On the Kirchhoff equation with prescribed mass and general nonlinearities

Abstract

In the present paper, we apply a global branch approach to study the existence, non-existence and multiplicity of positive normalized solutions $(λ_c, u_c)\in \mathbb{R}\times H^1(\mathbb{R}^N)$ to the following Kirchhoff problem $$ -\left(a+b\int_{\mathbb{R}^N}|\nabla u|^2dx\right)Δu+λu=g(u)~\hbox{in}~\mathbb{R}^N,\;N\geq 1 $$ satisfying the normalization constraint $ \displaystyle\int_{\mathbb{R}^N}u^2=c, $ which appears in free vibrations of elastic strings. The parameters $a,b>0$ are prescribed as well as the mass $c>0$. Due to the presence of the non-local term $b\int_{\mathbb{R}^N}|\nabla u|^2dx Δu$, such problems lack the mountain pass geometry in the higher dimension case $N\geq 5$. Our result seems to be the first attempt in this aspect.

Figures (3)

• Figure 1: The case of $N\leq 3$
• Figure 2: The case of $N=4$
• Figure 3: The case of $N\geq 5$

Theorems & Definitions (17)

• Remark 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Lemma 2.3