Existence and symmetry results for a Schrödinger type problem involving the fractional Laplacian

By Serena Dipierro, Giampiero Palatucci, Enrico Valdinoci. 2012

258 citations recorded by the source corpus.

This paper deals with the following class of nonlocal Schrodinger equations (-\Delta)^s u + u = |u|^{p-1}u in \mathbb{R}^N, for s\in (0,1). We prove existence and symmetry results for the solutions $u$ in the fractional Sobolev space H^s(\mathbb{R}^N). Our results are in clear accordance with those for the classical local counterpart, that is when s=1.

Subfields: Applied Mathematics, Computational Theory and Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/874de46d204347b691d0e8a63314343a40016db6

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In Applied Mathematics

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Citations in the corpus

Ground state of scalar field equations involving a fractional Laplacian with general nonlinearity (2013, background): “However, it should be noted that the arguments in [23] are not applicable in our setting due to the minimax arguments.”
Asymptotic method of moving planes for fractional parabolic equations (2020, background): “During the last decade, elliptic nonlocal equations, especially those involving fractional Laplacians, have been studied by more and more scholars and a series of results have been obtained, such as [9] for symmetry and nonexistence of solutions, [12, 16] for Liouville Theorems with indefinite nonli”
Existence of solutions for perturbed fractional p-Laplacian equations (2016, background): “3) For standing wave solutions of fractional Schrödinger equations in R , we refer to [11,14,16, 20,21,26] and the references therein.”
Ground states for fractional Schrödinger equations with critical growth (2014, background): “Recently, the existence of solutions for problem (2) involving subcritical growth with ε = 1 was considered, such as [5, 11, 14, 23, 24].”
Existence and symmetry of solutions for critical fractional Schrödinger equations with bounded potentials (2016, background): “Here we would like to point out some interesting models involving the fractional Laplacian, such as, the fractional Schrödinger equation (see [14, 15, 22, 23, 24]), the fractional Kirchhoff equation (see [16, 32, 33, 46, 47]), the fractional porous medium equation (see [9, 45]), the fractional Yamab”
Multiple solutions for nonhomogeneous Schrödinger–Kirchhoff type equations involving the fractional p-Laplacian in RN\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setl (2015, background): “See, for instance, [20,22,42] and the references therein for some results.”
Multiplicity results for the non-homogeneous fractional p-Kirchhoff equations with concave–convex nonlinearities (2015, background): “5) where the nonlinearity f satisfies some general conditions, for instance, see [28–30] for related discussions.”
Ground state solutions of scalar field fractional Schrödinger equations (2015, background): “Contrary to the classical Laplacian case that is widely investigated, the situation seems to be in a developing state when s ∈ (0, 1) and of an increasing interest (see, for instance, [19,21,24,26]).”