This paper is concerned with the nonlinear Schrödinger equation in \(\mathbb {R}^N\) , where \(N\ge 3\) , \(h:\mathbb {R}\rightarrow \mathbb {R}\) is a function with critical or supercritical growth at infinity, V changes sign, but is not periodic, and f(s)/s, \(s\ne 0\) , does not possess any kind of monotonicity. As the nonlinearity growth can be critical or supercritical in this paper, we first investigate the existence of a nontrivial solution for problem \((P)\) with \(h(s)=|s|^{2^*-2}s\) . The core of our work lies in the condition \(f(s)\ge \Lambda s^{r-1}\) , \(s\ge 0\) and for some \(\Lambda >0\) , which is crucial for our results. We show that the problem at infinity associated to (P) has a nontrivial positive solution \(u_0\) with adequate exponential decay and, hence, benefiting from the general linking theorem and making some interaction between \(u_0\) and problem (P), we establish the existence of a nontrivial solution. Furthermore, by analyzing the \(L^\infty \) -norm of this solution, we study the existence of nontrivial solutions of problem \((P)\) for \(h\) with any kind of growth at infinity.
