On A Class Of Semilinear Elliptic Equations

This dissertation investigates a class of semilinear elliptic equations:divandwhere n≥3, is the n-dimension Laplacian operator, p > 1 and p_i > 1, l_i > -2, c_i > 0, i = 1,2, ...,k. A(|x|) > 0, K(|x|) and f(|x|) are Holder continuous nonnegative functions.The equations arise in Riemannian geometry, are said to be the confformal scalar curvature equations. The first equation after simplication isFor the physical reasons and because of the results on the symmetry of positive solutions, most of mathematicians study the radial solutions of them, such as the existence of positive radial solutions and asymptotic behavior at infinity etc.The underlying obstacles result from the presence of the second terms (K(|x|) or c_i|x|^li,i = 1,2, ...,k, inhomogeneity and the lack of compactness due to the unbounded domain (entire space). This dissertation deserve to the radial solutions, with r = |x|, the class of equations reduce to the following respectively :andDenote the positive radial solutions of the class of equatons u_α(r) with initial value u(0) =α. For everyα> 0, there is a positive solution u_α(r) for Eq. (0.1.4) and (0.1.5) with some assumptions on p_i,l_i, 1≤i≤k and A(|x|), K(|x|) respectively. But there is only multiple many positive entire solutions for Eq. (0.1.6). This dissertation deserve to study the existence of positive solutions respective to the initial value u(0) = α > 0 of Eq. (0.1.6) and the monotonicity property with respective to the initial values of positive radial solutions of above three equations. and their asymptotic behaviors at infinity respectively. Meanwhile, we get some results about the existence, momotonicity property and the singularity at zero of the singular solutions of Eq (0.1.4) and a hardy equation. As for Eq. (0.1.6), we also study the existence, singularity at zero and asymptotic behavior at infinity of singular solutions (positive solutions in Rⁿ/{0}).