Multiplicity and concentration behavior of positive solutions for a Schrodinger-Kirchhoff type problem via penalization method

TL;DR

This paper investigates the existence, multiplicity, and concentration of positive solutions for a Schrödinger-Kirchhoff type problem involving a nonlocal operator, using a penalization method to handle the problem's complexities.

Contribution

It introduces a novel approach to analyze positive solutions of a nonlocal elliptic problem with concentration phenomena, extending previous methods to a Kirchhoff-type setting.

Findings

Multiple positive solutions are established for small epsilon.

Solutions exhibit concentration behavior around certain points.

The penalization method effectively handles the nonlocal operator.

Abstract

In this paper we are concerned with questions of multiplicity and concentration behavior of positive solutions of the elliptic problem $$\left\{\begin{array}{rcl} \mathcal{L}_{\varepsilon}u = f(u) \ \ \mbox{in} \ \ \mathbb{R}^3,\\ u>0 \ \ \mbox{in} \ \ \mathbb{R}^3,\\ u \in H^1 (\mathbb{R}^3), \end{array} \right.$$ where $\varepsilon$ is a small positive parameter, $f:\mathbb{R}\rightarrow \mathbb{R}$ is a continuous function, $\mathcal{L}_{\varepsilon}$ is a nonlocal operator defined by $$ \mathcal{L}_{\varepsilon} u = M \left(\frac{1}{\varepsilon} \int_{\mathbb{R}^3} |\nabla u|^2 + \frac{1}{\varepsilon^3} \int_{\mathbb{R}^3} V(x) u^{2}\right) \left[-\varepsilon^2\Delta u + V(x)u \right],$$ $M:\mathbb{R}_+\to \mathbb{R}_+$ and $V:\mathbb{R}^3 \to \mathbb{R}$ are continuous functions which verify some hypotheses.