Entire Solutions of an Integral Equation in

We will study the entire positive solution of the geometrically and analytically interesting integral equation: with in . We will show that only when , there are positive entire solutions which are given by the closed form up to dilation and translation. The paper consists of two parts. The first part is devoted to showing that must be equal to 11 if there exists a positive entire solution to the integral equation. The tool to reach this conclusion is the well-known Pohozev identity. The amazing cancelation occurred in Pohozev’s identity helps us to conclude the claim. It is this exponent which makes the moving sphere method work. In the second part, as normal, we adopt the moving sphere method based on the integral form to solve the integral equation. 1. Introduction In this paper, we will study a very special type of the integral equation. From an analytical point of view, such equation is interesting to be studied. We should point out that even for radius case, the analysis of the equation has been already difficult. The method of moving planes has become a very powerful tool in the study of nonlinear elliptic equations; see Alexandrov [1], Serrin [2], Gidas et al. [3], and others. The moving plane method can be applied to prove the radial symmetry of solutions, and then one only needs to classify radial solutions. The method of moving planes in the integral form has been developed by Chen and Li [4]. This technique requires not only to prove that the solutions are radially symmetric, but also to take care of a possible singularity at origin. We note that in [5], Li and Zhang have given different proofs to some previous established Liouville type theorems based on the method of moving sphere. It is this method that we applied to the integral equation to capture the solutions directly. As usual, in order for such a method to work, Pohozev’s identity is a must. In our argument, we derive and use this powerful identity to conclude that the negative exponent . Then it is the method of moving spheres in the integral form which helps us to deduce the exact solution to the integral equation in this paper, instead of only getting radius solutions or proving radial symmetry of solutions. The organization of the paper is as follows. In Section 2, we prove that . In this part, we use integration by parts to derive Pohozev’s identity in the form . It is easy to observe that if solves our integral equation, then also solves the differential equation . Thus we can put the integral form into Pohozev’s identity to calculate every term. It turns out that the boundary