Positivity of solutions of elliptic equations with nonlocal terms

In this paper we study a nonlocal problem for a second-order partial differential equation which depends on a parameter eta. We prove the existence 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 than or equal to <(eta)over bar> and for all non-negative right-hand sides, 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 corresponding eigenvalue problem. At the end of the paper, we consider the application of our results to the nonlinear system describing the distribution of temperature and potential in a microsensor.