Multiple nontrivial solutions for nonlinear eigenvalue problems
HTML articles powered by AMS MathViewer
- by D. Motreanu, V. V. Motreanu and N. S. Papageorgiou PDF
- Proc. Amer. Math. Soc. 135 (2007), 3649-3658 Request permission
Abstract:
In this paper we study a nonlinear eigenvalue problem driven by the $p$-Laplacian. Assuming for the right-hand side nonlinearity only unilateral and sign conditions near zero, we prove the existence of three nontrivial solutions, two of which have constant sign (one is strictly positive and the other is strictly negative), while the third one belongs to the order interval formed by the two opposite constant sign solutions. The approach relies on a combination of variational and minimization methods coupled with the construction of upper-lower solutions. The framework of the paper incorporates problems with concave-convex nonlinearities.References
- Antonio Ambrosetti, Haïm Brezis, and Giovanna Cerami, Combined effects of concave and convex nonlinearities in some elliptic problems, J. Funct. Anal. 122 (1994), no. 2, 519–543. MR 1276168, DOI 10.1006/jfan.1994.1078
- A. Anane and N. Tsouli, On the second eigenvalue of the $p$-Laplacian, Nonlinear partial differential equations (Fès, 1994) Pitman Res. Notes Math. Ser., vol. 343, Longman, Harlow, 1996, pp. 1–9. MR 1417265, DOI 10.1093/nq/43.1.1
- Siegfried Carl and Kanishka Perera, Sign-changing and multiple solutions for the $p$-Laplacian, Abstr. Appl. Anal. 7 (2002), no. 12, 613–625. MR 1950611, DOI 10.1155/S1085337502207010
- M. Cuesta, D. de Figueiredo, and J.-P. Gossez, The beginning of the Fučik spectrum for the $p$-Laplacian, J. Differential Equations 159 (1999), no. 1, 212–238. MR 1726923, DOI 10.1006/jdeq.1999.3645
- Johanna Schoenenberger-Deuel and Peter Hess, A criterion for the existence of solutions of non-linear elliptic boundary value problems, Proc. Roy. Soc. Edinburgh Sect. A 74 (1974/75), 49–54 (1976). MR 440191, DOI 10.1017/s030821050001653x
- J. P. García Azorero, I. Peral Alonso, and Juan J. Manfredi, Sobolev versus Hölder local minimizers and global multiplicity for some quasilinear elliptic equations, Commun. Contemp. Math. 2 (2000), no. 3, 385–404. MR 1776988, DOI 10.1142/S0219199700000190
- Leszek Gasiński and Nikolaos S. Papageorgiou, Nonsmooth critical point theory and nonlinear boundary value problems, Series in Mathematical Analysis and Applications, vol. 8, Chapman & Hall/CRC, Boca Raton, FL, 2005. MR 2092433
- Zhiren Jin, Multiple solutions for a class of semilinear elliptic equations, Proc. Amer. Math. Soc. 125 (1997), no. 12, 3659–3667. MR 1443158, DOI 10.1090/S0002-9939-97-04199-3
- Gary M. Lieberman, Boundary regularity for solutions of degenerate elliptic equations, Nonlinear Anal. 12 (1988), no. 11, 1203–1219. MR 969499, DOI 10.1016/0362-546X(88)90053-3
- Paul H. Rabinowitz, Minimax methods in critical point theory with applications to differential equations, CBMS Regional Conference Series in Mathematics, vol. 65, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1986. MR 845785, DOI 10.1090/cbms/065
- Peter Tolksdorf, Regularity for a more general class of quasilinear elliptic equations, J. Differential Equations 51 (1984), no. 1, 126–150. MR 727034, DOI 10.1016/0022-0396(84)90105-0
- J. L. Vázquez, A strong maximum principle for some quasilinear elliptic equations, Appl. Math. Optim. 12 (1984), no. 3, 191–202. MR 768629, DOI 10.1007/BF01449041
Additional Information
- D. Motreanu
- Affiliation: Département de Mathématiques, Université de Perpignan, 66860 Perpignan, France
- Email: motreanu@univ-perp.fr
- V. V. Motreanu
- Affiliation: Département de Mathématiques, Université de Perpignan, 66860 Perpignan, France
- Email: viorica@univ-perp.fr
- N. S. Papageorgiou
- Affiliation: Department of Mathematics, National Technical University, Athens 15780, Greece
- MR Author ID: 135890
- Email: npapg@math.ntua.gr
- Received by editor(s): August 1, 2006
- Received by editor(s) in revised form: September 6, 2006
- Published electronically: August 7, 2007
- Communicated by: David S. Tartakoff
- © Copyright 2007 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 135 (2007), 3649-3658
- MSC (2000): Primary 35J60; Secondary 35J70
- DOI: https://doi.org/10.1090/S0002-9939-07-08927-7
- MathSciNet review: 2336581