Positivity 10 (2006), 343–363
© 2006 Birkh
auser Verlag Basel/Switzerland
1385-1292/020343-21, published online April 26, 2006
Positive Periodic and Homoclinic Solutions
for Nonlinear Differential Equations
with Nonsmooth Potential
and NIKOLAOS S. PAPAGEORGIOU
Department of Mathematics, Southwest Missouri State University, Springﬁeld, MO
65804, USA. E-mail: email@example.com
Department of Mathematics, National Technical University, Athens 15780, Greece.
Received 5 August 2004; accepted 18 November 2004
Abstract. We study the existence of positive solutions and of positive homoclinic (to zero)
solutions for a class of periodic problems driven by the scalar ordinary p-Laplacian and
having a nonsmooth potential. Our approach is variational based on the nonsmooth crit-
ical point theory and our results extend the recent works of Korman–Lazer (Electronic
JDE (1994)) and of Grossinho–Minhos–Tersian (J. Math. Anal. Appl. 240 (1999)).
Mathematics Subject Classiﬁcation 2000: 34B15, 34C25, 34C37, 34A60
Key words: Ordinary p-Laplacian, nonsmooth critical point theory, Mountain Pass lemma,
In this paper we study the following nonlinear second order periodic equa-
tion with a nonsmooth potential
x(t)∈ ∂j (t,x(t)) a.e. on T = [0,b],
x(0) = x(b), x
(b), 1 <p<∞.
We also consider the following corresponding homoclinic problem;
namely, we investigate
x(t) ∈ ∂j (t, x(t)) a.e. on R,
|x(t)|→0 and |x
(t)|→0as|t|→∞, 1 <p<∞.
For problems (1) and (2), we investigate the existence of positive peri-
odic solutions and of positive homoclinic solutions, respectively. Thus far,