Positive solutions for some asymptotically linear and superlinear weighted problems

Abstract

In this paper we study the following nonlinear elliptic problem
$$-\mbox{div}(a(x)\nabla u)= f(x,u),~~ x\in \Omega~~u\in H^{1}_{0}(\Omega) \eqno(P) $$
where $\Omega$ is a regular bounded domain in $\R^{N}$, $N\geq 2$, $a(x)$ a bounded positive function and the nonlinear reaction source is strongly asymptotically linear in the following sense $$\displaystyle \lim_{t\rightarrow +\infty}\frac{f(x,t)}{t}= q(x)$$
uniformly in $x\in \Omega$.\\
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 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.