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New Conditions for the Existence of Infinitely Many Solutions for a Quasi-Linear Problem

Part of: Elliptic equations and systems Equations and inequalities involving nonlinear operators

Published online by Cambridge University Press:  15 December 2015

Francesca Faraci  and
Csaba Farkas
Francesca Faraci
Affiliation:
Department of Mathematics and Computer Science, University of Catania, Viale A. Doria, 95125 Catania, Italy ([email protected])
Csaba Farkas
Affiliation:
Fcaultatea de Matematică şi Informatică, Universitatea Babeş-Bolyai Cluj-Napoca, Str. Mihail Kogalniceanu nr. 1, 400084, Cluj-Napoca, Romania ([email protected])
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Abstract

In this paper we study a quasi-linear elliptic problem coupled with Dirichlet boundary conditions. We propose a new set of assumptions ensuring the existence of infinitely many solutions.

Keywords

p-Laplacianinfinitely many solutionsDirichlet problemvariational methods

MSC classification

Secondary: 35J20: Variational methods for second-order elliptic equations 35J92: Quasilinear elliptic equations with $p$-Laplacian 47J10: Nonlinear spectral theory, nonlinear eigenvalue problems
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 59 , Issue 3 , August 2016 , pp. 655 - 669
Copyright
Copyright © Edinburgh Mathematical Society 2015 

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References

1. Ambrosetti, A. and Rabinowitz, P. H., Dual variational methods in critical point theory and applications, J. Funct. Analysis 14 (1973), 349–381.CrossRefGoogle Scholar
2. Anello, G. and Cordaro, G., Infinitely many arbitrarily small positive solutions for the Dirichlet problem involving the p-Laplacian, Proc. R. Soc. Edinb. A 132 (2002), 511–519.CrossRefGoogle Scholar
3. Cammaroto, F., Chinnì, A. and Di Bella, B., Infinitely many solutions for the Dirichlet problem involving the p-Laplacian, Nonlin. Analysis 61(1–2) (2005), 41–49.CrossRefGoogle Scholar
4. Cammaroto, F., Chinnì, A. and Di Bella, B., Infinitely many solutions for the Dirichlet problem via a variational principle of Ricceri, in Variational analysis and applications, Nonconvex Optimization and Its Applications, Volume 79, pp. 215–229 (Springer, 2005).CrossRefGoogle Scholar
5. de Figueiro, D. G., On the uniqueness of positive solutions of the Dirichlet problem –Δu = ƛ sin u , in Nonlinear partial differential equations and their applications, College de France Seminar, Vol. VII, Paris, 1983–1984, Research Notes in Mathematics Series, Volume 122, pp. 80–83 (Pitman, Boston, MA, 1985).Google Scholar
6. Faraci, F. and Kristály, A., One-dimensional scalar field equations involving an oscillatory nonlinear term, Discrete Contin. Dynam. Syst. 18 (2007), 107–120.CrossRefGoogle Scholar
7. Faraci, F. and Livrea, R., Infinitely many periodic solutions for a second order nonautonomous system, Nonlin. Analysis 54 (2003), 417–429.CrossRefGoogle Scholar
8. Hirano, N. and Zou, W., A perturbation method for multiple sign-changing solutions, Calc. Var. PDEs 37 (2010), 87–98.CrossRefGoogle Scholar
9. Kristály, A. and Morosanu, G., New competition phenomena in Dirichlet problems, J. Math. Pures Appl. 94 (2010), 555–570.CrossRefGoogle Scholar
10. Omari, P. and Zanolin, F., Infinitely many solutions of a quasilinear elliptic problem with an oscillatory potential, Commun. PDEs 21 (1996), 721–733.CrossRefGoogle Scholar
11. Omari, P. and Zanolin, F., An elliptic problem with arbitrarily small positive solutions, Electron. J. Diff. Eqns Conference 5 (2000), 301–308.Google Scholar
12. Ricceri, B., A general variational principle and some of its applications: fixed point theory with applications in nonlinear analysis, J. Computat. Appl. Math. 113 (2000), 401–410.CrossRefGoogle Scholar
13. Ricceri, B., Infinitely many solutions of the Neumann problem for elliptic equations involving the p-Laplacian, Bull. Lond. Math. Soc. 33 (2001), 331–340.CrossRefGoogle Scholar
14. Ricceri, B., A new existence and localization theorem for the Dirichlet problem, Dynam. Syst. Applic. 22(2–3) (2013), 317–324.Google Scholar
15. Saint Raymond, J., On the multiplicity of solutions of the equation –Δu = ƛ · f(u), J. Diff. Eqns 180 (2002), 65–88.CrossRefGoogle Scholar
16. Tehrani, H. T., Infinitely many solutions for indefinite semilinear elliptic equations without symmetry, Commun. PDEs 21 (1996), 541–557.CrossRefGoogle Scholar