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

Abstract

top
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 ( - Δ ₚ , W ₀ 1 , p ( Z ) ) . We prove the existence of five nontrivial smooth solutions, two positive, two negative and the fifth nodal.

How to cite

top

Leszek 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. [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. [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. [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. [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. [5] F.H. Clarke, Optimization and Nonsmooth Analysis (Wiley, New York, 1983). Zbl0582.49001
  6. [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. [7] N. Dancer and Y. Du, On sign-changing solutions of certain semilinear elliptic problems, Appl. Anal. 56 (1995), 193-206. Zbl0835.35051
  8. [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. [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. [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. [11] L. Gasiński and N.S. Papageorgiou, Nonlinear Analysis (Chapman and Hall/ CRC Press, Boca Raton, FL, 2006). 
  12. [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. [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. [14] G.M. Lieberman, Boundary regularity for solutions of degenerate elliptic equations, Nonlinear Anal. 12 (1988), 1203-1219. Zbl0675.35042
  15. [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. [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. [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. [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. [19] J.L. Vázquez, A strong maximum principle for some quasilinear elliptic equations, Appl. Math. Optim. 12 (1984), 191-202. Zbl0561.35003
  20. [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

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.