Multiplicity and concentration of positive solutions for fractional nonlinear Schrodinger equations with critical growth
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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.
CitationShang, X., & Zhang, J. (2019). Multiplicity and concentration of positive solutions for fractional nonlinear Schrodinger equations with critical growth. Electronic Journal of Differential Equations, 2019(24), pp. 1-22.
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