  Given a pseudoconvex domain U with C^1-boundary in P^n, n>2, we show that if
H^{2n-2}_\dR}(U)\not=0, then there is a strictly psh function in a neighborhood
of boundary U. We also solve the \dbar-equation in X=P^n\ U, for data smooth
(0,1) forms on X. We also discuss Levi-flat domains in surfaces. If Z is a real
algebraic hypersurface in P^2, (resp a real-analytic hypersurface with a point
of strict pseudoconvexity), then there is a strictly psh function in a
neighborhood of Z.
