Seven solutions with sign information for sublinear equations with unbounded and indefinite potential and no symmetries (Q466100)

In this paper, the Dirichlet problem: \[ \begin{cases} -\Delta u(z) + \beta(z)u(z) = f(z,u(z)) \quad \text{in} \;\Omega, \\ u|_{\partial \Omega} = 0 \end{cases} \] is considered, where \(\Omega \subset {\mathbb R}^N\) is a bounded domain with a \(C^2\)-boundary \(\partial \Omega\), \(N \geq 3\), \(\beta \in L^s(\Omega)\), \(s>N\) and, in general, it is indefinite (sign changing) and unbounded from below. It is shown that the problem has at least seven nontrivial smooth solutions and provide sign information for all of them.