## Topological Methods in Nonlinear Analysis

### Resonant nonlinear periodic problems with the scalar $p$-Laplacian and a nonsmooth potential

#### Abstract

We study periodic problems driven by the scalar $p$-Laplacian with a nonsmooth potential. Using the nonsmooth critical point theory for locally Lipschitz functions, we prove two existence theorems under conditions of resonance at infinity with respect to the first two eigenvalues of the negative scalar $p$-Laplacian with periodic boundary conditions.

#### Article information

Source
Topol. Methods Nonlinear Anal., Volume 27, Number 2 (2006), 269-288.

Dates
First available in Project Euclid: 13 May 2016

https://projecteuclid.org/euclid.tmna/1463144523

Mathematical Reviews number (MathSciNet)
MR2237455

Zentralblatt MATH identifier
1141.34010

#### Citation

Aizicovici, Sergiu; Papageorgiou, Nikolaos S.; Staicu, Vasile. Resonant nonlinear periodic problems with the scalar $p$-Laplacian and a nonsmooth potential. Topol. Methods Nonlinear Anal. 27 (2006), no. 2, 269--288. https://projecteuclid.org/euclid.tmna/1463144523

#### References

• S. Ahmad and A. Lazer, Critical point theory and a theorem of Amaral and Pera , Boll. Un. Mat. Ital. B(6), 3 , no. 3(1984), 583–598 \ref\key 2
• P. Drabek and R. Manasevich, On the closed solution to some nonhomogeneous eigenvalue problems with $p$-Laplacian , Differential Integral Equations, 12 (1999), 773–788 \ref\key 3
• C. Fabry and A. Fonda, Periodic solutions of nonlinear differential equations with double resonance , Ann. Mat. Pura Appl., CLVII , 99–116 (1990) \ref\key 4
• A. Fonda and D. Lupo, Periodic solutions of second order ordinary differential equations , Boll. Un. Mat. Ital. A(7), 3 , no. 3, 291–299 (1989) \ref\key 5
• L. Gasinski and N. S. Papageorgiou, Nonsmooth Critical Point Theory and Nonlinear Boundary Value Problems, Chapman and Hall, CRC Press, Boca Raton (2005) \ref\key 6
• J. P. Gossez and P. Omari, Periodic solutions of a second order ordinary differential equation: a necessary and sufficient condition for resonance , J. Differential Equations, 94 , 67–82 (1991) \ref\key 7
• R. Iannacci and M. Nkashama, Nonlinear boudary value problems at resonance , Nonlinear Anal., 166 , 455–473 (1987) \ref\key 8
• J. Mawhin, Remarks on the preceding paper of Ahmad and Lazer on periodic solutions , Boll. Un. Mat. Ital. A(6), 3 , no. 2, 229–238 (1984) \ref\key 9
• N. S. Papageorgiou and N. Yannakakis, Periodic solutions for second order equations with the scalar $p$-Laplacian and nonsmooth potential , Funkcial. Ekvac., 47 , 107–117 (2004)