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Remarks on Ricceri’s variational principle and applications to the \(p(x)\)-Laplacian equations. (English) Zbl 1134.35035
Summary: We give some remarks on a variational principle of Ricceri in the case of sequentially weakly lower semi-continuous functionals defined on a reflexive real Banach space. In particular we introduce the notions of Ricceri block and Ricceri box which are more convenient in some applications than the weakly connected components. Using the variational principle of Ricceri and a local mountain pass lemma, we study the multiplicity of solutions of the \(p(x)\)-Laplacian equations with Neumann, Dirichlet or no-flux boundary condition, and under appropriate hypotheses, in which the integral functionals need not satisfy the \((PS)\) condition on the global space, we prove that the problem has at least seven solutions.

MSC:
35J20 Variational methods for second-order elliptic equations
35J65 Nonlinear boundary value problems for linear elliptic equations
35J70 Degenerate elliptic equations
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[1] Anello, G., A multiplicity theorem for critical points of functionals on reflexive Banach spaces, Arch. math., 82, 172-179, (2004) · Zbl 1071.35039
[2] Anello, G., Existence of infinitely many weak solutions for a Neumann problem, Nonlinear anal., 57, 199-209, (2004) · Zbl 1058.35071
[3] Anello, G.; Cordaro, G., Existence of solutions of the Neumann problem for a class of equations involving the \(p\)-Laplacian via a variational principle of, Ricceri, arch. math., 79, 274-287, (2002) · Zbl 1091.35025
[4] Anello, G.; Cordaro, G., An existence theorem for the Neumann problem involving the \(p\)-Laplacian, J. convex. anal., 10, 185-198, (2003) · Zbl 1091.35026
[5] Bonanno, G.; Candito, P., Three solutions to a Neumann problem for elliptic equations involving the \(p\)-Laplacian, Arch. math., 80, 424-429, (2003) · Zbl 1161.35382
[6] L. Diening, P. Hästö, A. Nekvinda, Open problems in variable exponent Lebesgue and Sobolev spaces, in: P. Drábek, J. Rákosník (Eds.), FSDONA04 Proceedings, Milovy, Czech Republic, 2004, pp. 38-58
[7] Fan, X.L., Solutions for \(p(x)\)-Laplacian Dirichlet problems with singular coefficients, J. math. anal. appl., 312, 464-477, (2005) · Zbl 1154.35336
[8] Fan, X.L.; Han, X.Y., Existence and multiplicity of solutions for \(p(x)\)-Laplacian equations in \(R^N\), Nonlinear anal., 59, 173-188, (2004) · Zbl 1134.35333
[9] Fan, X.L.; Shen, J.S.; Zhao, D., Sobolev embedding theorems for spaces \(W^{k, p(x)}(\Omega)\), J. math. anal. appl., 262, 749-760, (2001) · Zbl 0995.46023
[10] Fan, X.L.; Zhang, Q.H., Existence of solutions for \(p(x)\)-Laplacian Dirichlet problems, Nonlinear anal., 52, 1843-1852, (2003) · Zbl 1146.35353
[11] Fan, X.L.; Zhao, D., On the spaces \(L^{p(x)}(\Omega)\) and \(W^{m, p(x)}(\Omega)\), J. math. anal. appl., 263, 424-446, (2001) · Zbl 1028.46041
[12] Fan, X.L.; Zhao, Y.Z.; Zhang, Q.H., A strong maximum principle for \(p(x)\)-Laplace equations, Chinese J. contemp. math., 24, 3, 277-282, (2003)
[13] Faraci, F., Multiplicity results for a Neumann problem involving the \(p\)-Laplacian, J. math. anal. appl., 277, 180-189, (2003) · Zbl 1092.35033
[14] Faraci, F., Multiple solutions for two nonlinear problems involving the \(p\)-Laplacian, Nonlinear anal., 63, e1017-e1029, (2005) · Zbl 1224.35152
[15] Faraci, F.; Iannizzotto, A., A multiplicity theorem for a perturbed second order nonautonomous system, Proc. edinb. math. soc., 49, 267-275, (2006) · Zbl 1106.34025
[16] Kováčik, O.; Rákosník, J., On spaces \(L^{p(x)}(\Omega)\) and \(W^{k, p(x)}(\Omega)\), Czechoslovak math. J., 41, 592-618, (1991) · Zbl 0784.46029
[17] Lê, A., Eigenvalue problems for the \(p\)-Laplacian, Nonlinear anal., 64, 1057-1099, (2006) · Zbl 1208.35015
[18] Megginson, R.E., An introduction to Banach space theory, (1998), Springer-Verlag New York · Zbl 0910.46008
[19] Musielak, J., Orlicz spaces and modular spaces, () · Zbl 0557.46020
[20] Ricceri, B., A general variational principle and some of its applications, J. comput. appl. math., 113, 401-410, (2000) · Zbl 0946.49001
[21] Ricceri, B., Infinitely many solutions of the Neumann problem for elliptic equations involving the \(p\)-Laplacian, Bull. London math. soc., 33, 331-340, (2001) · Zbl 1035.35031
[22] Ricceri, B., Sublevel sets and global minima of coercive functionals and local minima of their perturbations, J. nonlinear convex anal., 52, 157-168, (2004) · Zbl 1083.49004
[23] Ricceri, B., A multiplicity theorem for the Neumann problem, Proc. amer. math. soc., 134, 1117-1124, (2006) · Zbl 1134.35345
[24] Růz˘ic˘ka, M., Electrorheological fluids: modeling and mathematical theory, (2000), Springer-Verlag Berlin · Zbl 0968.76531
[25] Samko, S., On a progress in the theory of Lebesgue spaces with variable exponent: maximal and singular operators, Integral transforms spec. funct., 16, 461-482, (2005) · Zbl 1069.47056
[26] Willem, M., Minimax theorems, (1996), Birkhauser Boston · Zbl 0856.49001
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