Nontrivial Solution for the Fractional p-Laplacian Equations via Perturbation Methods

Abstract

We study the existence of nontrivial solution of the following equation without compactness:where , ≥ 2, ∈ (0, 1), (−Δ) is the fractional -Laplacian, and the subcritical -superlinear term ∈ (R × R) is 1-periodic in for = 1, 2, . . . , . Our main difficulty is that the weak limit of (PS) sequence is not always the weak solution of fractional -Laplacian type equation. To overcome this difficulty, by adding coercive potential term and using mountain pass theorem, we get the weak solution of perturbation equations. And we prove that → as → 0. Finally, by using vanishing lemma and periodic condition, we get that is a nontrivial solution of fractional -Laplacian equation.