Advances in Mathematical Physics (Jan 2017)

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

  • Huxiao Luo,
  • Shengjun Li,
  • Xianhua Tang

DOI
https://doi.org/10.1155/2017/5317213
Journal volume & issue
Vol. 2017

Abstract

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