Multiplicity and concentration of positive solutions for fractional nonlinear Schrodinger equations with critical growth
Date
2019-02-12
Authors
Shang, Xudong
Zhang, Jihui
Journal Title
Journal ISSN
Volume Title
Publisher
Texas State University, Department of Mathematics
Abstract
In this article we consider the multiplicity and concentration behavior of positive solutions for the fractional nonlinear Schrödinger equation
ε2s (-Δ)su + V(x)u = u2*s-1 + ƒ(u), x ∈ ℝN, u ∈ Hs(ℝN), u(x) > 0,
where ε is a positive parameter, s ∈ (0, 1), N > 2s and 2*s = 2N/N-2s is the fractional critical exponent, and ƒ is a C1 function satisfying suitable assumptions. We assume that the potential V(x) ∈ C(ℝN) satisfies infℝN V(x) > 0, and that there exits k points xj ∈ ℝN such that for each j = 1,..., k, V(xj) are strictly global minimum. By using the variational method, we show that there are at least k positive solutions for a small ε > 0. Moreover, we establish the concentration property of solutions as ε tends to zero.
Description
Keywords
Fractional Schrödinger equations, Multiplicity of solutions, Critical growth, Variational method
Citation
Shang, X., & Zhang, J. (2019). Multiplicity and concentration of positive solutions for fractional nonlinear Schrodinger equations with critical growth. <i>Electronic Journal of Differential Equations, 2019</i>(24), pp. 1-22.
Rights
Attribution 4.0 International