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If there exists a positive bounded solution to the overdetermined elliptic problem $$ \\left\\{\\begin{array} {ll} \\Delta u + f(u) = 0 & \\mbox{in }\\; \\Omega\n  \\\\ u= 0\\, \\, \\, , \\, \\, \\, \\frac{\\partial u}{\\partial \\vec{\\nu}}=1 &\\mbox{on }\\; \\partial \\Omega \\end{array}\\right. $$ we prove that $\\Omega$ is a half-plane. In particular, we obtain a partial answer to a question raised by H. Berestycki, L. Caffarelli and L. 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