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- Title
Nontrivial Solution for the Fractional p-Laplacian Equations via Perturbation Methods.
- Authors
Luo, Huxiao; Li, Shengjun; Tang, Xianhua
- Abstract
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.
- Subjects
LAPLACIAN matrices; COMPACT spaces (Topology); MATHEMATICAL sequences; MATHEMATICS theorems; PERTURBATION theory
- Publication
Advances in Mathematical Physics, 2017, p1
- ISSN
1687-9120
- Publication type
Article
- DOI
10.1155/2017/5317213