| This thesis consists of two parts. In the first part we consider the equation -Deltau = K(x) un+2n-2 on Rn with K(x) being positive and periodic in each variable. Using a nonlinear version of the Liapunov-Schmidt method, we construct positive multi-bump solutions with each bump centered near a critical point of K(x). In the second part, we obtain symmetry or asymmetry properties for solutions of some subcritical and critical elliptic equations on Rn and Sn using moving sphere method and bifurcation argument. |
