Hostname: page-component-78c5997874-ndw9j Total loading time: 0 Render date: 2024-11-10T21:10:02.571Z Has data issue: false hasContentIssue false
  • English
  • Français

Sequences of Weak Solutions for Non-Local Elliptic Problems with Dirichlet Boundary Condition

Published online by Cambridge University Press:  16 April 2014

Giovanni Molica Bisci  and
Pasquale F. Pizzimenti
Giovanni Molica Bisci
Affiliation:
Department PAU, University of Reggio Calabria, Via Melissari, 89124 Reggio Calabria, Italy, (xlink:href="gmolica@unirc.it">gmolica@unirc.it)
Pasquale F. Pizzimenti
Affiliation:
Department of Mathematics, University of Messina, Contrada Papardo, Salita Sperone 31, 98166 Messina, Italy, (xlink:href="ppizzimenti@unime.it">ppizzimenti@unime.it)
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In this paper the existence of infinitely many solutions for a class of Kirchhoff-type problems involving the p-Laplacian, with p > 1, is established. By using variational methods, we determine unbounded real intervals of parameters such that the problems treated admit either an unbounded sequence of weak solutions, provided that the nonlinearity has a suitable behaviour at ∞, or a pairwise distinct sequence of weak solutions that strongly converges to 0 if a similar behaviour occurs at 0. Some comparisons with several results in the literature are pointed out. The last part of the work is devoted to the autonomous elliptic Dirichlet problem.

Keywords

Critical pointweak solutionsKirchhoff-type problemsmultiple solutionsPrimary 35J9235J6046N20
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 57 , Issue 3 , October 2014 , pp. 779 - 809
Copyright
Copyright © Edinburgh Mathematical Society 2014 

References

1.Alves, C. O., correa, F. S. J. A. and Ma, T. F., Positive solutions for a quasilinear elliptic equation of Kirchhoff type, Comput. Math. Appl. 49 (2005), 85–93.Google 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(3) (2002), 511–519 (corrigendum: Proc. R. Soc. Edinb. A 133(1) (2003)).Google Scholar
3.Arosio, A. and Panizzi, S., On the well-posedness of the Kirchhoff string, Trans. Am. Math. Soc. 348 (1996), 305–330.Google Scholar
4.Bonanno, G. and Molica Bisci, G., Infinitely many solutions for a boundary-value problem with discontinuous nonlinearities, Bound. Value Probi. 2009 (2009), 1–20.Google Scholar
5.Bonanno, G. and Molica Bisci, G., Infinitely many solutions for a Dirichlet problem involving the p-Laplacian, Proc. R. Soc. Edinb. A 140 (2009), 737–752.Google Scholar
6.Bonanno, G., Molica Bisci, G. and O'regan, D., Infinitely many weak solutions for a class of quasilinear elliptic systems, Math. Comput. Modelling 52 (2010), 152–160.Google Scholar
7.Bonanno, G., Molica Bisci, G. and Radulescu, V., Infinitely many solutions for a class of nonlinear eigenvalue problems in Orlicz-Sobolev spaces, C. R. Acad. Sci. Paris Ser. I 349 (2011), 263–268.CrossRefGoogle Scholar
8.Bonanno, G., Molica Bisci, G. and Rădulescu, V., Quasilinear elliptic non-homogeneous Dirichlet problems through Orlicz-Sobolev spaces, Nonlin. Analysis 75(12) (2011), 4441–4456.CrossRefGoogle Scholar
9.Bonanno, G., Molica Bisci, G. and Rădulescu, V., Variational analysis for a nonlinear elliptic problem on the Sierpiński gasket, ESAIM: Control Optim. Calc. Variations 18(4) (2012), 941–953.Google Scholar
10.Bonanno, G., Molica Bisci, G. and Rădulescu, V., Arbitrarily small weak solutions for a nonlinear eigenvalue problem in Orlicz-Sobolev spaces, Monatsh. Math. 165 (2012), 305–318.CrossRefGoogle Scholar
11.Bonanno, G., Molica Bisci, G. and Radulescu, V., Infinitely many solutions for a class of nonlinear elliptic problems on fractals, C. R. Acad. Sci. Paris Ser. I 350 (2012), 187–191.Google Scholar
12.Cammaroto, F., Chinni, A. and Di Bella, B., Infinitely many solutions for the Dirichlet problem involving the p-Laplacian, Nonlin. Analysis TMA 61 (2005), 41–49.CrossRefGoogle Scholar
13.Cavalcanti, M. M., Domingos Cavalcanti, V. N. and Soriano, J. A., Global existence and uniform decay rates for the Kirchhoff-Carrier equation with nonlinear dissipation, Adv. Diff. Eqns 6 (2001), 701–730.Google Scholar
14.Cheng, B. and Wu, X., Existence results of positive solutions of Kirchhoff-type problems, Nonlin. Analysis 71 (2009), 4883–4892.Google Scholar
15.D'agui, G. and Molica Bisci, G., Infinitely many solutions for perturbed hemivaria-tional inequalities, Bound. Value Probl. 2010 (2010), 363518.Google Scholar
16.D'agui, G. and Molica Bisci, G., Existence results for an elliptic Dirichlet problem, Le Matematiche 66(1) (2011), 133–141.Google Scholar
17.Dai, G. and Liu, D., Infinitely many positive solutions for a p(x)-Kirchhoff-type equation, J. Math. Analysis Applic. 359(2) (2009), 704–710.CrossRefGoogle Scholar
18.Dai, G. and Wei, J., Infinitely many non-negative solutions for a p(x)-Kirchhoff-type problem with Dirichlet boundary condition, Nonlin. Analysis 73 (2010), 3420–3430.Google Scholar
19.D'ancona, P. and Spagnolo, S., Global solvability for the degenerate Kirchhoff equation with real analytic data, Invent. Math. 108 (1992), 247–262.CrossRefGoogle Scholar
20.Evans, L. C., Partial differential equations, Graduate Studies in Mathematics, Volume 19 (American Mathematical Society, Providence, RI, 1998).Google Scholar
21.Faraci, F., A note on the existence of infinitely many solutions for the one-dimensional prescribed curvature equation, Stud. Univ. Babes-Bolyai Math. 4 (2010), 83–90.Google Scholar
22.Gilbarg, D. and Trudinger, N. S., Elliptic partial differential equations of second order, 2nd edn (Springer, 1983).Google Scholar
23.He, X. and Zou, W., Infinitely many positive solutions for Kirchhoff-type problems, Nonlin. Analysis 70 (2008), 1407–1414.CrossRefGoogle Scholar
24.Kirchhoff, G., Mechanik (Teubner, Leipzig, 1883).Google Scholar
25.Kristaly, A., Infinitely many solutions for a differential problem in ℝN, J. Diff. Eqns 220 (2006), 511–530.Google Scholar
26.Kristaly, A., Detection of arbitrarily many solutions for perturbed elliptic problems involving oscillatory terms, J. Diff. Eqns 245 (2008), 3849–3868.CrossRefGoogle Scholar
27.Kristaly, A., Asymptotically critical problems on higher-dimensional spheres, Discrete Contin. Dynam. Syst. 23 (2008), 919–935.Google Scholar
28.Kristaly, A. and Morosanu, G., New competition phenomena in Dirichlet problems, J. Math. Pures Appl. 94(9) (2010), 555–570.Google Scholar
29.Kristaly, A., Rădulescu, V. and Varga, CS., Variational principles in mathematical physics, geometry, and economics: qualitative analysis of nonlinear equations and unilateral problems, Encyclopedia of Mathematics and Its Applications, Volume 136 (Cambridge University Press, Cambridge, 2010).Google Scholar
30.Lions, J.-L., On some questions in boundary value problems of mathematical physics, in Contemporary developments in continuum mechanics and partial differential equations, North-Holland Mathematics Studies, Volume 30, pp. 284–346 (North-Holland, Amsterdam, 1978).Google Scholar
31.Ma, T. F. and Muñoz rivera, J. E., Positive solutions for a nonlinear nonlocal elliptic trasmission problem, Appl. Math. Lett. 16 (2003), 243–248.CrossRefGoogle Scholar
32.Marcus, M. and Mizel, V., Every superposition operator mapping one Sobolev space into another is continuous, J. Funct. Analysis 33 (1979), 217–229.Google Scholar
33.Njoku, F. I. and Zanolin, F., Positive solutions for two-point BVPs: existence and multiplicity results, Nonlin. Analysis 13 (1989), 1329–1338.Google Scholar
34.Njoku, F. I., Omari, P. and Zanolin, F., Multiplicity of positive radial solutions of a quasilinear elliptic problem in a ball, Adv. Diff. Eqns 5 (2000), 1545–1570.Google Scholar
35.Obersnel, F. and Omari, P., Positive solutions of elliptic problems with locally oscillating nonlinearities, J. Math. Analysis Applic. 323 (2006), 913–929.Google Scholar
36.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
37.Perera, K. and Zhang, Z. T., Nontrivial solutions of Kirchhoff-type problems via the Yang index, J. Diff. Eqns 221 (2006), 246–255.CrossRefGoogle Scholar
38.Ricceri, B., A general variational principle and some of its applications, J. Computat. Appl. Math. 113 (2000), 401–410.CrossRefGoogle Scholar
39.Ricceri, B., On an elliptic Kirchhoff-type problem depending on two parameters, J. Global Optim. 46 (2010), 543–549.Google Scholar
40.Saint Raymond, J., On the multiplicity of the solutions of the equation — Δu = λf (u), J. Diff. Eqns 180 (2002), 65–88.Google Scholar
41.Zhang, Z. T. and Perera, K., Sign changing solutions of Kirchhoff-type problems via invariant sets of descent flow, J. Math. Analysis Applic. 317 (2006), 456–463.Google Scholar