Existence of Solutions and of Multiple Solutions for Nonlinear Nonsmooth Periodic Systems

Nonlinear Periodic Systems with thep-Laplacian: Existence and Multiplicity Results

“…Moreover we do not assume any polynomial growth of the subdifferential ∂ j (t,x). The result so obtained extends the analogous and just mentioned results of [13][14][15], in the sense that there exist potential functions satisfying our hypotheses but not those of the mentioned theorems (see Remark 3.5).…”

“…In particular in Theorem 3.14 we make an Ambrosetti-Rabinowitz-type assumption (see H( j) 4 (iv)) which, together with the other hypotheses, implies a growth condition on j(t,x) strictly less than p. Moreover, Theorem 4.1 gives us the existence of multiple solutions in the setting of local, nonuniform anticoercive potential function. All the last three theorems extend, in the sense explained above, analogous results of [13][14][15].…”

“…Remark 3.5. We want to observe that our Theorem 3.3 extends the analogous existence results given in [14,15] Consider now the following hypotheses on j(t,x):…”

“…where σ > p > 1. This function satisfies H( j) 5 but note the hypotheses of [13,Theorem 3.4.11] and of [14,Theorem 10].…”

“…In the last years, using variational methods on the nonsmooth critical point theory (see [13]), the problem (I) has been studied in order to obtain existence and multiplicity results. We can mention the papers of E. H. Papageorgiou and N. S. Papageorgiou [14,15], or the numerous results collected in the book of Gasiński and Papageorgiou (see [13,Section 3.4.1]).…”