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In this paper, we investigate a singular Moser–Trudinger inequality involving $$L^{n}$$ norm in the entire Euclidean space. The blow-up procedures are used for the maximizing sequence. Then we obtain the existence of extremal functions for this singular geometric inequality in whole space. In general, $$W^{1,n}({\mathbb {R}}^n)\hookrightarrow L^q({\mathbb {R}}^n)$$ is a continuous embedding but not compact. But in our case we can prove that $$W^{1,n}({\mathbb {R}}^n)\hookrightarrow L^n({\mathbb {R}}^n)$$ is a compact embedding. Combining the compact embedding $$W^{1,n}({\mathbb {R}}^n)\hookrightarrow L^q({\mathbb {R}}^n, |x|^{-s}dx)$$ for all $$q\ge n$$ and $$0

In this paper, we develop the blow-up analysis and establish the energy quantization for solutions to super-Liouville type equations on Riemann surfaces with conical singularities at the boundary. In other problems in geometric analysis, the blow-up analysis usually strongly utilizes conformal invariance, which yields a Noether current from which strong estimates can be derived. Here, however, the conical singularities destroy conformal invariance. Therefore, we develop another, more general, method that uses the vanishing of the Pohozaev constant for such solutions to deduce the removability of boundary singularities.

In this paper, we investigate a sharp Moser-Trudinger inequality which involves the anisotropic Sobolev norm in unbounded domains. Under this anisotropic Sobolev norm, we establish the Lions type concentration-compactness alternative firstly. Then by using a blow-up procedure, we obtain the existence of extremal functions for this sharp geometric inequality. In particular, we combine the low dimension case of \begin{document}$ n = 2 $\end{document} and the high dimension case of \begin{document}$ n\geq 3 $\end{document} to prove the existence of the extremal functions, which is different from the arguments of isotropic case, see [ 5 , 19 ].

In this paper, we study the nonhomogeneous Dirichlet problem concerning general semilinear elliptic equations in divergence form. We establish that the boundary Lipschitz regularity of solutions under some more weaker conditions on the coefficients, the boundary, the boundary function and the nonhomogeneous term. In particular, we assume that the nonhomogeneous term satisfies Dini continuity condition and Lipschitz Newtonian potential condition, which will be the optimal conditions to obtain the boundary Lipschitz regularity of solutions., arXiv admin note: text overlap with arXiv:2105.06271

We investigate a sharp Moser–Trudinger inequality which involves the anisotropic Dirichlet norm ( ∫ Ω F N ( ∇ u ) d x ) 1 N on W 0 1 , N ( Ω ) for N ≥ 2 . Here F is convex and homogeneous of degree 1, and its polar F o represents a Finsler metric on R N . Under this anisotropic Dirichlet norm, we establish the Lions type concentration-compactness alternative. Then by using a blow-up procedure, we obtain the existence of extremal functions for this sharp geometric inequality.

Arising in problems of prescribed Gaussian and Geodesic curvatures, we consider a blow-up sequence of solutions to the following exponential Neumann boundary value problem: − Δ u n = V n ( x ) | x | 2 α n e 2 u n in B R + , ∂ u n ∂ ν = h n ( x ) | x | α n e u n on ∂ B R + ⋂ ∂ R + 2 , where ν denotes the outer unit normal vector to ∂ B R + ⋂ ∂ R + 2 and α n → α ∈ ( − 1 , + ∞ ) . We establish a uniform estimate for this blow-up sequence of solutions near an isolated singular blow up point. We treat two cases α ∉ N and α ∈ N + separately, and obtain two different uniform pointwise estimates.

In this paper, we will analyze the blow-up behaviors for solutions to the Laplacian equation with exponential Neumann boundary condition. In particular, the boundary value is with a kind of singular data. We show a Brezis–Merle type concentration-compactness theorem, calculate the blow up value at the blow-up point, and give a point-wise estimate for the profile of the solution sequence at the blow-up point.

In this work, a boronate-affinity sandwich electrochemiluminescence (ECL) sensor was constructed to detect alpha-fetoprotein (AFP) based on a multiple signal amplification strategy. Gold nanoparticles (AuNPs) were utilized and modified on the surface with chitosan in order to facilitate electron transfer. The composite of the molecularly imprinted polymer (MIP) enhanced the selectivity of alpha-fetoprotein detection. 4-mercaptophenylboronic acid (MPBA) was used as the tracing tag for capture of alpha-fetoprotein. SiO2 nanoparticles carried carbon quantum dots (CQDs) labeled with gold nanoparticles and produced an ECL signal. Under the optimum experimental conditions, the linear range for alpha-fetoprotein was between 0.001 and 1000 ng/mL with a correlation coefficient of 0.9952, and the detection limit was 0.0004 ng/mL (S/N = 3). This proposed ECL sensor displayed several advantages, including outstanding selectivity, fine reproducibility, high sensitivity, low detection limit and wide linear range. Furthermore, the newly constructed boronate-affinity sandwich ECL sensor was successfully applied to the determination of alpha-fetoprotein in serum samples, indicating great potential for application in clinical diagnostics.

In this paper, we study the radial symmetry of the solution to the following system of integral form: \begin{document}$\left\{ {\begin{array}{*{20}{l}}{{u_i}(x) = \int_{{{\bf{R}}^n}} {\frac{1}{{|x - y{|^{n - \alpha }}}}} {f_i}(u(y))dy,\;\;x \in {{\bf{R}}^n},\;\;i = 1, \cdots ,m,}\\{0 Here \begin{document}$f_i(s)∈ C^1(\mathbf{R^m_+})\bigcap$\end{document} \begin{document}$ C^0(\mathbf{\overline{R^m_+}})$\end{document} \begin{document}$(i = 1,2,···,m)$\end{document} are real-valued functions, nonnegative and monotone nondecreasing with respect to the variables \begin{document}$s_1$\end{document} , \begin{document}$s_2$\end{document} , \begin{document}$···$\end{document} , \begin{document}$s_m$\end{document} . We show that the nonnegative solution \begin{document}$u = (u_1,u_2,···,u_m)$\end{document} is radially symmetric in the general condition that \begin{document}$f_i$\end{document} satisfies monotonicity condition which contains the critical and subcritical homogeneous degree as special cases. The main technique we use is the method of moving planes in an integral form. Due to our condition here is more general, the more subtle method is needed to deal with this difficulty.

In this paper, we demonstrate that, for general second order uniformly elliptic equations of non-divergence form in unbounded cylinders with zero boundary condition, the space of solutions with a given exponential growth is of finite dimension. In particular, we generalize a result of Hang and Lin (1999) [4] in the non-divergence's setting by using a different approach. This allows us to consider the general lower order terms. Furthermore, a sharp estimate of the dimension is established by using the mean value inequality.

In this paper, we classify all solutions ofwith the finite conditionsHere c is a positive number and β > −1.

In this paper, we analyze the asymptotic behavior of solution sequences of the Liouville-type equation with Neumann boundary condition. In particular, we will obtain a sharp mass quantization result for the solution sequences at a blow-up point.

In this paper, we investigate the Moser-Trudinger inequality when it involves a Finsler-Laplacian operator that is associated with functionals containing \begin{document}$F^2(\nabla u)$\end{document} . Here \begin{document}$F$\end{document} is convex and homogeneous of degree 1, and its polar \begin{document}$F^o$\end{document} represents a Finsler metric on \begin{document}$\mathbb{R}^n$\end{document} . We obtain an existence result on the extremal functions for this sharp geometric inequality.

A simple and economical hydrothermal strategy was developed for the synthesis of nitrogen-doped graphitic quantum dots (N-GQDs) using citric acid as carbon source and melamine as nitrogen source. The as-synthesized N-GQDs possess many excellent features, such as strong blue fluorescence, excitation-independent emission, high monodispersity, good stability, and excellent water solubility with a fluorescence quantum yield of 18%. The analytical application of this method as an effective sensor for direct analysis of metacycline (MTC) revealed its selective response via a fluorescence quenching mechanism. This sensing system exhibits excellent sensitivity and selectivity toward MTC with the detection limit of 20 nM, which was successfully applied in the detection of MTC in mice plasma.

In this paper, we consider the positive viscosity solutions for certain fully nonlinear uniformly elliptic equations in unbounded cylinder with zero boundary condition. After establishing an Aleksandrov-Bakelman-Pucci maximum principle, we classify all positive solutions as three categories in unbounded cylinder. Two special solution spaces (exponential growth at one end and exponential decay at the another) are one dimensional, independently, while solutions in the third solution space can be controlled by the solutions in the other two special solution spaces under some conditions, respectively.

This paper investigates the positive solutions for second order linear elliptic equation in unbounded cylinder with zero boundary condition. We prove there exist two special positive solutions with exponential growth at one end while exponential decay at the other, and all the positive solutions are linear combinations of these two.

In this paper we will analyze the blow-up behaviors of solutions to the singular Liouville type equation with exponential Neumann boundary condition. We generalize the Brezis–Merle type concentration-compactness theorem to this Neumann problem. Then along the line of the Li–Shafrir type quantization property we show that the blow-up value $$m(0) \in 2\pi \mathbb N\cup \{ 2\pi (1+\alpha )+2\pi (\mathbb N\cup \{0\})\}$$ if the singular point 0 is a blow-up point. In the end, when the boundary value of solutions has an additional condition, we can obtain the precise blow-up value $$m(0)=2\pi (1+\alpha )$$ .

In this work, a boronate-affinity sandwich electrochemiluminescence (ECL) sensor was constructed to detect alpha-fetoprotein (AFP) based on a multiple signal amplification strategy. Gold nanoparticles (AuNPs) were utilized and modified on the surface with chitosan in order to facilitate electron transfer. The composite of the molecularly imprinted polymer (MIP) enhanced the selectivity of alpha-fetoprotein detection. 4-mercaptophenylboronic acid (MPBA) was used as the tracing tag for capture of alpha-fetoprotein. SiO

A novel method for the chemical vapor generation (CVG) of bismuth in non-aqueous phase based on cloud point extraction (CPE) coupled with thermospray flame furnace atomic absorption spectrometric (TS-FF-AAS) determination was developed. The method is based on the complex of bismuth with ammonium pyrrolidine dithiocarbamate (APDC). When the temperature of the system was higher than the cloud point of Triton X-100, Bi-APDC complex was extracted into Triton X-100 and separation of the analyte from the matrix was achieved. Volatile species of Bi were effectively generated through reduction of the extraction organic phase with solid KBH 4 in non-aqueous phase. Some parameters that influenced CVG, extraction and subsequent determination were evaluated in detail, such as the concentrations of Triton X-100 and APDC, pH and extraction time, flow rates of carrier gas, as well as interferences. Under optimized conditions, the detection limit (LOD) for Bi was 8 ng mL − 1 . Relative standard deviation for five replicate determinations of the standard solution containing 200 ng mL − 1 Bi was 3.8%. The proposed method was successfully applied to the determination of Bi in natural water samples, ore samples and certified reference materials with satisfactory results.

Conformal invariance of two-dimensional variational problems is a condition known to enable a blow-up analysis of solutions and to deduce the removability of singularities. In this paper, we identify another condition that is not only sufficient, but also necessary for such a removability of singularities. This is the validity of the Pohozaev identity. In situations where such an identity fails to hold, we introduce a new quantity, called the {\it Pohozaev constant}, which on one hand measures the extent to which the Pohozaev identity fails and on the other hand provides a characterization of the singular behavior of a solution at an isolated singularity. We apply this to the blow-up analysis for super-Liouville type equations on Riemann surfaces with conical singularities, because in the presence of such singularities, conformal invariance no longer holds and a local singularity is in general non-removable unless the Pohozaev constant is vanishing., To appear in J. Differential Geom

We introduce the super-Toda system on Riemann surfaces and study the blow-up analysis for a sequence of solutions to the super-Toda system on a closed Riemann surface with uniformly bounded energy. In particular, we show the energy identities for the spinor parts of a blow-up sequence of solutions for which there are possibly four types of bubbling solutions, namely, finite energy solutions of the super-Liouville equation or the super-Toda system defined on R 2 or on R 2 ∖ { 0 } . This is achieved by showing some new energy gap results for the spinor parts of these four types of bubbling solutions.

In this paper, we will analyze the blow-up behavior of solution sequences satisfying a conformal invariant equation defined on a compact 2-dimensional surface ( M , g ) with boundary. We will provide some accurate point-wise estimates for the profile of these sequences.

Continuing our work on the super-Liouville equations, a variational problem motivated by the supersymmetric extension of the Liouville functional in quantum field theory, we study the profile of blow-up solutions near the blow-up point and establish a local estimate for the bubbling sequences.

In this paper, we give a new proof of the Onofri-type inequalityon the sphere S with Gaussian curvature 1 and with conical singularities divisor for β ∈ (–1, 0); here p1 and p2 are antipodal.

This paper is concerned with the mean field equation of the equilibrium turbulence on a compact Riemannian surface. The solution existence is shown in the super-critical case without any assumption on the topology and the geometry of the surface.

In this paper, we study the super Liouville equations, a natural generalization of the Liouville equation. We establish energy identities and a precise blow-up analysis for solutions of the super Liouville equations.

This paper is concerned with the mean field equation of the equilibrium turbulence on a closed Riemannian surface. The existences of solution are shown in the critical and supercritical case.

Motivated by the supersymmetric extension of Liouville theory in the recent physics literature, we couple the standard Liouville functional with a spinor field term. The resulting functional is conformally invariant. We study geometric and analytic aspects of the resulting Euler-Lagrange equations, culminating in a blow up analysis.

We adopt a continuous model to estimate the Grothendieck constants. An analytical formula to compute the lower bounds of Grothendieck constants has been explicitly derived for arbitrary orders, which improves previous bounds. Moreover, our lower bound of the Grothendieck constant of order three gives a refined bound of the threshold value for the nonlocality of the two-qubit Werner states., 5 pages

Several theorems on the finiteness of energy for quasi-harmonic spheres are proved, some counter-examples which state that the energy of quasi-harmonic sphere may be infinite are given. The results support some conditions of a question posed by Lin Fanghua and Wang Changyou.

Continuing our work on the boundary value problem for super-Liouville equation, we study the qualitative behavior of boundary blow-ups. The boundary condition is derived from the chirality conditions in the physics literature, and is geometrically natural. In technical terms, we derive a new Pohozaev type identity and provide a new alternative, which also works at the boundary, to the classical method of Brezis–Merle.

The nonlocal properties of arbitrary dimensional bipartite quantum systems are investigated. A complete set of invariants under local unitary transformations is presented. These invariants give rise to both sufficient and necessary conditions for the equivalence of quantum states under local unitary transformations: two density matrices are locally equivalent if and only if all these invariants have equal values., Comment: 5 pages

We use PDE methods as developed for the Liouville equation to study the existence of conformal metrics with prescribed singularities on surfaces with boundary, the boundary condition being constant geodesic curvature. Our first result shows that a disk with two corners admits a conformal metric with constant Gauss curvature and constant geodesic curvature on its boundary if and only if the two corners have the same angle. In fact, we can classify all the solutions in a more general situation, that of the 2-sphere cut by two planes., to appear in Annales de l'IHP-ANL

In this paper, using a geometric method we show that the blow-up values of the elliptic sinh-Gordon equation are multiples of 8π.