Existence results for some functional elliptic equations

We consider nonlocal elliptic boundary value problems of the form $$ -\mathcal{A}(x,u) \bigtriangleup u = \lambda f(u) $$ with Dirichlet boundary conditions. We show that if $f$ has $n$ loops, then the problem has at least $n$ distinct solutions, by using very simple comparison principles. Also, we study the asymptotic behavior of such solutions as the parameter $\lambda $ tends to $\infty$.