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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.

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 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.

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 $$m(0)=2\pi (1+\alpha )$$ .

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.

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.

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