Abstract
In this paper, we establish the existence of three weak solutions for quasilinear elliptic equations in an Orlicz-Sobolev space via an abstract result recently obtained by Ricceri in [13].
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Adams, R., Fournier, J.F. Sobolev Spaces (Second Edition). Acad. Press, New York, 2003
Ambrosetti, A., Azorero, G.J., Peral, I. Multiplicity for some nonlinear elliptic equations. J. Funct. Anal., 137 (1): 219–242 (1996)
Clément, Ph., García-Huidobro, M., Manásevich, K. Schmitt, R. Moutain pass type solutions for quasilinear elliptic equations. Calc. Var. Partial Differential Equations, 11 (1): 33–62 (2000)
Damascelli, L., Sciunzi, B. Regularity, monotonicity and symmetry of positive solutions of m-Laplace equations. J. Differential Equations, 206 (2): 483–515 (2004)
Damascelli, L., Sciunzi, B. Harnack inequalities, maximum and comparison principles, and regularity of positive solutions of m-Laplace equations. Calc. Var. Partial Differential Equations, 25 (2): 139–159 (2005)
Degiovanni, M., Lancelotti, S. Linking over cones and nontrivial solutions for p–Laplace equations with p-superlinear nonlinearity. Ann. Inst. H. Poincaré: Analyse Non Linéaire, 24 (6): 907–919 (2007)
Fukagai, N., Ito, M., Narukawa, K. Positive solutions of quasilinear elliptic equations with critical Orlicz- Sobolev nonlinearity on RN.Funkcialaj Ekvacioj, 49 (2): 235–267 (2006)
Fukagai, N., Narukawa, K. On the existence of Multiple positive solutions of quasilinear elliptic eigenvalue problems. Annadli di Matematica, 186 (3): 539–564 (2007)
García-Huidobro, M., Le, V., Manásevich, R., Schmitt, K. On principal eigenvalues for quasilinear elliptic differential operators: An Orlicz-Sobolev space setting. Nonlinear Differential Equations and Appl., 6 (2): 207–225 (1999)
Liu, Q. Existence of three solutions for p(x)-Laplacian equations. Nonlinear Anal., 68 (7): 2119–2127 (2008)
Rao, M.M., Ren, Z.D. Theory of Orlicz Spaces. Marcel Dekker, New York, 1985
Liu, S.B. Existence of solutions to a superlinear p-Laplacian equation. Electron. J. Differential Eq., 66: 1–6 (2001)
Ricceri, B. A further three critical points theorem. Nonlinear Anal., 71 (9): 4151–4157 (2009)
Ricceri, B. On an elliptic Kirchhoff-type problem depending on two parameters. J. Glob. Optim., 46 (4): 543–549 (2010)
Sun, J. T., Chen, H.B., Nieto, J.J., Otero-Novoa, M. The multiplicity of solutions for perturbed secondorder Hamiltonian systems with impulsive effects. Nonlinear Anal., 72 (12): 4575–4586 (2010)
Acknowledgements
The authors would like to thank the anonymous referees for carefully reading the paper and for giving us valuable comments.
Author information
Authors and Affiliations
Corresponding author
Additional information
Supported by the National Natural Science Foundation of China (Grant No. 11626038).
Rights and permissions
About this article
Cite this article
Fang, F., Tan, Z. Existence of three solutions for quasilinear elliptic equations: an Orlicz-Sobolev space setting. Acta Math. Appl. Sin. Engl. Ser. 33, 287–296 (2017). https://doi.org/10.1007/s10255-017-0659-0
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10255-017-0659-0