Small order asymptotics for nonlinear fractional problems

Abstract

We study the limiting behavior of solutions to boundary value nonlinear problems involving the fractional Laplacian of order 2s when the parameter s tends to zero. In particular, we show that least-energy solutions converge (up to a subsequence) to a nontrivial nonnegative least-energy solution of a limiting problem in terms of the logarithmic Laplacian, i.e. the pseudodifferential operator with Fourier symbol ln(|ξ| 2 ). These results are motivated by some applications of nonlocal models where a small value for the parameter s yields the optimal choice. Our approach is based on variational methods, uniform energy-derived estimates, and the use of a new logarithmic-type Sobolev inequality.

A Direct Method of Moving Planes for Logarithmic Schrödinger Operator

“…In particular, Schrödinger equations arise in quantum field theory and in the Hartree-Fock theory (see [5,17,18,21]). Recently, there is a surge of interest to investigate integrodifferential operators of order close to zero and associated linear and nonlinear integrodifferential equations (see [1,4,14,16,19,20]). In particular, the logarithmic Laplacian and the logarithmic Schrödinger operator are two interesting examples of such class of operators.…”

A direct method of moving planes for the Logarithmic Laplacian

“…In particular, Schrödinger equations arise in quantum field theory and in the Hartree-Fock theory (see [5,15,16,19]). Recently, there is a surge of interest to investigate integrodifferential operators of order close to zero and associated linear and nonlinear integrodifferential equations (see [1,4,12,14,17,18]). In particular, the logarithmic Laplacian and the logarithmic Schrödinger operator are two interesting examples of such class of operators.…”

Spectral characteristics of Dirichlet problems for nonlocal operators

“…While the result of Chapter 2 essentially deals with nonlocal operato of order 2s, the rest of the thesis, namely, Chapter 3, 4 and 5 deals more or less with nonlocal operators of small order. These operators are getting nowadays, increasing interest in the study of linear and nonlinear nonlocal partial differential equations [13,29,30,32,86] and also, are motivated by some applications to nonlocal models where small order of the operator captures the optimal accuracy and the efficiency of the model [4,81].…”

“…Integrodifferential operators of order close to zero are getting increasing interest in the study of linear and nonlinear integrodifferential equations, see for e.g. [29,30,63,72,79,86] with references therein. In particular, the logarithmic Schrödinger operator (I − ∆) log has the same singular local behavior as that of the logarithmic Laplacian L ∆ studied in [29], while it eliminates the integrability problem of L ∆ at infinity.…”

“…In particular, we are interested in the study of opertators with order near zero. Motivated by some applications to nonlocal models, where a small order of the operator captures the optimal efficiency of the model [4,81], nonlocal operators with possibly differential order close to zero have been studied in linear and nonlinear integrodifferential equations, see [29,30,43,45,47,86,94] and references in there. From a stochastic point of view, general classes of nonlocal operators appear as the generator of jump processes, where the jump behavior is modelled through types of Lévy measures and properties of associated harmonic functions have been studied, see [56,58,63,79] and there references in there.…”