Concentration phenomena for fractional magnetic NLS equations

We study the multiplicity and concentration of complex-valued solutions for a fractional magnetic Schrödinger equation involving a scalar continuous electric potential satisfying a local condition and a continuous nonlinearity with subcritical growth. The main results are obtained by applying a penalization technique, generalized Nehari manifold method and Ljusternik–Schnirelman theory. We also prove a Kato's inequality for the fractional magnetic Laplacian which we believe to be useful in the study of other fractional magnetic problems.

Some inequalities for trace class operators via a Kato’s result

By the use of the celebrated Kato’s inequality, we obtain in this paper some new inequalities for trace class operators on a complex Hilbert space [Formula: see text] Natural applications for functions defined by power series of normal operators are given as well.

APPLICATIONS OF KATO'S INEQUALITY TO OPERATOR-VALUED INTEGRALS ON HILBERT SPACES

By the use of the celebrated Kato's inequality we obtain in this paper some inequalities for operator-valued integrals on a complex Hilbert space H. Among others, we show that [Formula: see text] for any x, y ∈ H, provided [Formula: see text] and p : E → [0, ∞) are μ-measurable functions on E and such that [Formula: see text] and [Formula: see text] are Bochner integrable on E for some α ∈ [0, 1]. Natural applications for various norms and numerical radii associated with the Bochner integral of operator-valued functions and some examples for the operator exponential are presented as well.