A singular semilinear equation in $L^{1}(\textbf {R})$

Abstract:

Let $\beta$ be a positive and nondecreasing function on R. The boundary-value problem $\beta (u) - u'' = f,u’( \pm \infty ) = 0$ is considered for $f \in {L^1}({\mathbf {R}})$. It is shown that this problem can have a solution only if $\beta$ is integrable near $- \infty$, and that if this is the case, then the problem has a solution exactly when $\smallint _{ - \infty }^\infty f(x)dx > 0$.