Semilinear elliptic problems on the half-space with a sign-changing nonlinearity

Semilinear elliptic problems of the form

$ \begin{equation*} \left\{ \begin{array}{ll} -\Delta w = g+\mu f\left( ., w\right) , & \text{in }\mathbb{R}_{+}^{n}, \\ w>0, & \text{in }\mathbb{R}_{+}^{n}, \\ \lim\limits_{y_{n}\rightarrow 0}w(y) = \phi (y^{\prime }), & \\ \underset{\left\vert y\right\vert \rightarrow \infty }{\lim }w(y) = 0, & \end{array} \right. \end{equation*} $

are considered, where $ \mathbb{R}_{+}^{n} = \left\{ y = (y^{\prime }, y_{n})\in \mathbb{R}^{n}:y_{n} > 0\right\} (n\geq 2), $ $ \mu > 0, $ and $ \phi $ is a nonnegative continuous function in $ \mathbb{R}^{n-1} $ vanishing at infinity$. $ The source term $ g $ is assumed to be a nontrivial nonnegative function belonging to a suitable Kato class, whereas the nonlinear term $ f $ may change sign. The existence and uniqueness of a positive continuous solution are established for a suitable range of $ \mu $ under Lipschitz-type assumptions on $ f $. The global behavior of this solution is also described. These results go beyond several earlier contributions by allowing a wider range of sign-changing nonlinearities, while preserving full qualitative information about the solution.