EXISTENCE RESULTS FOR A CLASS OF SEMILINEAR ELLIPTIC BOUNDARY VALUE PROBLEMS

We consider the semilinear elliptic boundary value problem { - deltau(x) =deltaf(U(X)); u(x) = 0; x element of omega xolement of partial omega where lambda 0 is a parameter, omega is a bounded region in RN with a smooth boundary, and f is a smooth function. We prove, under some additional conditions, the existence of a positive solution for A large. We prove that our solution u for lambda, large is such that I u 1:= SUp Iu(x) I~ infinity as lambda - infinity. xelement of omega Also, in the case of N =1, we use a bifurcation theory to show that the solution is unstable