Gradient Estimate of Solutions to a Class of Mean Curvature Equations with Prescribed Contact Angle Boundary Problem

This paper studies the prescribed contact angle boundary value problem of a certain type of mean curvature equation.Applying the maximum principle and the moving frame method and based on the location of the maximum point,the boundary gradient estimation of the solutions to the equation is obtained.

MULTIPLE POSITIVE SOLUTIONS FOR A CLASS OF SEMIPOSITONE NEUMANN PROBLEMSWITH SINGULARφ-LAPLACIAN

We study the existence of multiple positive solutions for a Neumann problem with singular φ-Laplacian{-(φ（u′）)′= λf（u）, x ∈（0, 1）,u′（0） = 0 = u′（1）,where λ is a positive parameter, φ（s） =s/（1-s;）;, f ∈ C;（[0, ∞）, R), f′（u） > 0 for u > 0, and for some 0 < β < θ such that f（u） < 0 for u ∈ [0, β)（semipositone） and f（u） > 0 for u > β.Under some suitable assumptions, we obtain the existence of multiple positive solutions of the above problem by using the quadrature technique. Further, if f ∈ C;（[0, β） ∪（β, ∞）, R),f′′（u） ≥ 0 for u ∈ [0, β) and f′′（u） ≤ 0 for u ∈（β, ∞）, then there exist exactly 2 n + 1 positive solutions for some interval of λ, which is dependent on n and θ. Moreover, We also give some examples to apply our results.

A priori estimates versus arbitrarily large solutions for fractional semi-linear elliptic equations with critical Sobolev exponent

We study positive solutions to the fractional semi-linear elliptic equation(−∆)σu=K(x)u n+2σn−2σin B2\{0}with an isolated singularity at the origin,where K is a positive function on B2,the punctured ball B2\{0}⊂Rn with n>2,σ∈(0,1),and(−∆)σis the fractional Laplacian.In lower dimensions,we show that for any K∈C1(B2),a positive solution u always satisfies that u(x)6 C|x|−(n−2σ)/2 near the origin.In contrast,we construct positive functions K∈C1(B2)in higher dimensions such that a positive solution u could be arbitrarily large near the origin.In particular,these results also apply to the prescribed boundary mean curvature equations on B n+1.

The area minimizing problem in conformal cones,Ⅱ

In this paper we continue to study the connection among the area minimizing problem,certain area functional and the Dirichlet problem of minimal surface equations in a class of conformal cones with a similar motivation from[15].These cones are certain generalizations of hyperbolic spaces.We describe the structure of area minimizing n-integer multiplicity currents in bounded C^2 conformal cones with prescribed C^1 graphical boundary via a minimizing problem of these area functionals.As an application we solve the corresponding Dirichlet problem of minimal surface equations under a mean convex type assumption.We also extend the existence and uniqueness of a local area minimizing integer multiplicity current with star-shaped infinity boundary in hyperbolic spaces into a large class of complete conformal manifolds.