On the existence of five nontrivial solutions for resonant problems with p-Laplacian
Leszek Gasiński ; Nikolaos S. Papageorgiou
Discussiones Mathematicae, Differential Inclusions, Control and Optimization, Tome 30 (2010), p. 169-189 / Harvested from The Polish Digital Mathematics Library
Access to full text
Full (PDF)

In this paper we study a nonlinear Dirichlet elliptic differential equation driven by the p-Laplacian and with a nonsmooth potential. The hypotheses on the nonsmooth potential allow resonance with respect to the principal eigenvalue λ₁ > 0 of (-Δₚ,W₀1,p(Z)). We prove the existence of five nontrivial smooth solutions, two positive, two negative and the fifth nodal.

Publié le : 2010-01-01
EUDML-ID : urn:eudml:doc:271202 
@article{bwmeta1.element.bwnjournal-article-doi-10_7151_dmdico_1118,
     author = {Leszek Gasi\'nski and Nikolaos S. Papageorgiou},
     title = {On the existence of five nontrivial solutions for resonant problems with p-Laplacian},
     journal = {Discussiones Mathematicae, Differential Inclusions, Control and Optimization},
     volume = {30},
     year = {2010},
     pages = {169-189},
     zbl = {1216.35035},
     language = {en},
     url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_7151_dmdico_1118}
}

Leszek Gasiński; Nikolaos S. Papageorgiou. On the existence of five nontrivial solutions for resonant problems with p-Laplacian. Discussiones Mathematicae, Differential Inclusions, Control and Optimization, Tome 30 (2010) pp. 169-189. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_7151_dmdico_1118/

[000] [1] A. Ambrosetti, J. García Azorero and I. Peral, Multiplicity results for some nonlinear elliptic equations, J. Funct. Anal. 137 (1996), 219-242. doi: 10.1006/jfan.1996.0045

[001] [2] S. Carl and D. Motreanu, Constant sign and sign-changing solutions for nonlinear eigenvalue problems, doi: 10.1016/j.na.2007.02.013, 2007. | Zbl 1212.35109

[002] [3] S. Carl and K. Perera, Sign-changing and multiple solutions for the p-Laplacian, Abstr. Appl. Anal. 7 (2002), 613-625. | Zbl 1106.35308

[003] [4] K.-C. Chang, Infinite-Dimensional Morse Theory and Multiple Solution Problems, volume 6 of Progress in Nonlinear Differential Equations and Their Applications (Birkhäuser Verlag, Boston, MA, 1993).

[004] [5] F.H. Clarke, Optimization and Nonsmooth Analysis (Wiley, New York, 1983). | Zbl 0582.49001

[005] [6] M. Cuesta, D. de Figueiredo and J.-P. Gossez, The beginning of the Fučik spectrum for the p-Laplacian, J. Differ. Equ. 159 (1999), 212-238. | Zbl 0947.35068

[006] [7] N. Dancer and Y. Du, On sign-changing solutions of certain semilinear elliptic problems, Appl. Anal. 56 (1995), 193-206. | Zbl 0835.35051

[007] [8] N. Dunford and J.T. Schwartz, Linear Operators I, General Theory, volume 7 of Pure and Applied Mathematics (Wiley, New York, 1958). | Zbl 0084.10402

[008] [9] J. García Azorero, J. Manfredi and I. Peral Alonso, Sobolev versus Hölder local minimizers and global multiplicity for some quasilinear elliptic equations, Commun. Contemp. Math. 2 (2000), 385-404. | Zbl 0965.35067

[009] [10] L. Gasiński and N.S. Papageorgiou, Nonsmooth Critical Point Theory and Nonlinear Boundary Value Problems (Chapman and Hall/CRC Press, Boca Raton, FL, 2005). | Zbl 1058.58005

[010] [11] L. Gasiński and N.S. Papageorgiou, Nonlinear Analysis (Chapman and Hall/ CRC Press, Boca Raton, FL, 2006).

[011] [12] Q.-S. Jiu and J.-B. Su, Existence and multiplicity results for Dirichlet problems with p-Laplacian, J. Math. Anal. Appl. 281 (2003), 587-601. doi: 10.1016/S0022-247X(03)00165-3 | Zbl 1146.35358

[012] [13] O.A. Ladyzhenskaya and N. Uraltseva, Linear and Quasilinear Elliptic Equations, volume 46 of Mathematics in Science and Engineering (Academic Press, New York, 1968).

[013] [14] G.M. Lieberman, Boundary regularity for solutions of degenerate elliptic equations, Nonlinear Anal. 12 (1988), 1203-1219. | Zbl 0675.35042

[014] [15] J.-Q. Liu and S.-B. Liu, The existence of multiple solutions to quasilinear elliptic equations, Bull. London Math. Soc. 37 (2005), 592-600. | Zbl 1122.35033

[015] [16] S.-B. Liu, Multiple solutions for coercive p-Laplacian equations, J. Math. Anal. Appl. 316 (2006), 229-236. doi: 10.1016/j.jmaa.2005.04.034 | Zbl 1148.35321

[016] [17] M. Marcus and V.J. Mizel, Every superposition operator mapping one Sobolev space into another is continuous, J. Funct. Anal. 33 (1979), 217-229. | Zbl 0418.46024

[017] [18] E.H. Papageorgiou and N.S. Papageorgiou, A multiplicity theorem for problems with the p-Laplacian, J. Funct. Anal. 244 (2007), 63-77. | Zbl 1231.35085

[018] [19] J.L. Vázquez, A strong maximum principle for some quasilinear elliptic equations, Appl. Math. Optim. 12 (1984), 191-202. | Zbl 0561.35003

[019] [20] Z. Zhang, J.-Q. Chen and S.-J. Li, Construction of pseudo-gradient vector field and sign-changing multiple solutions involving p-Laplacian, J. Differ. Equ. 201 (2004), 287-303. | Zbl 1079.35035