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Solutions to a nonlinear Maxwell equation with two competing nonlinearities in R3

Bieganowski Bartosz

Bulletin of the Polish Academy of Sciences. Mathematics

2021

Vol. 69, no. 1

37--60

We are interested in the nonlinear, time-harmonic Maxwell equation ∇(∇E)+V(x)E=h(x,E) in R3 with sign-changing nonlinear term h, i.e. we assume that h is of the form h(x,αw)=f(x,α)w−g(x,α)w for w∈R3, |w|=1 and α∈R. In particular, we can consider the nonlinearity consisting of two competing powers, h(x,E)=|E|p−2E−|E|q−2E with 2

Eigenvalue asymptotics for the Sturm-Liouville operator with potential having a strong local negative singularity

Nursultanov M., Rozenblum G.

Opuscula Mathematica

2017

Vol. 37, no. 1

109--139

We find asymptotic formulas for the eigenvalues of the Sturm-Liouville operator on the finite interval, with potential having a strong negative singularity at one endpoint. This is the case of limit circle in H. Weyl sense. We establish that, unlike the case of an infinite interval, the asymptotics for positive eigenvalues does not depend on the potential and it is the same as in the regular case. The asymptotics of the negative eigenvalues may depend on the potential quite strongly, however there are always asymptotically fewer negative eigenvalues than positive ones. By unknown reasons this type of problems had not been studied previously.

Widely Separated Deformations of Singular Potential

Kondej S.

Computational Methods in Science and Technology

2010

Vol. 16, No. 2

165-167

We study the two dimensional quantum system governedby the Schrodinger operator with delta type potential. The interaction is supported by line �ˇ which coincides with a straight at infinity and which admits two widely separated deformations. The aim of this paper is to express the number of bound states of our system by the number of bound states of the system with single deformation.