In this paper we prove the existence of non-positive or non-radial solutions to semilinear elliptic problems on $\mathbf{S}^2$ with a small hole. When the hole is sufficiently small, we prove that the multiplicity of eigenvalues to the corresponding linearized problem is $1$ or $2$. Thus, by using the result, we show those eigenvalues are bifurcation points, and the corresponding bifurcating solutions are not positive except for a bifurcating solution which is corresponding to the first eigenvalue. Moreover if the multiplicity of a eigenvalue is $2$, then the corresponding bifurcating solution is not radially symmetric.
back