On positive solutions for some semilinear periodic parabolic eigenvalue problems. (Q1869003)

Let \(\Omega\) be a bounded domain in \(\mathbb R^N\), \(N\geq2\). Consider positive \(T\)-periodic solutions of the problem \(Lu=\lambda g(x,t,u)\) in \(\Omega\times\mathbb R\), \(u=0\) on \(\partial\Omega\times\mathbb R\), where \(L\) is a linear parabolic operator and \(g\) is \(T\)-periodic in \(t\). Denote \(G(x,t,\xi)=g(x,t,\xi)/\xi\) and assume that \(G\) is nonincreasing in \(\xi\) and \(\int_0^T\text{ess}\,\)sup