  This paper deals with semilinear elliptic problems of the type \[ \left\{
  \begin{array}{ll}
  -\Delta u+\alpha(x)u= \beta (x)|u|^{p-1}u \quad \hbox{in }\mathbb{R}^N,
u(x)>0\quad\hbox{in } \mathbb{R}^N, \qquad u \in H^1(\mathbb{R}^N),
  \end{array} \right. \] where $p$ is superlinear but subcritical and the
coefficients $\alpha$ and $\beta$ are positive functions such that $\alpha(x)
\to a_\infty > 0$ and $\beta(x)\to b_\infty > 0$, as $|x| \to \infty$. Aim of
this work is to describe some phenomena that can occur when the coefficients
are "competing".
