Transactions of the London Mathematical Society (Dec 2021)

Positive solutions for the fractional Schrödinger equations with logarithmic and critical non‐linearities

  • Haining Fan,
  • Zhaosheng Feng,
  • Xingjie Yan

DOI
https://doi.org/10.1112/tlm3.12034
Journal volume & issue
Vol. 8, no. 1
pp. 206 – 242

Abstract

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Abstract In this paper, we study a class of fractional Schrödinger equations involving logarithmic and critical non‐linearities on an unbounded domain, and show that such an equation with positive or sign‐changing weight potentials admits at least one positive ground state solution and the associated energy is positive (or negative). By applying the Nehari manifold method and Ljusternik–Schnirelmann category, we investigate how the weight potential affects the multiplicity of positive solutions, and obtain the relationship between the number of positive solutions and the category of some sets related to the weight potential.

Keywords