Radial Symmetry of Entire Solutions of a Biharmonic Equation with Supercritical Exponent

Necessary and sufficient conditions for a regular positive entire solution u of a biharmonic equation\Delta^{2}u=u^{p}\quad\text{in }\mathbb{R}^{N},\,N\geq 5,\,p>\frac{N+4}{N-4}to be a radially symmetric solution are obtained via the exact asymptotic behavior of u at {\infty} and the moving plane method (MPM). It is known that above equation admits a unique positive radial entire solution {u(x)=u(|x|)} for any given {u(0)>0}, and the asymptotic behavior of {u(|x|)} at {\infty} is also known. We will see that the behavior similar to that of a radial entire solution of above equation at {\infty}, in turn, determines the radial symmetry of a general positive entire solution {u(x)} of the equation. To make the procedure of the MPM work, the precise asymptotic behavior of u at {\infty} is obtained.

Existence and Exact Asymptotic Behavior of Positive Solutions for a Fractional Boundary Value Problem

We establish the existence and uniqueness of a positive solution for the fractional boundary value problem , with the condition , where , and is a nonnegative continuous function on that may be singular at or .

Online Selection of Alternating Subsequences from a Random Sample

We consider sequential selection of an alternating subsequence from a sequence of independent, identically distributed, continuous random variables, and we determine the exact asymptotic behavior of an optimal sequentially selected subsequence. Moreover, we find (in a sense we make precise) that a person who is constrained to make sequential selections does only about 12 percent worse than a person who can make selections with full knowledge of the random sequence.