Positivity 12 (2008), 733–750
2008 Birkh¨auser Verlag Basel/Switzerland
1385-1292/040733-18, published online October 4, 2008
Positive solutions for nonlinear periodic
, Nikolaos S. Papageorgiou and Vasile Staicu
Abstract. We consider a nonlinear periodic problem driven by the scalar
p-Laplacian and with a nonsmooth potential. Using the degree map for multi-
valued perturbations of (S)
-operators and the spectrum of a weighted eigen-
value problem for the scalar periodic p-Laplacian, we prove the existence of a
strictly positive solution.
Mathematics Subject Classiﬁcation (2000). 34B15, 34B18, 34C25.
Keywords. Scalar p-Laplacian, degree theory, (S)
-operator, Picone’s identity,
In this paper, we study the existence of positive solutions for the following nonlin-
ear periodic problem with a nonsmooth potential (hemivariational inequality):
∈ ∂j(t, x(t)) a.e. on T =[0,b],
x(0) = x(b),x
(0) = x
(b), 1 <p<∞.
In this problem the potential function j(t, x) is jointly measurable, x →
j(t, x) is locally Lipschitz and in general nonsmooth and by ∂j(t, x) we denote
the generalized subdifferential of the function x → j(t, x). The goal is to produce
positive solutions for problem (1.1). To achieve this, we adopt a degree theo-
retic approach based on the degree map for certain multivalued perturbations of
-operators introduced by Hu-Papageorgiou  (see also Hu-Papageorgiou
). Another basic tool in our analysis, is the spectrum of a weighted periodic
eigenvalue problem with the scalar p-Laplacian, studied recently by Zhang .
In the past the existence of positive solutions for differential equations driven
by the scalar p-Laplacian, was studied in the framework of Dirichlet or more gen-
erally Sturm-Liouville problems. We refer to the works of Ben Naoum-De Coster
Researcher supported by a grant of the National Scholarship Foundation of Greece (I.K.Y.)