Multiplicity results for a class of superlinear elliptic problems

We study a class of superlinear elliptic problems $-\Delta u = \lambda f(u)$ under the Dirichlet boundary condition on a bounded smooth domain in $\mathbb R^N$. Assuming that the nonlinearity $f(u)$ is superlinear in a neighborhood of $u=0$, we study the dependence of the number of signed and sign-changing solutions on the parameter $\lambda$.