Monotonicity and one-dimensional symmetry for the solutions of Δu+f(u)=0 with possibly discontinuous nonlinearity

In this Note we study monotonicity and one-dimensional symmetry properties for founded solutions of Delta u + f(u) = 0 in R-N, where the nonlinearity f can be a discontinuous function. We consider solutions u such that mu- less than or equal to u less than or equal to u(+) and u(x(1),...,x(N)) --> mu(+/-) as x(N) --> +/-infinity, uniformly with respect to x(1),...,x(N-1). We prove that the solutions are strictly increasing functions depending only on the variable x(N), whenever f belongs to a suitable class of functions F-0. Since Lipschitz-continuous functions belong to F-0 but F-0 is not included in C-0([mu(-), mu(+)]), we recover and improve upon all known results concerning the classification of the considered problem 2000 Academie des sciences/Editions scientifiques et medicales Elsevier SAS.