On Some Scalar Field Equations with Competing Coefficients

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".