Structure of the solution set for a vector valued elliptic boundary value problem with a Lipschitz nonlinear term

We consider the boundary value problem with the Dirichlet condition in a Banach space for a semilinear elliptic equation on a bounded domain in Rn whose nonlinear term satisfies the Lipschitz condition. If the Lipschitz constant L is less than λ1, then this problem has a unique solution, where λ1 is the least eigenvalue of the corresponding (real valued) eigenvalue problem. On the other hand, for any L>λ1 we can construct a nonlinear term with the Lipschitz constant L such that the solution set is homeomorphic to any prescribed closed subset of the Banach space.