Abstract
In this paper, we study the following nonlinear elliptic problem
-div(a(x)del u) = f (x, u), x is an element of Omega u is an element of H-0(1)(Omega) (P)
where Omega is a regular bounded domain in R-N, N >= 2, a(x) a bounded positive function and the nonlinear reaction source is strongly asymptotically linear in the following sense
lim(t ->infinity) f(x,t)/(t =q(x))
uniformly in x is an element of Omega.
We use a variant version of Mountain Pass Theorem to prove that the problem (P) has a positive solution for a large class of f (x, t) and q(x). Here, the existence of solution is proved without use neither the AmbrosettiRabionowitz condition nor one of its refinements. As a second result, we use the same techniques to prove the existence of solutions when f (x, t) is superlinear and subcritical on t at infinity.