Nonuniform Nonresonance of Semilinear Differential Equations

AbstractConsider the Dirichlet problem of nonlinear differential equations with the principal part the p-Laplacian. When the nonlinearity satisfies some semilinearity conditions, the usual nonuniform nonresonance conditions are obtained by comparing nonlinear equations with the classical eigenvalues. In this article, we will introduce some weighted eigenvalues. The nonuniform nonresonance conditions, proved in this article using weighted eigenvalues, will improve the usual ones significantly

On the Meromorphic Integrability of the Critical Systems for Optimal Sums of Eigenvalues

The popularity of estimation to bounds for sums of eigenvalues started from P. Li and S. T. Yau for the study of the P\'{o}lya conjecture. This subject is extended to different types of differential operators. This paper explores for the sums of the first $m$ eigenvalues of Sturm-Liouville operators from two aspects. Firstly, by the complete continuity of eigenvalues, we propose a family of critical systems consisting of nonlinear ordinary differential equations, indexed by the exponent $p\in(1,\infty)$ of the Lebesgue spaces concerned. There have profound relations between the solvability of these systems and the optimal lower or upper bounds for the sums of the first $m$ eigenvalues of Sturm-Liouville operators, which provides a novel idea to study the optimal bounds. Secondly, we investigate the integrability or solvability of the critical systems. With suitable selection of exponents $p$, the critical systems are equivalent to the polynomial Hamiltonian systems of $m$ degrees of freedom. Using the differential Galois theory, we perform a complete classification for meromorphic integrability of these polynomial critical systems. As a by-product of this classification, it gives a positive answer to the conjecture raised by Tian, Wei and Zhang [J. Math. Phys. 64, 092701 (2023)] on the critical systems for optimal eigenvalue gaps. The numerical simulations of the Poincar\'{e} cross sections show that the critical systems for sums of eigenvalues can appear complex dynamical phenomena, such as periodic trajectories, quasi-periodic trajectories and chaos

On the Structure of Periodic Eigenvalues of the Vectorial pp-Laplacian

In this paper we will solve an open problem raised by Man\'asevich and Mawhin twenty years ago on the structure of the periodic eigenvalues of the vectorial $p$-Laplacian. This is an Euler-Lagrangian equation on the plane or in higher dimensional Euclidean spaces. The main result obtained is that for any exponent $p$ other than $2$, the vectorial $p$-Laplacian on the plane will admit infinitely many different sequences of periodic eigenvalues with a given period. These sequences of eigenvalues are constructed using the notion of scaling momenta we will introduce. The whole proof is based on the complete integrability of the equivalent Hamiltonian system, the tricky reduction to $2$-dimensional dynamical systems, and a number-theoretical distinguishing between different sequences of eigenvalues. Some numerical simulations to the new sequences of eigenvalues and eigenfunctions will be given. Several further conjectures towards to the panorama of the spectral sets will be imposed.Comment: 35 pages, 10 figure

On the Second-order Frechet Derivatives of Eigenvalues of Sturm-Liouville Problems in Potentials

The works of V. A. Vinokurov have shown that eigenvalues and normalized eigenfunctions of Sturm-Liouville problems are analytic in potentials, considered as mappings from the Lebesgue space to the space of real numbers and the Banach space of continuous functions respectively. Moreover, the first-order Frechet derivatives are known and paly an important role in many problems. In this paper, we will find the second-order Frechet derivatives of eigenvalues in potentials, which are also proved to be negative definite quadratic forms for some cases.Comment: 10 page

Efficient double-quenching of electrochemiluminescence from CdS:Eu QDs by hemin-graphene-Au nanorods ternary composite for ultrasensitive immunoassay

A novel ternary composite of hemin-graphene-Au nanorods (H-RGO-Au NRs) with high electrocatalytic activity was synthesized by a simple method. And this ternary composite was firstly used in construction of electrochemiluminescence (ECL) immunosensor due to its double-quenching effect of quantum dots (QDs). Based on the high electrocatalytic activity of ternary complexes for the reduction of H(2)O(2) which acted as the coreactant of QDs-based ECL, as a result, the ECL intensity of QDs decreased. Besides, due to the ECL resonance energy transfer (ECL-RET) strategy between the large amount of Au nanorods (Au NRs) on the ternary composite surface and the CdS:Eu QDs, the ECL intensity of QDs was further quenched. Based on the double-quenching effect, a novel ultrasensitive ECL immunoassay method for detection of carcinoembryonic antigen (CEA) which is used as a model biomarker analyte was proposed. The designed immunoassay method showed a linear range from 0.01 pg mL(−1) to 1.0 ng mL(−1) with a detection limit of 0.01 pg mL(−1). The method showing low detection limit, good stability and acceptable fabrication reproducibility, provided a new approach for ECL immunoassay sensing and significant prospect for practical application