Concentrating solutions for a class of nonlinear fractional Schrödinger equations in RN\mathbb R^N

  • Vincenzo Ambrosio

    Università Politecnica delle Marche, Ancona, Italy
Concentrating solutions for a class of nonlinear fractional Schrödinger equations in $\mathbb R^N$ cover
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Abstract

We deal with the existence of positive solutions for the following fractional Schrödinger equation:

ε2s(Δ)su+V(x)u=f(u)\mboxinRN,\varepsilon ^{2s} (-\Delta)^{s} u + V(x) u = f(u) \quad \mbox{in } \mathbb{R}^{N},

where ε>0\varepsilon>0 is a parameter, s(0,1)s\in (0, 1), N2N\geq 2, (Δ)s(-\Delta)^{s} is the fractional Laplacian operator, and V ⁣:RNRV\colon \mathbb{R}^{N}\rightarrow \mathbb{R} is a positive continuous function. Under the assumptions that the nonlinearity ff is either asymptotically linear or superlinear at infinity, we prove the existence of a family of positive solutions which concentrates at a local minimum of VV as ε\varepsilon tends to zero.

Cite this article

Vincenzo Ambrosio, Concentrating solutions for a class of nonlinear fractional Schrödinger equations in RN\mathbb R^N. Rev. Mat. Iberoam. 35 (2019), no. 5, pp. 1367–1414

DOI 10.4171/RMI/1086