Normalized solutions to Kirchhoff type equations with a critical growth nonlinearity

TL;DR

This paper investigates the existence and non-existence of normalized solutions to Kirchhoff type equations with critical growth nonlinearities, employing variational methods across different dimensions and growth conditions.

Contribution

It provides new results on normalized solutions for Kirchhoff equations with exponential and Sobolev critical growth, including existence, multiplicity, and non-existence results.

Findings

Existence of normalized mountain pass solutions for N=2 with exponential critical growth.

Existence of normalized ground state and mountain pass solutions for N≥4 with Sobolev critical growth.

Non-existence results for certain parameter regimes.

Abstract

In this paper, we are concerned with normalized solutions of the Kirchhoff type equation \begin{equation*} -M\left(\int_{\R^N}|\nabla u|^2\mathrm{d} x\right)\Delta u = \lambda u +f(u) \ \ \mathrm{in} \ \ \mathbb{R}^N \end{equation*} with $u \in S_c:=\left\{u \in H^1(\R^N): \int_{\R^N}u^2 \mathrm{d}x=c^2\right\}$. When $N=2$ and $f$ has exponential critical growth at infinity, normalized mountain pass type solutions are obtained via the variational methods. When $N \ge 4$, $M(t)=a+bt$ with $a$, $b>0$ and $f$ has Sobolev critical growth at infinity, we investigate the existence of normalized ground state solutions and normalized mountain pass type solutions. Moreover, the non-existence of normalized solutions is also considered.