NEHARI MANIFOLD AND MULTIPLICITY RESULTS FOR A CLASS OF FRACTIONAL BOUNDARY VALUE PROBLEMS WITH p-LAPLACIAN

In this work, we investigate the following fractional boundary value problems  tD T ( |0D t (u(t))|0Dt u(t) ) = ∇W (t, u(t)) + λg(t)|u(t)|q−2u(t), t ∈ (0, T ), u(0) = u(T ) = 0, where ∇W (t, u) is the gradient of W (t, u) at u and W ∈ C([0, T ]×Rn,R) is homogeneous of degree r, λ is a positive parameter, g ∈ C([0, T ]), 1 < r < p < q and 1 p < α < 1. Using the Fibering map and Nehari manifold, for some positive constant λ0 such that 0 < λ < λ0, we prove the existence of at least two non-trivial solutions.