Remarks on uniqueness of boundary blow-up solutions

Abstract

We show that there is at most one nonnegative boundary blow-up solution for the one-dimensional boundary blow-up problem ( | u ′ | p − 2 u ′ ) ′ = f ( u ) in ( 0 , 1 ) , u ( 0 ) = ∞ , u ( 1 ) = ∞ where p > 1 provided f ∈ C 1 ( 0 , ∞ ) ∩ C 0 [ 0 , ∞ ) with f ( s ) > 0 and f ′ ( s ) ≥ 0 for s ∈ ( 0 , ∞ ) . We see that the same result still holds for some equations with special nonlinearities satisfying f ( s ) > 0 and f ′ ( s ) ≥ 0 for s ∈ ( 0 , ∞ ) in higher dimensions, but we conjecture that the same result should be true for equations with general nonlinearities satisfying f ( s ) > 0 and f ′ ( s ) ≥ 0 for s ∈ ( 0 , ∞ ) in higher dimensions.