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 n . We prove the existence of critical values 0 < and 0 > such that for all ≦ɳ≦ and for all non-negative right-hand sides, our problem has nonnegative solutions. We obtain a formula for ɳ 0 , the maximal possible value of , and find the exact value of ɳ for spherical ɳ. 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.