Existence and blow up behavior of prescribed mass solutions on large smooth domains to the Kirchhoff equation with combined nonlinearities

TL;DR

This paper investigates the existence, multiplicity, and blow-up behavior of prescribed mass solutions to a nonlinear Kirchhoff equation with mixed nonlinearities on large smooth domains and the whole space, highlighting challenges due to potential V(x).

Contribution

It provides new results on the existence and blow-up behavior of solutions to a Kirchhoff equation with combined nonlinearities, overcoming limitations of traditional methods affected by potential V(x).

Findings

Existence of prescribed mass solutions on large domains and  in .

Analysis of blow-up behavior of solutions as parameters vary.

Identification of challenges posed by the potential V(x) in standard approaches.

Abstract

In this paper, we consider the existence, multiplicity and the asymptotic behavior of prescribed mass solutions to the following nonlinear Kirchhoff equation with mixed nonlinearities: -(a+b\int|\nabla u|^2\mathrm{d}x)\Delta u+V(x)u+\lambda u=|u|^{q-2}u+\beta|u|^{p-2}u \quad&\text{in}\ \Omega, \int_{\Omega}|u|^2\mathrm{d}x=\alpha, both on large bounded smooth star-shaped domain \Omega\subset\mathbb{R}^{3} and on \mathbb{R}^{3}, where 2<p<\frac{14}{3}<q<6 and V(x) is the potential. The standard approach based on the Pohozaev identity to obtain normalized solutions is invalid due to the presence of potential V(x).