On Some Scalar Field Equations with Competing Coefficients

Abstract

This paper deals with semilinear elliptic problems of the type \[ \left\{ \begin{array}{ll} -Δu+α(x)u= β(x)|u|^{p-1}u \quad \hbox{in }\mathbb{R}^N, u(x)>0\quad\hbox{in } \mathbb{R}^N, \qquad u \in H^1(\mathbb{R}^N), \end{array} \right. \] 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".

19 pages. Improved version