On the existence of solutions for the Schrödinger–Poisson equations

AbstractIn this paper, we are concerned with the system of Schrödinger–Poisson equations(*){−Δu+V(x)u+ϕu=f(x,u),in R3,−Δϕ=u2,in R3. Under certain assumptions on V and f, the existence and multiplicity of solutions for (*) are established via variational methods

Homoclinic Orbits for a Class of Nonperiodic Hamiltonian Systems

We study the following nonperiodic Hamiltonian system ż=JHz(t,z), where H∈C1(R×R2N,R) is the form H(t,z)=(1/2)B(t)z⋅z+R(t,z). We introduce a new assumption on B(t) and prove that the corresponding Hamiltonian operator has only point spectrum. Moreover, by applying a generalized linking theorem for strongly indefinite functionals, we establish the existence of homoclinic orbits for asymptotically quadratic nonlinearity as well as the existence of infinitely many homoclinic orbits for superquadratic nonlinearity

Integrating Overlapping Structures and Background Information of Words Significantly Improves Biological Sequence Comparison

Word-based models have achieved promising results in sequence comparison. However, as the important statistical properties of words in biological sequence, how to use the overlapping structures and background information of the words to improve sequence comparison is still a problem. This paper proposed a new statistical method that integrates the overlapping structures and the background information of the words in biological sequences. To assess the effectiveness of this integration for sequence comparison, two sets of evaluation experiments were taken to test the proposed model. The first one, performed via receiver operating curve analysis, is the application of proposed method in discrimination between functionally related regulatory sequences and unrelated sequences, intron and exon. The second experiment is to evaluate the performance of the proposed method with f-measure for clustering Hepatitis E virus genotypes. It was demonstrated that the proposed method integrating the overlapping structures and the background information of words significantly improves biological sequence comparison and outperforms the existing models

A Search for Light Super Symmetric Baryons

We have searched for the production and decay of light super-symmetric baryons produced in 800 GeV/c proton copper interactions in a charged hyperon beam experiment. We observe no evidence for the decays R+(uud \g^~) -> S(uds \g^~) pi+ and X-(ssd \g^~) -> S(uds \g^~) pi- in the predicted parent mass and lifetime ranges of 1700-2500 Mev/c2 and 50-500 ps. Production upper limits for R+ at xF=0.47, Pt=1.4 GeV/c2 and X- at xF=0.48, Pt=0.65 GeV/c2 of less than 10^-3 of all charged secondary particles produced are obtained for all but the highest masses and shortest lifetimes predicted.Comment: 9 pages, uuencoded postscript 4 figures uuencoded, tar-compressed file (submitted to PRL

Existence and multiplicity of normalized solutions for a class of fractional Schrödinger–Poisson equations

  We consider the fractional Schrödinger-Poisson equation $\begin{cases}(-\Delta)^su-\lambda u+\phi u=|u|^{p-2}u,& x\in\mathbb{R}^3,\\ (-\Delta)^t\phi=u^2,& x\in\mathbb{R}^3,\end{cases}$ where $s,t\in(0,1)$ satisfy 2s+2t>3, $p\in(\frac{4s+6}{3},2^*_s)$ and $\lambda\in\mathbb{R}$ is an undetermined parameter. We deal with the case where the associated functional is not bounded below on the $L^2$-unit sphere and show the existence of infinitely many solutions $(u,\lambda)$ with $u$ having prescribed $L^2$-norm

Infinitely many solutions for asymptotically linear periodic Hamiltonian elliptic systems

This paper is concerned with the following periodic Hamiltonian elliptic system $\{  -\Delta \varphi+V(x)\varphi=G_\psi(x,\varphi,\psi)\ \hbox{in }\mathbb{R}^N, \\ -\Delta \psi+V(x)\psi=G_\varphi(x,\varphi,\psi)\ \hbox{in }\mathbb{R}^N, \\ \varphi(x)\to 0\ \hbox{and }\psi(x)\to0\ \hbox{as }|x|\to\infty.$ Assuming the potential V is periodic and 0 lies in a gap of $\sigma(-\Delta+V)$, $G(x,\eta)$ is periodic in x and asymptotically quadratic in $\eta=(\varphi,\psi)$, existence and multiplicity of solutions are obtained via variational approach.

Infinitely many solutions to quasilinear elliptic equation with concave and convex terms

In this paper, we are concerned with the following quasilinear elliptic equation with concave and convex terms -\Delta u-{\frac12}u\Delta(|u|^2)=\alpha|u|^{p-2}u+\beta|u|^{q-2}u,\quad x\in \Omega, \leqno(\rom{P}) % where $\Omega\subset\mathbb{R}^N$ is a bounded smooth domain, $1< p< 2$, $4< q\leq 22^*$. The existence of infinitely many solutions is obtained by the perturbation methods