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Positivity

, Volume 22, Issue 3, pp 873–895 | Cite as

The positive solutions to a quasi-linear problem of fractional p-Laplacian type without the Ambrosetti–Rabinowitz condition

  • Bin Ge
  • Ying-Xin Cui
  • Liang-Liang Sun
  • Massimiliano Ferrara
Article
  • 94 Downloads

Abstract

In this paper, we study the existence of nontrivial solution to a quasi-linear problem where \( (-\Delta )_{p}^{s} u(x)=2\lim \nolimits _{\epsilon \rightarrow 0}\int _{\mathbb {R}^N \backslash B_{\varepsilon }(X)} \frac{|u(x)-u(y)|^{p-2} (u(x)-u(y))}{| x-y | ^{N+sp}}dy, \) \( x\in \mathbb {R}^N\) is a nonlocal and nonlinear operator and \( p\in (1,\infty )\), \( s \in (0,1) \), \( \lambda \in \mathbb {R} \), \( \Omega \subset \mathbb {R}^N (N\ge 2)\) is a bounded domain which smooth boundary \(\partial \Omega \). Using the variational methods based on the critical points theory, together with truncation and comparison techniques, we show that there exists a critical value \(\lambda _{*}>0\) of the parameter, such that if \(\lambda >\lambda _{*}\), the problem \((P)_{\lambda }\) has at least two positive solutions, if \(\lambda =\lambda _{*}\), the problem \((P)_{\lambda }\) has at least one positive solution and it has no positive solution if \(\lambda \in (0,\lambda _{*})\). Finally, we show that for all \(\lambda \ge \lambda _{*}\), the problem \((P)_{\lambda }\) has a smallest positive solution.

Keywords

Fractional p-Laplacian Quasi-linear Local minimizers Mountain-pass theorem Nontrivial solution 

Mathematics Subject Classification

35P15 35P30 35R11 

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Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  • Bin Ge
    • 1
  • Ying-Xin Cui
    • 1
  • Liang-Liang Sun
    • 1
  • Massimiliano Ferrara
    • 2
  1. 1.Department of Applied MathematicsHarbin Engineering UniversityHarbinPeople’s Republic of China
  2. 2.Department of Law and EconomicsUniversity Mediterranea of Reggio CalabriaPalazzo ZaniItaly

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