Multiple solutions for asymptotically linear elliptic equations with sign-changing weight (Q500826)

In this paper, the following semilinear Dirichlet problem \[ -\Delta u(z) = \beta(z)f(u(z)) \quad \text{in} \;\Omega, \quad u|_{\partial \Omega}=0 \] is considered, where \(\Omega \subset {\mathbb R^N}\) is a bounded domain with a \(C^2\)-boundary \(\partial \Omega\), \(\beta \in L^\infty(\Omega)\) is sign changing, \(f\in C^1({\mathbb R})\), \(f(0)=0\) and \(f\) exhibits linear growth near \(\pm\infty\) and is superlinear near \(0\). It is shown that the problem has at least three nontrivial solutions.