Abstract
The purpose of this paper is to investigate the existence of periodic solutions for a class of nonlocal p(t)-Laplacian systems. When the nonlinear term is p^{+}-superlinear at infinity, some new solvability conditions of nontrivial periodic solutions are obtained by using a version of the local linking theorem. A major point is that we ensure compactness without the well-known Ambrosetti–Rabinowitz type superlinearity condition. In addition, by applying the saddle point theorem, we established the existence of at least one periodic solution for such problems with a p^{-}-subquadratic potential.
Highlights
Introduction and main resultsConsider the non-autonomous second order Hamiltonian systems ⎧⎨u(t) + ∇V (t, u(t)) = 0, a.e. t ∈ [0, T],⎩u(0) – u(T) = u (0) – u (T) = 0, (1.1)where T > 0, u(t) ∈ RN, ∇V (t, u) denotes the gradient of V (t, u) in u
In the classical monograph [1], Mawhin and Willem investigated the existence of periodic solutions for problem (1.1) via critical point theory
Problems with variable exponent growth conditions arise in the description of the physical phenomena with “pointwise different properties” which first arose from the nonlinear elasticity theory; see [2]
Summary
Where T > 0, M(s) : [0, +∞) → (0, +∞) is a continuous function, and assume that V (t, u) satisfies the following assumption: (V0) V (t, u) is measurable in t for each u ∈ RN and continuously differentiable in u for a.e. t ∈ [0, T], and there exist a ∈ C(R+, R+), b ∈ L1([0, T]; R+) such that Throughout the paper, we assume that p(t) appearing in problem (1.3) satisfies (P) p(t) ∈ C([0, T], R+), p(t) = p(t + T) and p(t) fulfills the following hypothesis: 1 < p– := min p(t) ≤ p+ := max p(t) < +∞.
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