Nontrivial Solution for the Fractional -Laplacian Equations via Perturbation Methods

We study the existence of nontrivial solution of the following equation without compactness: ,   where ,  ,   is the fractional -Laplacian, and the subcritical -superlinear term is 1-periodic in for . 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 . Finally, by using vanishing lemma and periodic condition, we get that is a nontrivial solution of fractional -Laplacian equation.

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