Abstract
In this paper, we prove that if Q and f satisfy some suitable conditions, then − Δ u + u = Q ( x ) f ( u ) in R + N with the boundary condition u ( y , 0 ) = λ g ( y ) has at least two positive solutions if 0 < λ < λ ∗ , a unique positive solution if λ = λ ∗ and no positive solution if λ > λ ∗ .
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