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A typical nonlinearity we consider is $f(x,u) = \\mu_0 (\\phi (x)) u(1-u)$, where $\\mu_0$ is a 1-periodic function and $\\phi$ is a $\\mathcal{C}^1$ increasing function that satisfies $\\lim_{x\\to +\\infty} \\phi (x) = +\\infty$ and $\\lim_{x\\to +\\infty} \\phi' (x) = 0$. Although quite specific, the choice of such a reaction term is motivated by its highly heterogeneous nature. 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