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arXiv · 2501.04071

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

Xiaolu Lin·Zongyan Lv
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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).

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Xiaolu Lin, Zongyan Lv. 2025-01-07. Existence and blow up behavior of prescribed mass solutions on large smooth domains to the Kirchhoff equation with combined nonlinearities. https://arxiv.org/abs/2501.04071

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