Positive solutions for some asymptotically linear and superlinear weighted problems

Abstract

In this paper, we study the following nonlinear elliptic problem -div(a(x) ?u) = f (x,u), x ? ? u ? H10(?) (P) where ? is a regular bounded domain in RN, N ? 2, a(x) a bounded positive function and the nonlinear reaction source is strongly asymptotically linear in the following sense lim t?+? f(x,t)/t = q(x) uniformly in x ? ?. 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 Ambrosetti-Rabionowitz 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.