Multiple solutions for nonlinear Dirichlet problems via bifurcation and additional results

In this paper we study the existence of multiple solutions for semilinear elliptic boundary value problems, first when the nonlinearity is asymptotically linear and then when it has an arbitrary behavior for large values of the argument. Our proofs use extensively the global bifurcation theorem and bifurcation from infinity. Additionally, when we can apply the Lyapunov–Schmidt reduction method, we show the existence of multiple solutions, we give an exact number of solutions, and we provide qualitative properties of these solutions.