# 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 (2010)

- Volume: 30, Issue: 2, page 169-189
- ISSN: 1509-9407

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topLeszek Gasiński, and Nikolaos S. Papageorgiou. "On the existence of five nontrivial solutions for resonant problems with p-Laplacian." Discussiones Mathematicae, Differential Inclusions, Control and Optimization 30.2 (2010): 169-189. <http://eudml.org/doc/271202>.

@article{LeszekGasiński2010,

abstract = {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 $(-Δₚ,W₀^\{1,p\}(Z))$. We prove the existence of five nontrivial smooth solutions, two positive, two negative and the fifth nodal.},

author = {Leszek Gasiński, Nikolaos S. Papageorgiou},

journal = {Discussiones Mathematicae, Differential Inclusions, Control and Optimization},

keywords = {p-Laplacian; Clarke subdifferential; linking sets; upper-lower solutions; second eigenvalue; nodal and constant sign solutions; second deformation theorem; -Laplacian},

language = {eng},

number = {2},

pages = {169-189},

title = {On the existence of five nontrivial solutions for resonant problems with p-Laplacian},

url = {http://eudml.org/doc/271202},

volume = {30},

year = {2010},

}

TY - JOUR

AU - Leszek Gasiński

AU - Nikolaos S. Papageorgiou

TI - On the existence of five nontrivial solutions for resonant problems with p-Laplacian

JO - Discussiones Mathematicae, Differential Inclusions, Control and Optimization

PY - 2010

VL - 30

IS - 2

SP - 169

EP - 189

AB - 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 $(-Δₚ,W₀^{1,p}(Z))$. We prove the existence of five nontrivial smooth solutions, two positive, two negative and the fifth nodal.

LA - eng

KW - p-Laplacian; Clarke subdifferential; linking sets; upper-lower solutions; second eigenvalue; nodal and constant sign solutions; second deformation theorem; -Laplacian

UR - http://eudml.org/doc/271202

ER -

## References

top- [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
- [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. Zbl1212.35109
- [3] S. Carl and K. Perera, Sign-changing and multiple solutions for the p-Laplacian, Abstr. Appl. Anal. 7 (2002), 613-625. Zbl1106.35308
- [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).
- [5] F.H. Clarke, Optimization and Nonsmooth Analysis (Wiley, New York, 1983). Zbl0582.49001
- [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. Zbl0947.35068
- [7] N. Dancer and Y. Du, On sign-changing solutions of certain semilinear elliptic problems, Appl. Anal. 56 (1995), 193-206. Zbl0835.35051
- [8] N. Dunford and J.T. Schwartz, Linear Operators I, General Theory, volume 7 of Pure and Applied Mathematics (Wiley, New York, 1958). Zbl0084.10402
- [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. Zbl0965.35067
- [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). Zbl1058.58005
- [11] L. Gasiński and N.S. Papageorgiou, Nonlinear Analysis (Chapman and Hall/ CRC Press, Boca Raton, FL, 2006).
- [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 Zbl1146.35358
- [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).
- [14] G.M. Lieberman, Boundary regularity for solutions of degenerate elliptic equations, Nonlinear Anal. 12 (1988), 1203-1219. Zbl0675.35042
- [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. Zbl1122.35033
- [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 Zbl1148.35321
- [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. Zbl0418.46024
- [18] E.H. Papageorgiou and N.S. Papageorgiou, A multiplicity theorem for problems with the p-Laplacian, J. Funct. Anal. 244 (2007), 63-77. Zbl1231.35085
- [19] J.L. Vázquez, A strong maximum principle for some quasilinear elliptic equations, Appl. Math. Optim. 12 (1984), 191-202. Zbl0561.35003
- [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. Zbl1079.35035

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