Qualitative theory of solutions for semilinear elliptic partial differential equations

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1.We study semilinear elliptic equations in a ball or annulus of n-dimensional Euclid space. Let G be a closed subgroup of the orthogonal group. A solution is called G invariant if it is invariant under G action. Since G is a closed subgroup of the orthogonal group, it is a transformation group on the unit sphere. It is proved that there exists a G invariant non-radial solution if and only if G is not transitive on the unit sphere.2.We study the nodal solution, which is a radially symmetric solution having zeros, for the second order sublinear elliptic equations. We obtain the necessary and sufficient condition for the existence and uniqueness of a k-nodal solution for each integer k. The result means that the radially symmetric solution of a sublinear elliptic equation is uniquely determined by its number of zeros. This gives an important information in the study of group invariant solutions.3.In sublinear elliptic equations, it is proved that there exist infinitely many solutions without the assumption that the nonlinear term is odd. In this case, the Lagrangean functional associated with the elliptic equation is not even, however it is considered as a perturbation from an even functional. The existence of multiple solutions has been studied for the superlinear elliptic equations. However, little is known about the multiple solutions of the sublinear elliptic equations.