The Existence Of Positive Solutions On Semilinear Singular Elliptic Equations With Robin Boundary Condition

This paper deals with the equation:where Q is a bounded domain in Rn with C1 boundary (n > 3), and u< u*,u* is a positive constant, is the unit outward normal to is a positive function.Recently many people are interested in semilinear singular elliptic equations. In 2000 Ghoussoub investigated the equation,where are positive parameters, is a smooth bounded domain in Rn containing 0 in its interior, 1 < p < n,p < q < p* (s) =. He investigated the solutions of the equation when p, q, are different values (a positive solution, a sign-changing solution, infinitely many solutions). Yinbin Deng ang Dongsheng Kang investigated the equation,What they considered are all singular elliptic equations with Dirichlet boundary condition, while this paper considers singular elliptic equations with Robin boundary condition.The main idea of this paper is as the following. Firstly we prove a general existence theorem about this equation. Secondly we give an estimate on the equation's energy function, verify it satisfy the conditions of the general existence theorem, and thereby prove the existence of the positive solution of the equation. In the proof of the general existence theorem, because the exponent 2*(s) is critical, the function J(u) isnot satisfy the (PS) conditions, and the singularity also attributes to some difficulty, we cannot apply the standard variational methods directly. By the achieving function of the best Sobolev-Hardy constant and the means of straightening the boundary, we find the equation's energy function satisfy the (PS)c conditions, so we can prove the theorem.