| Popis: |
We consider the semilinear elliptic problem (0.1) { − Δ u = f ( u ) in R + N u = 0 on ∂ R + N where the nonlinearity f is assumed to be C 1 and globally Lipschitz with f ( 0 ) 0 , and R + N = { x ∈ R N : x N > 0 } stands for the half-space. Denoting by u 0 the unique solution of the one-dimensional problem − u ″ = f ( u ) with u ( 0 ) = u ′ ( 0 ) = 0 , we show that nonnegative solutions u of (0.1) which verify u ( x ) ≥ u 0 ( x N ) in R + N either are positive and monotone in the x N direction or coincide with u 0 . As a particular instance, when f ( t ) = t − 1 , we prove that the unique nonnegative (not necessarily bounded) solution of (0.1) is u ( x ) = 1 − cos x N . This solves in a strengthened form a conjecture posed by Berestycki, Caffarelli and Nirenberg in 1997. |
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