An autonomous Kirchhoff-type equation with general nonlinearity in ${\mathbb{R}}^N$

Abstract

We consider the following autonomous Kirchhoff-type equation \begin{equation*} -\left(a+b\int_{\mathbb{R}^N}|\nabla{u}|^2\right)Δu= f(u),~~~~u\in H^1(\mathbb{R}^N), \end{equation*} where $a\ge 0,b>0$ are constants and $N\ge 1$. Under general Berestycki-Lions type assumptions on the nonlinearity $f$, we establish the existence results of a ground state and multiple radial solutions for $N\ge 2$, and obtain a nontrivial solution and its uniqueness, up to a translation and up to a sign, for $N=1$. The proofs are mainly based on a rescaling argument, which is specific for the autonomous case, and a new description of the critical values in association with the level sets argument.

20 pages. Major changes and added references

References in corpus (2)

• Multiple solutions for a Kirchhoff-type equation with general nonlinearity
• A note on the elliptic Kirchhoff equation in R^N perturbed by a local nonlinearity

Cited by in corpus (4)

• Nonlinear scalar field equations with general nonlinearity
• Berestycki-Lions conditions on ground state solutions for Kirchhoff-type problems with variable potentials
• On the Kirchhoff equation with prescribed mass and general nonlinearities
• Multiple solutions for a Kirchhoff-type equation with general nonlinearity