Abstract
In this note, we deal with the Helmholtz equation -Delta u + cu = lambda f (u) with Dirichlet boundary condition in a smooth bounded domain Omega of R-n, n > 1. The nonlinearity is superlinear that is lim(t ->infinity) f(t)/t infinity and f is a positive, convexe and C-1 function defined on [0, infinity). We establish existence of regular solutions for lambda small enough and the bifurcation phenomena. We prove the existence of critical value lambda* such that the problem does not have solution for lambda > lambda* even in the weak sense.
We also prove the existence of a type of stable solutions u* called extremal solutions. We prove that for f (t) = e(t), Omega = B-1 and n <= 9, u* is regular.