Singularly perturbed elliptic problems with nonautonomous asymptotically linear nonlinearities

TL;DR

This paper studies singularly perturbed elliptic equations with spatially varying nonlinearities, focusing on the existence and localization of solutions that concentrate near minima of potential and saturation functions.

Contribution

It introduces conditions for solution concentration points in elliptic problems with nonautonomous asymptotically linear nonlinearities, considering potential and saturation effects.

Findings

Existence of nontrivial nonnegative solutions concentrating at specific points.

Necessary conditions for the location of concentration points.

Analysis of the influence of potential and saturation functions on solution behavior.

Abstract

We consider a class of singularly perturbed elliptic problems with nonautonomous asymptotically linear nonlinearities. The dependence on the spatial coordinates comes from the presence of a potential and of a function representing a saturation effect. We investigate the existence of nontrivial nonnegative solutions concentrating around local minima of both the potential and of the saturation function. Necessary conditions to locate the possible concentration points are also given.