POSITIVITY OF SOLUTIONS OF ELLIPTIC-EQUATIONS WITH NONLOCAL TERMS

In this paper we study a nonlocal problem for a second-order partial d ifferential equation which depends on a parameter eta. We prove the ex istence of critical values 0 < <(eta)over bar> and 0 > <(eta)under bar > such that for all <(eta)under bar> less than or equal to eta less th an or equal to <(eta)over bar> and for all non-negative right-hand sid es, our problem has nonnegative solutions. We obtain a formula for eta (0), the maximal possible value of <(eta)over bar>, and find the exact value of <eta(>)0 for spherical Omega. We also study the correspondin g eigenvalue problem. At the end of the paper, we consider the applica tion of our results to the nonlinear system describing the distributio n of temperature and potential in a microsensor.