In this talk I will discuss a polynuclear growth model in half space with two external sources. I will present the strategy developed to study the model in the full space setting (Baik-Rains '00), which relies on algebraic and orthogonal polynomials identities, and Riemann-Hilbert techniques, and which led to a limit distribution formulated in terms of the solution to Painlevé II equation; then I will underline the differences with respect to the half space case. This result also proves a conjecture by Barraquand-Krajenbrink-Le Doussal '22 on the distribution of the stationary KPZ equation on the half line. Based on joint work with M. Cafasso, D. Ofner, H. Walsh.