nonlinear functional analysis and its applications. (1985) · Zbl 0583.47051
[48] Zhang, G.; Zhang, W.; Liu, S.: Multiplicity result for a discrete eigenvalue problem with discontinuous nonlinearities. J. math. Anal. appl. 328, 362-375 (2007) · Zbl 1190.39004
[49] Zhang, G.; Liu, S.: Three symmetric solutions for a class of elliptic equations involving the p-Laplacian with discontinuous nonlinearities in RN. Nonlinear anal. 67, 2232-2239 (2007) · Zbl 1155.35323