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.
Highlights
The aim of this paper is to study how the weight potential affects the existence of ground state solutions and the number of positive solutions of the fractional Schrodinger equation: (−Δ)αu + u = λa(x)u ln |u| + b(x)|u|2∗α−2u, x ∈ RN, (1.1)
We show that ωλ+(x, 0) is a positive ground state solution of equation (1.1)
By virtue of the maximum principle for the fractional elliptic equations [33], we obtain that the problem (2.2) admits at least catMδ (M ) + 1 positive solutions
Summary
Haining Fan Zhaosheng Feng The University of Texas Rio Grande Valley Xingjie Yan. Follow this and additional works at: https://scholarworks.utrgv.edu/mss_fac Part of the Mathematics Commons.
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