*(English)*Zbl 0844.35035

Let ${\Omega}$ be a bounded domain in ${\mathbb{R}}^{N}$. The paper is concerned with the semilinear elliptic problem

where $g(x,u)=\alpha {u}^{+}+\beta {u}^{-}+{g}_{0}(x,u)$, ${g}_{0}(x,u)/u\to 0$ as $\left|u\right|\to \infty $, ${e}_{1}$ is the positive eigenvalue of the Laplacian and $\alpha ,\beta ,t\in \mathbb{R}$. To $(*)$ there corresponds a functional

in ${H}_{0}^{1}\left({\Omega}\right)$ and critical points of ${f}_{t}$ are solutions of $(*)$. It is shown that for $(\alpha ,\beta )$ in certain regions of ${\mathbb{R}}^{2}$, if $t$ is large enough, then $(*)$ has at least one, two, three, respectively four solutions. Existence of one solution is shown by using a variant of the saddle point theorem of Rabinowitz. Two and three solutions are obtained by linking-type arguments where careful estimates are needed in order to show that certain linking levels are different. An additional argument gives a fourth critical point. It should also be noted that a rather general sufficient condition for ${f}_{t}$ to satisfy the Palais-Smale condition is given in this paper.