Abstract. In this work, we discuss the existence and the non-existence of principal eigenvalue in an unbounded domain of R N for some potentials which change sign. We also give certain properties of this principal eigenvalue.

Abstract. The aim of this paper is to investigate the existence of solutions of a nonlocal parabolic problem. The method of upper and lower solutions and the classical maximum principle are used to obtain our results. IntroductionIn [4], Deng studied the following nonlocal boundary-value problemwhere is a bounded domain in K n , dQ. £ C 2 , f (x,u) is in x and C 1 in u with f(x, 0) = 0, and y) is a continuous function defined for x € dQ., yen.He first established the comparison principle for (-Pi). Then he showed the local existence of the solution and he discussed its long time behavior, assumingThe results obtained in [4] generalize the result of [6].In [10], Yin considered a problem similar to (Pi), namely,' ut -Lu = f (t,x,u),for allx G Q;

The aim of this paper is to investigate the existence of solutions of a nonlocal parabolic problem. The method of upper and lower solutions and the classical maximum principle are used to obtain our results.