Non-linear second-order periodic systems with non-smooth potential

Abstract

In this paper we study second order non-linear periodic systems driven by the ordinary vector p-Laplacian with a non-smooth, locally Lipschitz potential function. Our approach is variational and it is based on the non-smooth critical point theory. We prove existence and multiplicity results under general growth conditions on the potential function. Then we establish the existence of non-trivial homoclinic (to zero) solutions. Our theorem appears to be the first such result (even for smooth problems) for systems monitored by the p-Laplacian. In the last section of the paper we examine the scalar non-linear and semilinear problem. Our approach uses a generalized Landesman-Lazer type condition which generalizes previous ones used in the literature. Also for the semilinear case the problem is at resonance at any eigenvalue.

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):…”

“…Remark 3. 15. Hypotheses H( j) 4 (vi) is assumed only to avoid the possibility that the solution x is trivial.…”

“…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]).…”