Existence of positive solutions for nonlinear Kirchhoff type problems in R^3 with critical Sobolev exponent and sign-changing nonlinearities

In this paper, we study the following nonlinear problem of Kirchhoff type with critical Sobolev exponent: -\left(a+b\ds\int_{\R^3}|D u|^2\right)\Delta u+u=f(x,u)+u^{5}, u\in H^1(\R^3), u>0, $x\in \R^3$ where a,b>0 are constants. Under certain assumptions on the sign-changing function $f(x,u)$, we prove the existence of positive solutions by variational methods. Our main results can be viewed as a partial extension of a recent result of He and Zou in [17] concerning the existence of positive solutions to the nonlinear Kirchhoff problem \left(\varepsilon^2a+\varepsilon b\ds\int_{\R^3}|D u|^2\right)\Delta u+V(x)u=f(u), u\in H^1(\R^3), u>0, $x\in \R^3$, where $\varepsilon>0$ is a parameter, $V(x)$ is a positive continuous potential and $f(u)\thicksim |u|^{p-2}u$ with $4<p<6$ and satisfies the Ambrosetti-Rabinowitz type condition

Existence of positive ground state solutions for the nonlinear Kirchhoff type equations in R3R^3

In this paper, we study the following nonlinear problem of Kirchhoff type with pure power nonlinearities: (a+b\ds\int_{\R^3}|D u|^2\right)\Delta u+V(x)u=|u|^{p-1}u, u\in H^1(\R^3), u>0, $x\in \R^3, where$a,$$b>0$are constants,$2<p<5$and$V:\R^3\rightarrow\R$. Under certain assumptions on$V, we prove that \eqref{0.1} has a positive ground state solution by using a monotonicity trick and a new version of global compactness lemma. Our main results can be viewed as a partial extension of a recent result of He and Zou in [16] concerning the existence of positive solutions to the nonlinear Kirchhoff problem (\varepsilon^2a+\varepsilon b\ds\int_{\R^3}|D u|^2\right)\Delta u+V(x)u=f(u), u\in H^1(\R^3), u>0, x\in \R^3$, where$\varepsilon>0$is a parameter,$V(x)$is a positive continuous potential and$f(u)\thicksim |u|^{p-1}u$with$3<p<5$ and satisfies the Ambrosetti-Rabinowitz type condition. Our main results extend also the arguments used in [7,36], which deal with Schr\"{o}dinger-Poisson system with pure power nonlinearities, to the Kirchhoff type problem

On quasi-static cloaking due to anomalous localized resonance in R3\mathbb{R}^3

This work concerns the cloaking due to anomalous localized resonance (CALR) in the quasi-static regime. We extend the related two-dimensional studies in [2,10] to the three-dimensional setting. CALR is shown not to take place for the plasmonic configuration considered in [2,10] in the three-dimensional case. We give two different constructions which ensure the occurrence of CALR. There may be no core or an arbitrary shape core for the cloaking device. If there is a core, then the dielectric distribution inside it could be arbitrary

On novel elastic structures inducing plasmonic resonances with finite frequencies and cloaking due to anomalous localized resonances

This paper is concerned with the theoretical study of plasmonic resonances for linear elasticity governed by the Lam\'e system in $\mathbb{R}^3$, and their application for cloaking due to anomalous localized resonances. We derive a very general and novel class of elastic structures that can induce plasmonic resonances. It is shown that if either one of the two convexity conditions on the Lam\'e parameters is broken, then we can construct certain plasmon structures that induce resonances. This significantly extends the relevant existing studies in the literature where the violation of both convexity conditions is required. Indeed, the existing plasmonic structures are a particular case of the general structures constructed in our study. Furthermore, we consider the plasmonic resonances within the finite frequency regime, and rigorously verify the quasi-static approximation for diametrically small plasmonic inclusions. Finally, as an application of the newly found structures, we construct a plasmonic device of the core-shell-matrix form that can induce cloaking due to anomalous localized resonance in the quasi-static regime, which also includes the existing study as a special case.Comment: 26 pages, comments are welcom

On the concentration phenomenon of L2L^2-subcritical constrained minimizers for a class of Kirchhoff equations with potentials

In this paper, we study the existence and the concentration behavior of minimizers for $i_V(c)=\inf\limits_{u\in S_c}I_V(u)$, here $S_c=\{u\in H^1(\R^N)|~\int_{\R^N}V(x)|u|^20\}$ and I_V(u)=\frac{1}{2}\ds\int_{\R^N}(a|\nabla u|^2+V(x)|u|^2)+\frac{b}{4}\left(\ds\int_{\R^N}|\nabla u|^2\right)^2-\frac{1}{p}\ds\int_{\R^N}|u|^{p}, where $N=1,2,3$ and $a,b>0$ are constants. By the Gagliardo-Nirenberg inequality, we get the sharp existence of global constraint minimizers for $2<p<2^*$ when $V(x)\geq0$, $V(x)\in L^{\infty}_{loc}(\R^N)$ and $\lim\limits_{|x|\rightarrow+\infty}V(x)=+\infty$. For the case $p\in(2,\frac{2N+8}{N})\backslash\{4\}$, we prove the global constraint minimizers $u_c$ behave like $$u_{c}(x)\approx \frac{c}{|Q_{p}|_2}\left(\frac{m_{c}}{c}\right)^{\frac{N}{2}}Q_p\left(\frac{m_{c}}{c}x-z_c\right).$$ for some $z_c\in\R^N$ when $c$ is large, where $Q_p$ is up to translations, the unique positive solution of $-\frac{N(p-2)}{4}\Delta Q_p+\frac{2N-p(N-2)}{4}Q_p=|Q_p|^{p-2}Q_p$ in $\R^N$ and $m_c=(\frac{\sqrt{a^2D_1^2-4bD_2i_0(c)}+aD_1}{2bD_2})^{\frac12}$, $D_1=\frac{Np-2N-4}{2N(p-2)}$ and $D_2=\frac{2N+8-Np}{4N(p-2)}$

On Anomalous Localized Resonance for the Elastostatic System

We consider the anomalous localized resonance due to a plasmonic structure for the elastostatic system in R^2. The plasmonic structure takes a general core-shell-matrix form with the metamaterial located in the shell. If there is no core, we show that resonance occurs for a very broad class of sources. If the core is nonempty and of an arbitrary shape, we show that there exists a critical radius such that anomalous localized resonance (ALR) occurs. Our argument is based on a variational technique by making use of the primal and dual variational principles for the elastostatic system, along with the construction of suitable test functions.Comment: 21 pages, no figur

On anomalous localized resonance and plasmonic cloaking beyond the quasistatic limit

In this paper, we give the mathematical construction of novel core-shell plasmonic structures that can induce anomalous localized resonance and invisibility cloaking at certain finite frequencies beyond the quasistatic limit. The crucial ingredient in our study is that the plasmon constant and the loss parameter are constructed in a delicate way that are correlated and depend on the source and the size of the plasmonic structure. As a significant byproduct of this study, we also derive the complete spectrum of the Neumann-Poinc\'are operator associated to the Helmholtz equation with finite frequencies in the radial geometry. The spectral result is the first one in its type and is of significant mathematical interest for its own sake

On three-dimensional plasmon resonance in elastostatics

We consider the plasmon resonance for the elastostatic system in $\mathbb{R}^3$ associated with a very broad class of sources. The plasmonic device takes a general core-shell-matrix form with the metamaterial located in the shell. It is shown that the plasmonic device in the literature which induces resonance in $\mathbb{R}^2$ does not induce resonance in $\mathbb{R}^3$. We then construct two novel plasmonic devices with suitable plasmon constants, varying according to the source term or the loss parameter, which can induce resonances. If there is no core, we show that resonance always occurs. If there is a core of an arbitrary shape, we show that the resonance strongly depends on the location of the source. In fact, there exists a critical radius such that resonance occurs for sources lying within the critical radius, whereas resonance does not occur for source lying outside the critical radius. Our argument is based on the variational technique by making use of the primal and dual variational principles for the elastostatic system, along with the highly technical construction of the associated perfect plasmon elastic waves.Comment: 25 pages, comments welcome. arXiv admin note: text overlap with arXiv:1601.0774

Efficient GAN-based method for cyber-intrusion detection

Ubiquitous anomalies endanger the security of our system constantly. They may bring irreversible damages to the system and cause leakage of privacy. Thus, it is of vital importance to promptly detect these anomalies. Traditional supervised methods such as Decision Trees and Support Vector Machine (SVM) are used to classify normality and abnormality. However, in some case the abnormal status are largely rarer than normal status, which leads to decision bias of these methods. Generative adversarial network (GAN) has been proposed to handle the case. With its strong generative ability, it only needs to learn the distribution of normal status, and identify the abnormal status through the gap between it and the learned distribution. Nevertheless, existing GAN-based models are not suitable to process data with discrete values, leading to immense degradation of detection performance. To cope with the discrete features, in this paper, we propose an efficient GAN-based model with specifically-designed loss function. Experiment results show that our model outperforms state-of-the-art models on discrete dataset and remarkably reduce the overhead

Reconstruction via the intrinsic geometric structures of interior transmission eigenfunctions

We are concerned with the inverse scattering problem of extracting the geometric structures of an unknown/inaccessible inhomogeneous medium by using the corresponding acoustic far-field measurement. Using the intrinsic geometric properties of the so-called interior transmission eigenfunctions, we develop a novel inverse scattering scheme. The proposed method can efficiently capture the cusp singularities of the support of the inhomogeneous medium. If further a priori information is available on the support of the medium, say, it is a convex polyhedron, then one can actually recover its shape. Both theoretical analysis and numerical experiments are provided. Our reconstruction method is new to the literature and opens up a new direction in the study of inverse scattering problems.Comment: 20 pages, 21 figure