On Some Scalar Field Equations with Competing Coefficients

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 $α$ and $β$ are positive functions such that $α(x) \to a_\infty > 0$ and $β(x)\to b_\infty > 0$, as $|x| \to \infty$. Aim of this work is to describe some phenomena that can occur when the coefficients are "competing".

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