On Some Scalar Field Equations with Competing Coefficients

TL;DR

This paper investigates semilinear elliptic equations with variable coefficients that compete, analyzing how their asymptotic behavior influences the existence and properties of positive solutions.

Contribution

It provides new insights into the effects of competing coefficients in scalar field equations, especially regarding solution existence and qualitative behavior.

Findings

Existence of positive solutions under certain coefficient conditions

Phenomena related to coefficient competition affecting solution properties

Asymptotic analysis of coefficients at infinity

Abstract

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".