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

TL;DR

The paper studies prescribed-mass (normalized) solutions to a Kirchhoff-type equation with a nonlocal gradient term and mixed nonlinearities on large bounded domains and in $\mathbb{R}^3$, under a mass constraint. It develops a monotonicity-trick variational framework to obtain constrained critical points despite the nonconstant potential, and proves the existence of ground-state and mountain-pass normalized solutions on large domains, then passes to the whole space. By domain-augmentation and concentration-compactness analyses, it derives the asymptotic behavior as the domain grows and as the prescribed mass $\alpha$ tends to zero, including blow-up scenarios and positivity of the Lagrange multiplier. The results extend normalized Kirchhoff theory to nonconstant potentials with combined nonlinearities and provide a rigorous bridge from large bounded domains to $\mathbb{R}^3$.

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)Δu+V(x)u+λu=|u|^{q-2}u+β|u|^{p-2}u \quad&\text{in}\ Ω, \int_Ω|u|^2\mathrm{d}x=α, both on large bounded smooth star-shaped domain Ω\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).

Theorems & Definitions (32)

• Theorem 1.1
• Theorem 1.2
• Remark 1.3
• Theorem 1.4
• Theorem 1.5
• Lemma 2.1
• Theorem 2.2
• Lemma 3.1
• proof
• Theorem 3.2