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Nontrivial Solution for the Fractional -Laplacian Equations via Perturbation Methods

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  • Huxiao Luo
  • Shengjun Li
  • Xianhua Tang

Abstract

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.

Suggested Citation

  • Huxiao Luo & Shengjun Li & Xianhua Tang, 2017. "Nontrivial Solution for the Fractional -Laplacian Equations via Perturbation Methods," Advances in Mathematical Physics, Hindawi, vol. 2017, pages 1-9, November.
    • Handle: RePEc:hin:jnlamp:5317213
    DOI: 10.1155/2017/5317213
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