A singular semilinear equation in 𝐿¹(𝑅)

Abstract

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